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["Mixed Radix Number Conversions"]
["by Dr. Irv Bromberg, University of Toronto, Canada"]
["irv.bromberg@utoronto.ca",null,null,"Revised 2022-03-08 (Gregorian)"]
["http://individual.utoronto.ca/intervalmath/mixed/radix.html"]
[]
["Syntax for Visual Basic for Applications (VBA) Macro Functions in the MixedRadix Module:"]
[]
["ToRadix( Bases As String, TheNumber As Variant, Optional ShowBases As Boolean = False, Optional Rounding As Variant = 0 ) As String"]
[]
["The Bases list is comma delimited, optionally enclosed in [ ] or ( ) or { } brackets. Spaces are ignored."]
["The Bases list can include one radix point, indicated by a semicolon instead of a comma delimiter. Bases to the left of the radix point refer to the integer part of TheNumber"]
["and bases to the right of the radix point refer to the fractional part of TheNumber."]
[]
["If only a single base is specified then a radix conversion will be returned, and if the base is <=36 then numbers and, if necessary, letters will be employed to represent the digits"]
["without any delimiters or brackets in the result. For example, {2} will convert the given number to a binary string, or {16} will convert it to a hexadecimal string."]
["With a single base, if the given number has a fractional part then the radix converted result will have a radix point and an appropriate number of radix values to its right."]
[]
["TheNumber is declared as Variant because it is allowed to contain a Double precision number or a String representing a mixed, proper, or improper fraction (spaces ignored)."]
["If TheNumber is negative then each of the radix values will be negative, or, in the case of a single base with symbolic rendering a single minus sign leftmost."]
[]
["This function will always return { } brackets around the result list (except in single base radix conversion mode), so it's a good idea to use different brackets for the list of bases."]
[]
["ShowBases = True switches on the showing of base values alongside the radix values. I realize that traditionally these are rendered as subscripted numbers, for example for"]
["a mixed radix conversion of a moment to separated time values the bases would be shown like { 132441603760 }, but VBA can’t return subscripted values between functions"]
["or back to worksheet cells, so I implemented the showing of plain text /bases like { 13/24, 41/60, 37/60 }. See also the ApplyUnits( ) function described below."]
["In single base mode ShowBases = True will cause this function to return the converted radix values as comma-delimited base 10 number groups."]
[]
["The optional Rounding variant parameter controls rounding of the least fractional radix value, with the following choices:"]
[]
[null,null,"Rounding Method","Code","also see the RoundToLeast( ) function described below"]
[null,null,"Unrounded","U","least fractional radix value won't be rounded --- this could return too many decimals, but you might want to see them"]
[null,null,"    Truncate","T","least fractional radix value will be floored, thus truncating any decimal points (as always happens for integer radix values)"]
[null,null,"Reduce Least","R","rounds the least fractional radix value to an integer and then reduces it (using its base as denominator), returning a fraction"]
[null,null,"Integer",0,"(digit zero) this is the default (least fractional radix value will be rounded to zero decimal points)"]
[null,null,"any higher digit","1 or more","rounds the least fractional radix value to that many decimal points"]
[]
["If desired, use the TrimZeros( ) function to remove leading and trailing zeros from the list of radix values returned by the ToRadix( ) function."]
[]
[]
["FromRadix( Bases As String, Radices As String, Optional FractType As String = \"D\" ) As Variant"]
[]
["Converts the given radix values to a base 10 number using the given radix bases."]
["The Radices string may contain one radix point (semicolon), which can be anywhere in the string (provided that it correlates with the position of the radix point in the given Bases list),"]
["including the beginning or end. If lead zeros were suppressed by ToRadix( ) and if the Radices string contains any fractional bases then they must be preceded by a radix point ( ; ),"]
["as would have been originally returned by the ToRadix( ) function call. By default the fractional part, if any, is returned as decimal points, but the user can specify other fraction types"]
["(as used for many of the Tibetan Calendar examples):"]
[]
[null,"Code","Meaning"]
[null,"P","Proper fraction"]
[null,"I","Improper fraction (will return as Proper if Numerator is less than Denominator)"]
[null,"M","Mixed fraction (will return as Proper if Whole part is zero)"]
[null,"D","(default) Decimal fraction"]
[]
["When returning a Whole number with or without a Decimal fraction the return type is Double, otherwise fractions are returned as type String (text)."]
["See also the Whole( ), Numerator( ), and Denominator( ) functions below."]
[]
[]
["Nth( List As String, Optional ItemNo As Integer, Optional RemoveSlashBase As Boolean = True ) As Variant"]
[]
["Returns the Nth comma-delimited numeric item from the given list (which can be a list of bases or radix values and can contain a radix point semicolon that will be treated like a comma)."]
["By default, /bases are removed, returning a Double number, but if RemoveSlashBase = False and there is a /base then it returns the fraction as a String (text)."]
[]
["If ItemNo is positive then it is counted from the left of the List, if it is negative then it is counted from the right of the list."]
["ItemNo = -1 is a convenient way to extract the least radix or least base from a list, without needing to know the number of items in the list. Least() is equivalent."]
["ItemNo = 0 or is omitted then Nth( ) returns the count of items in the list."]
["If ItemNo is out-of-range then that item doesn't exist in the list, in which case Nth( ) returns zero."]
[]
["Ten LISP-like functions call Nth( ) to do their work, and each take one List string as parameter: NthFirst(), NthSecond(), NthThird(), NthFourth(), and so on up to NthTenth()."]
["All of these functions that indirectly call the Nth( ) function also accept and pass on the optional RemoveSlashBase parameter."]
["(Unfortunately, the \"Nth\" prefix is necessary to avoid confusion with built-in worksheet functions, and for consistency.)"]
[]
["ApplyUnits( List As String, Units As String, Optional Plurals As Boolean = True, Optional SkipZeros As Boolean = False ) As String"]
[]
["Given a List of radix values, and a list of Units to apply to them, returns a string that lists each radix value alongside its respective unit."]
[]
["The units change to appropriate singular or plural forms unless Plurals = False. Clearly if Units are in a non-English language or are"]
["abbreviations or punctuation or symbols then you should pass Plurals as False to suppress pluralization."]
["The default plural form simply has an \"s\" appended to the unit, but if that is inappropriate then you can specify the single and plural forms,"]
["for example \"foot/feet\" or \"penny/pence\", or you can specify individual units to be the same in singular and plural forms, for example \"x $100/x $100\"."]
[]
["If the unit starts with the special symbol \"#\" then a numeric suffix will be prefixed to the unit (st, nd, rd, th …), without plurals."]
[]
["If TrimZeros( ) was applied to and shortened the list of radix values then ApplyUnits( ) will skip the units that correspond to the removed zeros."]
["If SkipZeros = True is passed then all radix values of zero will be omitted from the returned result, but because the positions are undefined for"]
["the remaining radix values after skipping zeros it won't be possible to use FromRadix( ) on the radices with units to recover the correct number."]
["In such a case use the original radices including zeros to recover the correct number, if required."]
[]
["If the Units list radix point \";\" is in the wrong position or is missing then ApplyUnits( ) will return an error message to that effect."]
[]
["Passing bases as the List is convenient for confirming their sequence, meanings, and maximum counts."]
[]
[]
["SigmaX( List1 As String, List2 As String ) As Double"]
[]
["Given two comma-delimited numeric lists containing the same number of elements, return the sum of the products of the corresponding elements."]
["If one list has fewer elements than the other then it will first be padded on the left with an appropriate number of zeros"]
[]
[]
["RoundToLeast( Bases as String, TheNumber As Double, Optional Decimals As Integer = 0 ) As Double"]
[]
["Returns TheNumber rounded to the least significant radix. This is the same as using the default Rounding = 0 when calling the ToRadix( ) function."]
["Normally the fractional bases to the right of the radix point are used for rounding, but if there are none then rounds TheNumber to the specified number of decimal points (default = zero)."]
[]
[]
["SpaceEvery( Interval As Integer, TheString As String) As String"]
[]
["Removes all spaces from TheString, then inserts one space at intervals from radix point to left, and radix point to right."]
["Useful for binary or hexadecimal strings to divide into nybbles or bytes or larger substrings."]
["TheString will be returned unchanged if it contains one or more comma delimiter(s)."]
[]
[]
["TrimZeros( Radices As String ) As String"]
[]
["Removes leading zeros to the left of the radix point and trailing zeros to the right of the radix point."]
["If no radix values remain after removing trailing zeros then it also removes the radix point."]
["It's probably not a good idea to trim zeros from a list intended for calendrical calculations or that will be passed to the SigmaX( ) function."]
[]
[]
["ReduceFraction( Numerator As Double, Denominator As Double, Optional ImproperFraction As Boolean = False, Optional compact As Boolean = True )"]
[]
["Returns a string containing the fraction Numerator / Denominator, reduced if possible. Numerator and Denominator must be integers --- they are Double for extended range."]
["If ImproperFraction = True then wil return an improper fraction if Numerator is greater than Denominator."]
["The compact flag controls the spacing around the \"/\" in the fraction. If set to False then slash will have leading and trailing space \" / \"."]
["Most often there is no need to specify the optional parameters."]
[]
["The parts of a fraction or mixed fraction can be extracted from the string returned from ReduceFraction(), ignoring all spaces:"]
["Whole( ) returns the whole part that is in front of the fraction, or zero if there is none."]
["Numerator( ) returns the numerator, or zero if there is none."]
["Denominator( ) returns the denominator, or one if there is none."]
[]
["ReduceFractStr( Fraction As String, Optional ImproperFraction As Boolean = False, Optional compact As Boolean = True) As String"]
[]
["This function is like ReduceFraction( ) but it accepts a Fraction string, which can be mixed, improper, or proper."]
["It just calls the Whole( ), Numerator( ), and Denominator( ) functions, described above, to get the components of the fraction,"]
[" and passes them on, along with the optional flags, to the ReduceFraction( ) function."]
[]
[]
["RoundIt( Result As Double, Decimals As Integer, SigFigs As Integer, Optional Direction As RoundingDirection = Simple) As Double"]
[]
["This available function isn't used by any of the demo examples nor internally by any of the above functions."]
["The intention was to use it, but eventually other ways were found to accomplish the desired performance."]
["Rounds to a specified number of decimal points or decimal figures --- positive rounds to right of decimal point, negative to the left."]
["Then rounds to the specified number of significant figures --- mandatory parameter, but if not desired then pass a large number like 20."]
[]
["Rounding Directions:"]
[null,"LowerBound = round lower bound outward (downward)"]
[null,"Simple (default) = normal rounding to closest value (used for non-intervals)"]
[null,"UpperBound = round upper bound outward (upward)"]
[]
["Outward rounding is particularly useful when rounding the bounds of intervals or analytical reference ranges or uncertainty ranges,"]
["to ensure that the rounded range includes the full unrounded range."]
["These rounding issues are admittedly beyond the scope of mixed radix conversions."]
[]
[]
["ContinuedFraction( TargetValue As Double, Optional MinConvergent As Integer = 2, Optional MaxConvergent As Integer = 12, Optional ErrorLimit As Double = 0.01) As String"]
[]
["Uses the continued fraction method to find a series of progressively better fractional approximations (called \"convergents\") to the given"]
["TargetValue. The search terminates when the a convergent's parts-per-trillion error is less than the ErrorLimit or when it reaches the"]
["specified maximum convergent. The final mixed or proper fraction is returned as a text string. If it returns a mixed fraction that you don't"]
["want then pass the inverse of the TargetValue or pass the fractional result to ReduceFractStr( ) with the ImproperFraction flag = True."]
[]
["If the fractional result has an absurd number of digits then try specifying a lower MaxConvergent or a greater ErrorLimit."]

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["Calendrical Examples and Validation of Mixed Radix Conversions"]
["Examples involving calendars, cycles, mean year or month, dates, times, angles"]
["http://individual.utoronto.ca/intervalmath/mixed/radix.html"]
["Note: The abbreviation \"CCUE\" refers to the book \"Calendrical Calculations: The Ultimate Edition\" (4th, 2018) by Reingold and Dershowitz,"]
["which introduced Mixed Radix conversions on page 27."]
[]
["Cells marked with a","yellow","background and bold test contain numbers that the user can change."]
[]
["Time as integers vs. fractions"]
["Bases","[ 24,60,60 ]",null,"[ ; 24,60,60 ]","spaces are ignored in bases lists"]
["Given Time",102341,"seconds",null,"and brackets are optional"]
[null,null,"/ 86400 seconds/day =",{"$f":"B11/86400","v":1.1845023148148148},"days"]
["Mixed Radix",{"$f":"ToRadix(B10,B11)","v":"{1, 4, 25, 41}"},null,{"$f":"ToRadix(D10,D12)","v":"{1; 4, 25, 41}"},"note radix point position"]
["Back to Number",{"$f":"FromRadix(B10,B13)","v":102341},"seconds",{"$f":"FromRadix(D10,D13)","v":1.1845023148148148},"days"]
["Show Bases",{"$f":"ToRadix(B10,B11,TRUE)","v":"{1, 4/24, 25/60, 41/60}"},null,{"$f":"ToRadix(D10,D12,TRUE)","v":"{1; 4/24, 25/60, 41/60}"}]
["Back to Number",{"$f":"FromRadix(B10,B15)","v":102341},"seconds",{"$f":"FromRadix(D10,D15)","v":1.1845023148148148},"days"]
["Above row shows that presence of /bases in {mixed radix list} doesn't interfere with conversion back to number"]
["Nth( List ) = item count",null,{"$f":"Nth(B13)","v":4},"count of items in mixed radix list (programmatically useful, not no much on worksheets)"]
["Extract Nth radix value",3,{"$f":"Nth(B15,B19)","v":25},"positive item# counts from left, any /base present is discarded"]
["Extract Nth radix value",-1,{"$f":"Nth(B15,B20)","v":41},"negative item# counts from right"]
[]
[]
["Example from p27 of CCUE"]
["Bases","[ 7 ; 24, 60, 60 ]"]
["Given Mixed Radix","{4, 1; 12, 44, 2.88}",null,"authors didn't provide mandatory radix point"]
["Convert to Number",{"$f":"FromRadix(B$24,B25)","v":{"v":29.53058888888889,"d":"29.5305888888889"}},null,"days"]
["Back to Mixed Radix",{"$f":"ToRadix(B$24,B$26)","v":"{4, 1; 12, 44, 3}"},null,null,"default rounding of least radix to integer"]
["Back to Number",{"$f":"FromRadix(B$24,B27)","v":{"v":29.530590277777776,"d":"29.5305902777778"}},null,"days","increased by rounding"]
["Show Bases",{"$f":"ToRadix(B24,B26,TRUE,2)","v":"{4, 1/7; 12/24, 44/60, 2.88/60}"},null,null,"least radix value rounded to 2 decimal points"]
["Back to Number",{"$f":"FromRadix(B$24,B29)","v":{"v":29.53058888888889,"d":"29.5305888888889"}},null,"days","original value preserved"]
["Mixed Radix",{"$f":"ToRadix(B$24,B$26,,3)","v":"{4, 1; 12, 44, 2.88}"},null,null,"least radix value rounded to 3 decimal points (unchanged)"]
["Append Hundredths","[ 7 ; 24, 60, 60, 100 ]"]
["Back to Mixed Radix",{"$f":"ToRadix(B$32,B$26)","v":"{4, 1; 12, 44, 2, 88}"},null,"exact"]
["Raw Mixed Radix",{"$f":"ToRadix(B$32,B$26,,0)","v":"{4, 1; 12, 44, 2, 88}"},null,"still exact"]
["Back to Number",{"$f":"FromRadix(B$32,B34)","v":{"v":29.53058888888889,"d":"29.5305888888889"}},null,"days","same as above"]
["Show Bases",{"$f":"ToRadix(B$32,B$26,TRUE)","v":"{4, 1/7; 12/24, 44/60, 2/60, 88/100}"}]
["Back to Number",{"$f":"FromRadix(B$32,B36)","v":{"v":29.53058888888889,"d":"29.5305888888889"}},null,"days","same as above"]
["Nth(List) counts items",{"$f":"Nth(B36)","v":6},null,"count of items in mixed radix list (programmatically useful, not no much on worksheets)"]
["Extract Nth radix value",5,{"$f":"Nth(B36,B39)","v":2},"Nth function treats radix point like a comma"]
[{"$nulls":3},"(If extracting the 5th item with all its decimal points, you have to format it with"]
[{"$nulls":3},"at least 10 decimal points to see more than 2.88 with or without trailing zeros.)"]
[]
[]
["https://en.wikipedia.org/wiki/Mixed_radix"]
["Given mixed radix values"]
[32,"weeks","w"]
[5,"days","d",7]
[7,"hours","h",24]
[45,"minutes","m",60,";","Why is radix point indicated after minutes?"]
[15,"seconds","s",60,null,"(Don't know, go with the flow…)"]
[500,"milliseconds","ms",1000]
[]
["[7, 24, 60; 60, 1000]",{"$nulls":3},"[7; 24, 60, 60, 1000]","shifted radix point left, after days"]
["{32,5,7,45;15,500}","from these figure out what was the number",null,null,"{32,5;7,45,15,500}","move the radix point to the corresponding position"]
[{"$f":"FromRadix(A53,A54)","v":330225.25833333336},"minutes (seconds and milliseconds in fraction)",null,null,{"$f":"FromRadix(E53,E54)","v":229.32309606481482},"mixed radices converted to days"]
[{"$f":"ToRadix(A53,A55)","v":"{32, 5, 7, 45; 15, 500}"},null,"got back radices",null,{"$f":"ToRadix(E53,E55)","v":"{32, 5; 7, 45, 15, 500}"},"converted back to mixes radices"]
[{"$f":"ToRadix(A53,A55,TRUE)","v":"{32, 5/7, 7/24, 45/60; 15/60, 500/1000}"},null,null,"show bases",{"$f":"ToRadix(E53,E55,TRUE)","v":"{32, 5/7; 7/24, 45/60, 15/60, 500/1000}"},null,"show bases"]
["Units to Apply","w,d,h,m;s,ms",null,null,{"$f":"-E55","v":-229.32309606481482},"what happens with a negative number?"]
["Apply Units (no plurals)",{"$f":"ApplyUnits(A56,B58,FALSE)","v":"32 w, 5 d, 7 h, 45 m ; 15 s, 500 ms"},{"$nulls":3},"each converted radix is a negative number"]
[{"$nulls":4},{"$f":"ToRadix(E53,E58)","v":"{-32, -5; -7, -45, -15, -500}"}]
[{"$nulls":4},{"$f":"FromRadix(E53,E60)","v":-229.32309606481482},"got back the negative number"]
[]
[]
["https://github.com/colwem/mixed-radix",null,null,"seconds/86400"]
[100000,"seconds",null,{"$f":"A65/86400","v":1.1574074074074074},"days"]
[1,"day"]
[3,"hours",24]
[46,"minutes",60]
[40,"seconds",60]
[]
["Bases","[ 24,60,60 ]",null,null,"[ ; 24,60,60 ]","radix point leftmost"]
["Convert # to radices",{"$f":"ToRadix(B71,A65)","v":"{1, 3, 46, 40}"},null,null,{"$f":"ToRadix(E71,D65)","v":"{1; 3, 46, 40}"}]
["Back to number",{"$f":"FromRadix(B71,B72)","v":100000},null,null,{"$f":"FromRadix(E71,E72)","v":1.1574074074074074},"days x 86400 seconds/day =",{"$f":"E73*86400","v":100000},"seconds"]
["Show Bases",{"$f":"ToRadix(B71,A65,TRUE)","v":"{1, 3/24, 46/60, 40/60}"},null,null,{"$f":"ToRadix(E71,D65,TRUE)","v":"{1; 3/24, 46/60, 40/60}"}]
["Back to number",{"$f":"FromRadix(B71,B74)","v":100000},null,null,{"$f":"FromRadix(E71,E74)","v":1.1574074074074074}]
["Units to Apply","d,h,m,s",null,null,"d;h,m,s"]
["Apply Units",{"$f":"ApplyUnits(B72,B76,FALSE)","v":"1 d, 3 h, 46 m, 40 s"},null,null,{"$f":"ApplyUnits(E72,E76,FALSE)","v":"1 d ; 3 h, 46 m, 40 s"}]
["Full Units to Apply","day,hour,minute,second",null,null,"day;hour,minute,second"]
["Apply Full Units",{"$f":"ApplyUnits(B72,B78)","v":"1 day, 3 hours, 46 minutes, 40 seconds"},null,null,{"$f":"ApplyUnits(E72,E78)","v":"1 day ; 3 hours, 46 minutes, 40 seconds"}]
[{"$nulls":4},"radix point position is critically important in units list"]
[]
["Mayan Calendar Cycles",null,"(reference: CCUE page 170)",null,{"$f":"365+B130","v":{"v":365.24203584437765,"d":"365.242035844378"}},"derived below"]
["Units of the Long Count (in blue):",null,null,"days","Mayan solar years"]
["kin",1,"day",1,"(approximate)"]
["uinal",20,"kin",{"$f":"B85*D84","v":20}]
["tun",18,"uinal",360,{"$f":"D86/E$82","v":{"v":0.9856477750917718,"d":"0.98565"}}]
["katun",20,"tun",7200,{"$f":"D87/E$82","v":{"v":19.712955501835435,"d":"19.713"}}]
["baktun",20,"katun",144000,{"v":394.2591100367087,"d":"394.26"}]
["pictun",20,"baktun",2880000,{"v":7885.182200734174,"d":"7885.2"}]
["calabtun",20,"pictun",57600000,{"v":157703.6440146835,"d":"157703.6"}]
["kinchiltun",20,"calabtun",1152000000,{"v":3154072.8802936696,"d":"3154072.9"}]
["alautun",20,"kinchiltun",23040000000,{"v":63081457.6058734,"d":"63081457.6"}]
[]
["Units to Apply","alautun, kinchiltun, calabtun, pictun, baktun, katun, tun, uinal, kin"]
["Confirm Limit / Unit",{"$f":"ApplyUnits(B97,B94)","v":"20 kinchiltuns, 20 calabtuns, 20 pictuns, 20 baktuns, 20 katuns, 20 tuns, 18 uinals, 20 kins"}]
[]
["Long Count Bases","[ 20, 20, 20, 20, 20, 20, 18, 20 ]",null,null,"reverse"]
["Long Count","{0, 6, 4, 17}",null,null,{"$f":"ToRadix(B97,E100)","v":"{1, 3, 10, 11}"}]
["Apply Units",{"$f":"ApplyUnits(B98,B94)","v":"0 katuns, 6 tuns, 4 uinals, 17 kins"},null,null,{"$f":"ApplyUnits(E98,B94)","v":"1 katun, 3 tuns, 10 uinals, 11 kins"}]
["Total Days in Count",{"$f":"FromRadix(B97,B98)","v":2257},null,null,8491]
[]
[null,null,"haab calendar cycle =",{"$f":"18*20+5","v":365},"days"]
[null,null,"tzolkin calendar cycle =",{"$f":"13*20","v":260},"days"]
[]
[46,"tzolkins =",405,"lunations =",{"$f":"A105*D103/C105","v":{"v":29.530864197530864,"d":"29.5308641975309"}},"days per Mayan mean lunation"]
[{"$nulls":4},"[ 7 ; 24, 60, 60, 1000 ]"]
[{"$nulls":3},"Least radix value truncated:",{"$f":"ToRadix(E106,E105)","v":"{4, 1; 12, 44, 26, 667}"}]
[{"$nulls":3},"Show Bases:",{"$f":"ToRadix(E106,E105,TRUE)","v":"{4, 1/7; 12/24, 44/60, 26/60, 667/1000}"}]
[{"$nulls":3},"The least radix value reduces but needs decimal points:",{"$f":"ReduceFraction(Nth(E107,-1),Nth(E106,-1))","v":"667/1000"},"(can't reduce with decimal points)"]
[{"$nulls":3},"Try reducing as big numbers:","66666666666667/100000000000000"]
[{"$nulls":3},"Can't reduce with 7 as last decimal point:",{"$f":"ReduceFractStr(E110)","v":"66666666666667/100000000000000"}]
[{"$nulls":3},"Try again reducing as big numbers, without 7:","66666666666666/100000000000000"]
[{"$nulls":3},"Reduction only by a factor of 2:",{"$f":"ReduceFractStr(E112)","v":"33333333333333/50000000000000"}]
[{"$nulls":3},"Try maximum number of repeating decimal points:",0.666666666666667]
[{"$nulls":3},"Continued Fraction function finds the exact fraction:",{"$f":"ContinuedFraction(E114)","v":"2/3"},"(zero error at 2nd convergent)"]
[{"$nulls":3},"Try only 2 decimal points:",{"v":0.67,"d":"0.670000000000000"}]
[{"$nulls":3},"Continued Fraction function finds the exact fraction:",{"$f":"ContinuedFraction(E116,,2)","v":"2/3"},"(if limited to 2nd convergent)"]
[{"$nulls":3},"Least radix value rounded to 6 decimals is obviously 2/3:",{"$f":"ToRadix(E106,E105,,6)","v":"{4, 1; 12, 44, 26, 666.666667}"}]
[{"$nulls":4},"= 4 weeks 1 day 12 hours 44 minutes 26+2/3 seconds"]
["Compare with mean astronomical duration of lunation near the beginning of the peak of the Mayan era (250 AD):"]
[{"$nulls":4},"4 weeks 1 day 12 hours 44 minutes 3+21/73 seconds"]
["Obviously the Mayan estimate is rather inaccurate. It is typical of what might have been based on solar eclipse studies."]
["It's flatly impossible to obtain the accurate duration of the mean lunation interval using any eclipse studies,"]
["over any span of time, even if the eclipses are sequential in a saros series, hence occurring in similar circumstances."]
["The triple-saros interval would yield the best similarity between eclipses."]
[]
["{7,13,0,0,0}","Mayan long count for 365-day haab to move twice through the solar year"]
[{"$f":"FromRadix(B97,A127)","v":1101600},"days",{"$f":"A128/D102","v":3018.082191780822},"haab cycles"]
["2 x haab / (habb cycles - 2) ="]
[null,{"$f":"2*365/(C128-2)","v":{"v":0.2420358443776286,"d":"0.242035844377629"}},null,"mean solar year fraction in excess of 365 days"]
[null,{"$f":"ContinuedFraction(B130)","v":"26645/110087"},null,"exact fraction for the decimal points above"]
[null,"[ ; 24, 60, 60, 1000 ]"]
[null,{"$f":"ToRadix(B132,B130)","v":"{; 5, 48, 31, 897}"}]
[null,{"$f":"ToRadix(B132,B130,TRUE)","v":"{; 5/24, 48/60, 31/60, 897/1000}"}]
[null,"= 365 days 5 hours 48 minutes 31+112/125 seconds"]
["Compare with mean astronomical solar years near the beginning of the peak of the Mayan era 200 AD):"]
["See chart:","http://individual.utoronto.ca/kalendis/leap/Solar-Year-Range.pdf"]
[]
[null,"Solar Year Type","365 days + ...","Comments (colors match the chart linked above)"]
[null,"Longest Solar Year (aphelion)","05:49:56","steadily decreasing due to tidal slowing of the Earth rotation rate"]
[null,"South Solstitial Year","05:49:53","just past beginning of rapidly declining era that continues to the present"]
[null,"Southward Equinoctial Year","05:49:18","near middle of rapidly declining era that continues to the present"]
[null,"Average Solar Year","05:49:05","steadily decreasing due to tidal slowing of the Earth rotation rate"]
[null,"Northward Equinoctial Year","05:48:53","near middle of stable era that continues to the present"]
[null,"North Solstitial Year","05:48:16","(closest to Mayan year) entering stable era that continues to the present"]
[null,"Shortest Solar Year (perihelion)","05:48:14","steadily decreasing due to tidal slowing of the Earth rotation rate"]
[]
["The Mayan territory was near the Tropic of Cancer, consistent with reckoning their solar year according to the north solstice."]
[]
[]
["CCUE p67 Gregorian calendar [ alt-gregorian-year-from-fixed() ] and p100 Icelandic calendar"]
["compute the number of elapsed periods of 4, 100, and 400 years using mixed-radix arithmetic"]
["[4, 25, 4]",{"$nulls":3},"{97,24,1,0}","leap days"]
[4,"centuries per 400-year cycle",null,null,97,"per 400-year cycle"]
[25,"quadrennia per century",null,null,24,"per century"]
[4,"years per quadrennium",null,null,1,"per quadrennium"]
[{"$nulls":4},0,"for additional years"]
[{"$f":"C169","v":"{5, 0, 5, 1}"},"SigmaX",{"$f":"E153","v":"{97,24,1,0}"},{"$f":"SigmaX(A158,C158)","v":490},"total leap days"]
[]
[{"$nulls":3},"days elapsed","same, calculated"]
[{"$nulls":3},"prior to year","using SigmaX( )","+1 = rata die of"]
["year","prior year","to radix","using Nth( )","365 x prior years + leap days","New Year Day"]
[100,{"$f":"A163-1","v":99},{"$f":"ToRadix(A$153,B163)","v":"{24, 3}"},{"$f":"B163*365+Nth(C163,-4)*E$154+Nth(C163,-3)*E$155+Nth(C163,-2)*E$156+Nth(C163,-1)*E$157","v":36159},{"$f":"B163*365+SigmaX(C163,E$153)","v":36159},{"$f":"D163+1","v":36160}]
[350,{"$f":"A164-1","v":349},"{3, 12, 1}",127469,{"$f":"B164*365+SigmaX(C164,E$153)","v":127469},{"$f":"D164+1","v":127470}]
[1600,{"$f":"A165-1","v":1599},"{3, 3, 24, 3}",584022,{"$f":"B165*365+SigmaX(C165,E$153)","v":584022},{"$f":"D165+1","v":584023}]
[1900,{"$f":"A166-1","v":1899},"{4, 2, 24, 3}",693595,693595,{"$f":"D166+1","v":693596}]
[2000,1999,"{4, 3, 24, 3}",730119,730119,{"$f":"D167+1","v":730120}]
[2021,2020,"{5, 0, 5, 0}",737790,737790,737791]
[2022,2021,"{5, 0, 5, 1}",738155,738155,738156]
[2200,2199,"{5, 1, 24, 3}",803168,803168,803169]
[2300,2299,"{5, 2, 24, 3}",839692,839692,839693]
[2400,2399,"{5, 3, 24, 3}",876216,876216,876217]
[]
[]
["Gregorian Calendar Mean Year"]
["Years Per Cycle",400,"years"]
["Days Per Common Year",365,"days"]
["Leap Days Per Cycle",{"$f":"B176/4-B176/100+1","v":97},"days","(calculated)"]
["Days Per Cycle",{"$f":"B176*B177+97","v":146097},"days","400 x 365 + 97"]
["Mean Year + Fraction",{"$f":"ReduceFraction(B179,B176)","v":"365+97/400"},null,"days"]
["Mean Year",{"$f":"B179/B176","v":365.2425},"days","Days Per Cycle / Years Per Cycle",null,"(exact decimal)"]
["Bases for Time","[ ; 24, 60, 60, 400 ]",null,"Start with least base = years per cycle"]
["Mixed Radix with bases",{"$f":"ToRadix(B182,B181,TRUE)","v":"{365; 5/24, 49/60, 12/60, 0/400}"},null,null,"Least radix value is zero, therefore time is exact to the second"]
["Trim Zeros",{"$f":"TrimZeros(B183)","v":"{365; 5/24, 49/60, 12/60}"},null,null,"Trailing zero dropped"]
["Bases for Time","[ ; 24, 60, 60 ]",null,"Drop least base"]
["Mixed Radix with bases",{"$f":"ToRadix(B185,B181,TRUE)","v":"{365; 5/24, 49/60, 12/60}"},null,null,"Exact mean year in time units"]
["Long Form",{"$f":"Nth(B186,1)&\" days, \"&Nth(B186,2)& \" hours, \"&Nth(B186,3)&\" minutes, and \"&Nth(B186,4)&\" seconds\"","v":"365 days, 5 hours, 49 minutes, and 12 seconds"}]
["Mean Year from Fraction",{"$f":"ToRadix(B185,B180)","v":"{365; 5, 49, 12}"}]
["Units to Apply","day;hour,minute,second"]
["Apply Units",{"$f":"ApplyUnits(B188,B189)","v":"365 days ; 5 hours, 49 minutes, 12 seconds"}]
[]
[]
["Dee Calendar Mean Year"]
["Years Per Cycle",33,"years"]
["Days Per Common Year",365,"days"]
["Leap Days Per Cycle",{"$f":"INT(B194/4)","v":8},"days"]
["Days Per Cycle",{"$f":"B194*B195+B196","v":12053},"days","33 x 365 + 8"]
["Mean Year + Fraction",{"$f":"ReduceFraction(B197,B194)","v":"365+8/33"},"days"]
["Mean Year",{"$f":"B197/B194","v":{"v":365.24242424242425,"d":"365.242424242424"}},null,"Days Per Cycle / Years Per Cycle",null,"(approximate, \"24\" repeats)"]
["Bases for Time","[ ; 24, 60, 60, 33 ]",null,"Start with least base = years per cycle, and try reducing the least radix value"]
["Mixed Radices",{"$f":"ToRadix(B200,B199,FALSE,\"R\")","v":"{365; 5, 49, 5, 5/11}"},null,{"$f":"\"Least radix value got reduced by a factor of \"&Least(B200)/Denominator(Least(B201,FALSE))","v":"Least radix value got reduced by a factor of 3"}]
["Long Form",{"$f":"Nth(B201,1)&\" days, \"&Nth(B201,2)& \" hours, \"&Nth(B201,3)&\" minutes, and \"&Nth(B201,4)&\"+\"&Least(B201,FALSE)&\" seconds\"","v":"365 days, 5 hours, 49 minutes, and 5+5/11 seconds"}]
["Mean Year from Fraction",{"$f":"ToRadix(B200,B198,FALSE,\"R\")","v":"{365; 5, 49, 5, 5/11}"},null,"denominator remains on reduced least radix value"]
[]
[]
["Symmetry454 and Symmetry010 Calendar Mean Year (for northward equinox)"]
["Years Per Cycle",293,"years"]
["Days Per Common Year",364,"days","52 weeks"]
["Leap Days Per Cycle",{"$f":"7*52","v":364},"days","7 x 52 leap weeks"]
["Days Per Cycle",{"$f":"B207*B208+B209","v":107016},"days","293 x 364 + 364 or simply 294 x 364"]
["Mean Year + Fraction",{"$f":"ReduceFraction(B210,B207)","v":"365+71/293"},null,"days"]
["Mean Year",{"$f":"B210/B207","v":{"v":365.24232081911265,"d":"365.242320819113"}},null,"Days Per Cycle / Years Per Cycle",null,"(approximate, the first 146 decimal digits repeat)"]
["Bases for Time","[ ; 24, 60, 60, 293 ]",null,"Start with least base = years per cycle, and try reducing the least radix value"]
["Mixed Radix with bases",{"$f":"ToRadix(B213,B212,FALSE,\"R\")","v":"{365; 5, 48, 56, 152/293}"},null,"Least radix value doesn't reduce"]
["Long Form",{"$f":"Nth(B214,1)&\" days, \"&Nth(B214,2)& \" hours, \"&Nth(B214,3)&\" minutes, and \"&Nth(B214,4)&\"+\"&Least(B214,FALSE)&\" seconds\"","v":"365 days, 5 hours, 48 minutes, and 56+152/293 seconds"}]
["Mean Year from Fraction",{"$f":"ToRadix(B213,B211,FALSE,\"R\")","v":"{365; 5, 48, 56, 152/293}"},null,{"$f":"D214","v":"Least radix value doesn't reduce"}]
[null,"Even though the mean year is limited to double floating point precision, that is good enough"]
[null,"to resolve the long form of the mean year down to the exact fraction of a second."]
[]
[]
["Symmetry454 and Symmetry010 Calendar Mean Year (alternate for northward equinox)"]
["Years Per Cycle",524,"years"]
["Days Per Common Year",364,"days","52 weeks"]
["Leap Days Per Cycle",{"$f":"7*93","v":651},"days","7 x 93 leap weeks"]
["Days Per Cycle",{"$f":"B222*B223+B224","v":191387},"days","293 x 364 + 364 or simply 294 x 364"]
["Mean Year + Fraction",{"$f":"ReduceFraction(B225,B222)","v":"365+127/524"},null,"days"]
["Mean Year",{"$f":"B225/B222","v":{"v":365.24236641221376,"d":"365.242366412214"}},null,"Days Per Cycle / Years Per Cycle",null,"(approximate, 0.24 is followed by 130 repeating digits)"]
["Bases for Time","[ ; 24, 60, 60, 524 ]",null,"Start with least base = years per cycle, and try reducing the least radix value"]
["Mixed Radices",{"$f":"ToRadix(B228,B227,FALSE,\"R\")","v":"{365; 5, 49, 0, 60/131}"},null,{"$f":"\"Least radix value got reduced by a factor of \"&Least(B228)/Denominator(Least(B229,FALSE))","v":"Least radix value got reduced by a factor of 4"}]
["Long Form",{"$f":"Nth(B229,1)&\" days, \"&Nth(B229,2)& \" hours, \"&Nth(B229,3)&\" minutes, and \"&Nth(B229,4)&\"+\"&Least(B229,FALSE)&\" seconds\"","v":"365 days, 5 hours, 49 minutes, and 0+60/131 seconds"}]
["Mean Year from Fraction",{"$f":"ToRadix(B228,B226,FALSE,\"R\")","v":"{365; 5, 49, 0, 60/131}"}]
[null,"Even though the mean year is limited to double floating point precision, that is good enough"]
[null,"to resolve the long form of the mean year down to the exact fraction of a second."]
[]
[]
["Symmetry454 and Symmetry010 Calendar Mean Year (for north solstice)"]
["Years Per Cycle",389,"years"]
["Days Per Common Year",364,"days","52 weeks"]
["Leap Days Per Cycle",{"$f":"7*69","v":483},"days","7 x 69 leap weeks"]
["Days Per Cycle",{"$f":"B237*B238+B239","v":142079},"days","389 x 364 + 483"]
["Mean Year + Fraction",{"$f":"ReduceFraction(B240,B237)","v":"365+94/389"},null,"days"]
["Mean Year",{"$f":"B240/B237","v":{"v":365.24164524421593,"d":"365.241645244216"}},null,"Days Per Cycle / Years Per Cycle",null,"(approximate, the first 388 decimal digits repeat)"]
["Bases for Time","[ ; 24, 60, 60, 389 ]",null,"Start with least base = years per cycle, and try reducing the least radix value"]
["Mixed Radices",{"$f":"ToRadix(B243,B242,FALSE,\"R\")","v":"{365; 5, 47, 58, 58/389}"},null,"Least radix value doesn't reduce"]
["Long Form",{"$f":"Nth(B244,1)&\" days, \"&Nth(B244,2)& \" hours, \"&Nth(B244,3)&\" minutes, and \"&Nth(B244,4)&\"+\"&Least(B244,FALSE)&\" seconds\"","v":"365 days, 5 hours, 47 minutes, and 58+58/389 seconds"}]
["Mean Year from Fraction",{"$f":"ToRadix(B243,B241,FALSE,\"R\")","v":"{365; 5, 47, 58, 58/389}"}]
[null,"Even though the mean year is limited to double floating point precision, that is good enough"]
[null,"to resolve the long form of the mean year down to the exact fraction of a second."]
[]
[]
["Georgian Calendar Mean Year"]
["Years Per Cycle",132,"years"]
["Days Per Common Year",365,"days","52 weeks + 1 day"]
["Leap Days Per Cycle",{"$f":"B252/4-1","v":32},"days","every 4 years except 132nd year (last year of each cycle)"]
["Days Per Cycle",{"$f":"B252*B253+B254","v":48212},"days","132 x 365 + 32"]
["Mean Year + Fraction",{"$f":"ReduceFraction(B255,B252)","v":"365+8/33"},null,"days"]
["Mean Year",{"$f":"B255/B252","v":{"v":365.24242424242425,"d":"365.242424242424"}},null,"Days Per Cycle / Years Per Cycle",null,"(approximate, the first 388 decimal digits repeat)"]
["Bases for Time","[ ; 24, 60, 60, 132 ]",null,"Start with least base = years per cycle, and use the option to reduce the least fractional radix value"]
["Mixed Radices",{"$f":"ToRadix(B258,B257,FALSE,\"R\")","v":"{365; 5, 49, 5, 5/11}"},null,{"$f":"\"Least radix value got reduced by a factor of \"&Least(B258)/Denominator(Least(B259,FALSE))","v":"Least radix value got reduced by a factor of 12"}]
["Bases for Time","[ ; 24, 60, 60, 11 ]",null,"Change least radix base to match reduced denominator of least fractional radix"]
["Mixed Radix with bases",{"$f":"ToRadix(B260,B257,TRUE)","v":"{365; 5/24, 49/60, 5/60, 5/11}"},null,null,"Below uses Least( ) with RemoveSlashBase = False to obtain fraction."]
["Long Form",{"$f":"Nth(B261,1)&\" days, \"&Nth(B261,2)& \" hours, \"&Nth(B261,3)&\" minutes, \"&Nth(B261,4)&\"+\"&Least(B261,FALSE)&\" seconds\"","v":"365 days, 5 hours, 49 minutes, 5+5/11 seconds"}]
["Mean Year from Fraction",{"$f":"ToRadix(B260,B256,TRUE)","v":"{365; 5/24, 49/60, 5/60, 5/11}"}]
[null,"Mean year is the same as the Dee 8/33 leap rule, but jitter is 1.75x greater because of the"]
[null,"omission of a leap every 132 years, whereas the leap years in the Dee cycle are smoothly spread."]
[]
[]
["500-Year Gregorian-Like Leap Day Cycle Mean Year"]
["Years Per Cycle",500,"years"]
["Days Per Common Year",365,"days","52 weeks + 1 day"]
["Leap Days Per Cycle",{"$f":"B269/4-B269/100+1","v":121},"days","every 4 years except century years except if divisible by 500"]
["Days Per Cycle",{"$f":"B269*B270+B271","v":182621},"days"]
["Mean Year + Fraction",{"$f":"ReduceFraction(B272,B269)","v":"365+121/500"},null,"days"]
["Mean Year",{"$f":"B272/B269","v":365.242},null,"Days Per Cycle / Years Per Cycle",null,"(exact decimal points)"]
["Bases for Time","[ ; 24, 60, 60, 500 ]",null,"Start with least base = years per cycle, and use the option to reduce the least fractional radix value"]
["Mixed Radices",{"$f":"ToRadix(B275,B274,FALSE,\"R\")","v":"{365; 5, 48, 28, 4/5}"},null,{"$f":"\"Least radix value got reduced by a factor of \"&Least(B275)/Denominator(Least(B276,FALSE))","v":"Least radix value got reduced by a factor of 100"}]
["Long Form",{"$f":"Nth(B276,1)&\" days, \"&Nth(B276,2)& \" hours, \"&Nth(B276,3)&\" minutes, \"&Nth(B276,4)&\"+\"&Least(B276,FALSE)&\" seconds\"","v":"365 days, 5 hours, 48 minutes, 28+4/5 seconds"}]
["Mean Year from Fraction",{"$f":"ToRadix(B275,B273,FALSE,\"R\")","v":"{365; 5, 48, 28, 4/5}"},null,{"$f":"D276","v":"Least radix value got reduced by a factor of 100"}]
[null,"This short mean year is presently a good approximation to the mean southward equinoctial year."]
[]
[]
["300-Year Gregorian-Like Leap Day Cycle Mean Year"]
["Years Per Cycle",300,"years"]
["Days Per Common Year",365,"days","52 weeks + 1 day"]
["Leap Days Per Cycle",{"$f":"B283/4-B283/100+1","v":73},"days","every 4 years except century years except if divisible by 300"]
["Days Per Cycle",{"$f":"B283*B284+B285","v":109573},"days"]
["Mean Year + Fraction",{"$f":"ReduceFraction(B286,B283)","v":"365+73/300"},null,"days"]
["Mean Year",{"$f":"B286/B283","v":{"v":365.24333333333334,"d":"365.243333333333"}},null,"Days Per Cycle / Years Per Cycle"]
["Bases for Time","[ ; 24, 60, 60, 300 ]",null,"Start with least base = years per cycle, and use the option to reduce the least fractional radix value"]
["Mixed Radices",{"$f":"ToRadix(B289,B288,FALSE)","v":"{365; 5, 50, 24, 0}"},null,"Least fractional radix value is zero, so time is exact to the second."]
["Bases for Time","[ ; 24, 60, 60 ]",null,"Drop least base"]
["Mixed Radix with bases",{"$f":"ToRadix(B291,B288,TRUE)","v":"{365; 5/24, 50/60, 24/60}"}]
["Units to Apply","day;hour,minute,second"]
["Apply Units",{"$f":"ApplyUnits(B292,B293)","v":"365 days ; 5 hours, 50 minutes, 24 seconds"}]
["Mean Year from Fraction",{"$f":"ToRadix(B291,B287,TRUE)","v":"{365; 5/24, 50/60, 24/60}"}]
[null,"This rather long mean year is appreciably longer than any of the equinoctial or solstitial mean years."]
[]
[]
["Mean Atomic Revolution Year (MARY)"]
["Years Per Cycle",128,"years"]
["Days Per Common Year",365,"days"]
["Leap Days Per Cycle",31,"days"]
["Days Per Cycle",{"$f":"B300*B301+B302","v":46751},"days","128 x 365 + 31"]
["Mean Year + Fraction",{"$f":"ReduceFraction(B303,B300)","v":"365+31/128"},null,"days"]
["Mean Year",{"$f":"B303/B300","v":365.2421875},null,"Days Per Cycle / Years Per Cycle",null,"(exact)"]
["Bases for Time","[ ; 24, 60, 60, 128 ]",null,"Start with least base = years per cycle, and try reducing the least radix value"]
["Mixed Radix with bases",{"$f":"ToRadix(B306,B305,TRUE)","v":"{365; 5/24, 48/60, 45/60, 0/128}"},null,null,"Least radix value is zero, time is exact to the second"]
["Trim Trailing Zero",{"$f":"TrimZeros(B307)","v":"{365; 5/24, 48/60, 45/60}"}]
["Units","day;hour,minute,second"]
["Apply Units",{"$f":"ApplyUnits(B308,B309)","v":"365 days ; 5 hours, 48 minutes, 45 seconds"}]
["Mean Year from Fraction",{"$f":"ToRadix(B306,B304,TRUE)","v":"{365; 5/24, 48/60, 45/60, 0/128}"}]
["Julian Mean Year",{"$f":"365+1/4","v":365.25},null,"exact"]
["Julian - MARY Year",{"$f":"B312-B305","v":{"v":7.8125E-3,"d":"0.0078125"}},null,"exact fraction of a day"]
["Convert to Seconds",{"$f":"B313*86400","v":675},"seconds"]
["Convert to Fraction",{"$f":"B314&\"/86400\"","v":"675/86400"}]
["Reduce Fraction",{"$f":"ReduceFraction(B314,86400)","v":"1/128"},"of a day"]
["Bases for Seconds","[ 24, 60, 60 ]",null,"no radix point when working in seconds"]
["Mixed Radix",{"$f":"ToRadix(B317,B314)","v":"{11, 15}"}]
["Units","hour,minute,second"]
["Apply Units",{"$f":"ApplyUnits(B318,B319)","v":"11 minutes, 15 seconds"}]
[{"$f":"\"MARY mean year is exactly \"&B316&\" \"&C316&\" = \"&B320&\" shorter than the Julian mean year.\"","v":"MARY mean year is exactly 1/128 of a day = 11 minutes, 15 seconds shorter than the Julian mean year."}]
[]
[]
["The Alpo Balognian calendar 9000-Year Cycle Mean Year (for northward equinox)"]
["http://the-light.com/cal/converter/calendarA.html"]
["Common years have 364 days, plus 1 feast day appended to the end of the year."]
["Every 5th year append an extra feast day, every 25th append another feast day,",{"$nulls":9},"Palmen-"]
["every 450th year append yet another feast day, and another in the 9000th year.",{"$nulls":4},"Reduced 3000-Year Cycle:",{"$nulls":4},"Modified"]
["Years Per Cycle",9000,"years",{"$nulls":3},{"$f":"B329/3","v":3000},{"$nulls":3},{"$f":"G329","v":3000}]
["Days Per Common Year",364,"days",{"$nulls":3},364,{"$nulls":3},{"$f":"G330","v":364}]
["Days for All Common Years",{"$f":"B329*B330","v":3276000},{"$nulls":4},{"$f":"G329*G330","v":1092000},{"$nulls":3},{"$f":"G331","v":1092000}]
["Feast Days of Common Years",{"$f":"B329","v":9000},"days","1 to 5 feast days at the end of year",null,null,{"$f":"G329","v":3000},{"$nulls":3},{"$f":"G332","v":3000}]
["Extra Feast Every 5th Year",{"$f":"B329/5","v":1800},"days",null,null,"every 5th year",{"$f":"G329/5","v":600},null,null,"+4th",{"$f":"K329/4","v":750}]
["Extra Feast Every 25th Year",{"$f":"B329/25","v":360},"days",null,null,"every 25th year",{"$f":"G329/25","v":120},null,null,"-100th",{"$f":"-K329/100","v":-30}]
["Extra Feast Every 450th Year",{"$f":"B329/450","v":20},"days",null,null,"every 500th year",{"$f":"G329/500","v":6},null,null,"+500th",{"$f":"G335","v":6}]
["Extra Feast 9000th Year",{"$f":"B329/9000","v":1},"day",null,null,"every 3000th year",{"$f":"3000/3000","v":1},null,null,"+3000th",{"$f":"G336","v":1}]
["Total Feast Days / Cycle",{"$f":"SUM(B332:B336)","v":11181},{"$nulls":4},{"$f":"SUM(G332:G336)","v":3727},{"$nulls":3},{"$f":"SUM(K332:K336)","v":3727}]
["Total Days Per Cycle",{"$f":"B331+B337","v":3287181},"days","Common Days + Feast Days",null,null,{"$f":"G331+G337","v":1095727},{"$nulls":3},{"$f":"K331+K337","v":1095727}]
["Mean Year + Fraction",{"$f":"ReduceFraction(B338,B329)","v":"365+727/3000"},null,"days",null,null,{"$f":"ReduceFraction(G338,G329)","v":"365+727/3000"},"same",null,null,"same"]
["Mean Year",{"$f":"B338/B329","v":{"v":365.2423333333333,"d":"365.242333333333"}},null,"Days Per Cycle / Years Per Cycle",null,{"$f":"G338/G329","v":{"v":365.2423333333333,"d":"365.242333333333"}},null,"same",null,null,"also same"]
["Bases for Time","[ ; 9000 ]",null,"Start with one base = years per cycle, and try reducing the radix value"]
["Mixed Radix with bases",{"$f":"ToRadix(B341,B$340,FALSE,\"R\")","v":"{365; 727/3000}"},null,{"$f":"\"Least radix value got reduced by a factor of \"&Least(B341)/Denominator(Least(B342,FALSE))","v":"Least radix value got reduced by a factor of 3"}]
["Confirm Mean Year",{"$f":"365+727/3000","v":{"v":365.2423333333333,"d":"365.242333333333"}},null,"=365+727/3000",{"$f":"B343=B340","v":true},"Yes, we recovered the original Mean Year"]
["Bases for Time","[ ; 24, 60, 60, 3000 ]",null,"Try again with least base set to the reduced denominator"]
["Mixed Radix with bases",{"$f":"ToRadix(B344,B$340,FALSE,\"R\")","v":"{365; 5, 48, 57, 3/5}"},null,{"$f":"\"Least radix value got reduced by a factor of \"&Least(B344)/Denominator(Least(B345,FALSE))","v":"Least radix value got reduced by a factor of 600"}]
["Long Form",{"$f":"Nth(B345,1)&\" days, \"&Nth(B345,2)& \" hours, \"&Nth(B345,3)&\" minutes, \"&Nth(B345,4)&\"+\"&Least(B345,FALSE)&\" seconds\"","v":"365 days, 5 hours, 48 minutes, 57+3/5 seconds"}]
[null,"Even though the mean year is limited to double floating point precision, that is good enough"]
[null,"to resolve the long form of the mean year down to the exact fraction of a second."]
["Cycle Jitter",{"$f":"4 - 0.242333333333333","v":{"v":3.757666666666667,"d":"3.757666666667"}},null,"days, jitter reckoned by Karl Palmen of the UK"]
["Base to Convert Decimal","[ ; 3000 ]"]
["Mixed Radix with bases",{"$f":"ToRadix(B350,B$349,TRUE)","v":"{3; 2273/3000}"}]
["Confirm Jitter",{"$f":"3+2273/3000","v":{"v":3.7576666666666667,"d":"3.757666666667"}},null,"=3+2273/3000",{"$f":"B352=B349","v":true},"Yes, we recovered Karl's jitter"]
["Exact Jitter",{"$f":"Nth(B351,1)&\"+\"&Nth(B351,-1)&\"/\"&Nth(B350,-1)","v":"3+2273/3000"},null,"days"]
[null,"This is a good mean year for the mean northward equinoctial year, but the cycle is absurdly long,"]
[null,"has excessive cycle jitter = 3+2273/3000 days, and its mean year is almost the same as the 293-year cycle."]
[null,"The number 9000 reduced to 3000 above, and days per cycle is divisible by 3, which implies"]
[null,"that the cycle is 3 times longer than it needs to be. The same mean year and jitter is obtained"]
[null,"by setting years per cycle = 3000 years and changing feast day every 450th year to every 500th year"]
[null,"and changing extra feast day every 9000th year to every 3000th year."]
[null,"The Palmen modification of the 3000-year variant retains the same mean year but reduces the"]
[null,"cycle jitter to 3+689/3000 days and limits the maximum number of Feast Days = 4."]
[]
[]
["Sidereal Calendar Mean Year"]
["Years Per Cycle",39,"years"]
["Days Per Common Year",365,"days"]
["Leap Days Per Cycle",10,"days"]
["Days Per Cycle",{"$f":"B365*B366+B367","v":14245},"days","39 x 365 + 10"]
["Mean Year + Fraction",{"$f":"ReduceFraction(B368,B365)","v":"365+10/39"},null,"days"]
["Mean Year",{"$f":"B368/B365","v":{"v":365.2564102564103,"d":"365.256410256410"}},null,"Days Per Cycle / Years Per Cycle",null,"(approximate, the first 6 decimal digits repeat)"]
["Bases for Time","[ ; 24, 60, 60, 39 ]",null,"Start with least base = years per cycle, and try reducing the least radix value"]
["Mixed Radix with bases",{"$f":"ToRadix(B371,B370,TRUE)","v":"{365; 6/24, 9/60, 13/60, 33/39}"},{"$nulls":3},"Least radix value reduces =",{"$f":"ReduceFraction(Nth(B372,-1),Nth(B371,-1))","v":"11/13"}]
["Reduced Bases for Time","[ ; 24, 60, 60, 13 ]",null,"Change least base to match reduced denominator"]
["Mixed Radix with bases",{"$f":"ToRadix(B373,B370,TRUE)","v":"{365; 6/24, 9/60, 13/60, 11/13}"},null,null,"Exact mean year in time units"]
["Long Form",{"$f":"Nth(B374,1)&\" days, \"&Nth(B374,2)& \" hours, \"&Nth(B374,3)&\" minutes, and \"&Nth(B374,4)&\"+\"&Nth(B374,-1)&\"/\"&Nth(B373,-1)&\" seconds\"","v":"365 days, 6 hours, 9 minutes, and 13+11/13 seconds"},{"$nulls":3},"a good approximation"]
["Mean Year from Fraction",{"$f":"ToRadix(B373,B369,TRUE)","v":"{365; 6/24, 9/60, 13/60, 11/13}"}]
[]
[]
["Ptolemy's mean synodic month as published in his Almagest treatise = 29 days 31' 50'' 8''' 20''''."]
["(Ptolemy cited Hipparchus as his source for the mean duration of the lunation.)"]
["That's 29 days 31 minutes 50 seconds 8 thirds and 20 fourths, where the numbers that represent the"]
["fraction of a day are in sexagesimal. Let's convert that to a decimal number:"]
[]
["Single base","[ 60 ]",null,null,"Using sexigesimal for radix conversion"]
["Given radix values","{ 29 ; 31, 50, 8, 20 }",null,null,"As above, from Ptolemy's Almagest"]
["Units to Apply","day; minute, second, third, fourth"]
["Apply Units",{"$f":"ApplyUnits(B385,B386)","v":"29 days ; 31 minutes, 50 seconds, 8 thirds, 20 fourths"}]
["Convert to Decimals",{"$f":"FromRadix(B384,B385)","v":{"v":29.53059413580247,"d":"29.5305941358025"}},null,"days","Limited by double precision floating point, but good enough."]
["Convert to Fraction",{"$f":"ContinuedFraction(B388)","v":"29+13753/25920"},null,"days","The accurate decimal value is: 29.530594135802469... days"]
[{"$nulls":4},"where the group of underscored digits repeat without end."]
["Time bases","[ ; 24, 60, 60, 3 ]",null,null,"Hours, minutes, seconds, and thirds of seconds"]
["Convert to time",{"$f":"ToRadix(B391,B388,TRUE)","v":"{29; 12/24, 44/60, 3/60, 1/3}"},null,null,"= Traditional Hebrew calendar molad interval"]
[null,"Even though the mean month is limited to double floating point precision, that is good enough"]
[null,"to resolve the mean month down to the exact number of seconds, including the third of a second."]
["Time bases with parts","[ ; 24, 1080 ]",null,null,"Hours, and parts (1080 parts per hour)"]
["Convert to time",{"$f":"ToRadix(B395,B388,TRUE)","v":"{29; 12/24, 793/1080}"},null,null,"= Traditional Hebrew calendar molad interval"]
["יחידות ליישום","מחזור/מחזורים, שבוע/שבועות, יום/ימים; שעה/שעות, חלק/חלקים"]
["להחיל יחידות",{"$f":"ApplyUnits(B396,B397)","v":"29 ימים ; 12 שעות, 793 חלקים"},null,null,"LibreOffice Calc incorrectly displays 29 leftmost."]
["Time bases","[ ; 24, 60, 18 ]",null,null,"Hours, minutes, and parts (18 parts per minute)"]
["Convert to time",{"$f":"ToRadix(B399,B388,TRUE)","v":"{29; 12/24, 44/60, 1/18}"},null,null,"= Traditional Hebrew calendar molad interval"]
["יחידות ליישום","מחזור/מחזורים, שבוע/שבועות, יום/ימים; שעה/שעות, דקה/דקות, חלק/חלקים"]
["להחיל יחידות",{"$f":"ApplyUnits(B400,B401)","v":"29 ימים ; 12 שעות, 44 דקות, 1 חלק"},null,null,"LibreOffice Calc incorrectly displays 29 leftmost."]
["Fraction numerator",{"$f":"Nth(B396,2)*1080+Nth(B396,3)","v":13753},"months having 30 days",null,"12 hours x 1080 parts + 793 parts"]
["Fraction denominator",{"$f":"Nth(B395,1)*Nth(B395,2)","v":25920},"months per cycle",null,"24 hours x 1080 parts"]
["Exact fraction",{"$f":"ReduceFraction(B403,B404)","v":"13753/25920"},null,"of a day","This fraction implies a 25920-month fixed lunar calendar cycle having"]
["Denominator - Numerator",{"$f":"B404-B403","v":12167},"months having 29 days",null,"13753 full 30-day months and 12167 deficient 29-day months per cycle."]
["Mean Month from Fraction",{"$f":"ToRadix(B399,B405,TRUE)","v":"{; 12/24, 44/60, 1/18}"},null,null,"Fractional part of mean month"]
["\"29+\" Put in Front",{"$f":"ToRadix(B399,\"29+\"&B405,TRUE)","v":"{29; 12/24, 44/60, 1/18}"},null,null,"Inserted \"29+\" in front of fraction"]
[]
[]
["The traditional molad interval together with the 19-year leap cycle determines the mean year of the Hebrew calendar:"]
["Mean Year Hebrew calendar",{"$f":"B388*235/19","v":{"v":365.24682220597794,"d":"365.246822205978"}},null,null,"molad interval x 235 months per cycle / 19 years per cycle"]
["Convert to Fraction",{"$f":"ContinuedFraction(B412,,10)","v":"365+24311/98496"},null,null,"10th continued fraction convergent"]
["Confirm Fraction",{"$f":"Whole(B413)+Numerator(B413)/Denominator(B413)","v":{"v":365.24682220597794,"d":"365.246822205978"}},null,null,"exactly the same decimal points"]
["Time bases with regaim","[ ; 24, 60, 18, 76 ]",null,null,"Hours, minutes, parts, and moments (76 moments per part)"]
["Exact time to the rega",{"$f":"ToRadix(B415,B$412,TRUE)","v":"{365; 5/24, 55/60, 7/18, 48/76}"},null,null,{"$f":"\"The moments fraction 48/76 reduces to \"&ReduceFraction(48,76)&\" of a part.\"","v":"The moments fraction 48/76 reduces to 12/19 of a part."}]
[null,"Even though the mean year is limited to double precision floating point, that is good enough"]
[null,"to resolve the mean year down to the exact number of moments (regaim)."]
["Bases with Nanoseconds","[; 24, 60, 60, 1000, 1000, 1000]",null,null,"(can't resolve to picoseconds)"]
["Time with Nanoseconds",{"$f":"ToRadix(B419,B413)","v":"{365; 5, 55, 25, 438, 596, 492}"}]
["Seconds Decimal Fraction",{"$f":"Nth(B420,-3)/1000+Nth(B420,-2)/1000000 + Nth(B420,-1)/1000000000","v":0.438596492},null,"= milliseconds/1000 + microseconds/1000000 + nanoseconds/1000000000"]
["Fraction of Seconds",{"$f":"ContinuedFraction(B421,,6)","v":"25/57"},null,"ContinuedFraction( ) 6th convergent yields the exact fraction of second."]
["Confirm Fraction",{"$f":"Numerator(B422)/Denominator(B422)","v":{"v":0.43859649122807015,"d":"0.438596491228"}},null,"slightly different because time only resolved to nanoseconds"]
["Time bases with seconds","[ ; 24, 60, 60, 57 ]",null,"set least base to match denominator of fraction of second"]
["Exact time to seconds",{"$f":"ToRadix(B424,B$412,TRUE)","v":"{365; 5/24, 55/60, 25/60, 25/57}"}]
["Mean Year from Fraction",{"$f":"ToRadix(B424,B413,TRUE)","v":"{365; 5/24, 55/60, 25/60, 25/57}"}]
[null,"Even though the mean year is limited to double floating point precision, that is good enough"]
[null,"to resolve the mean year down to the exact fraction of a second."]
[]
[]
["CCUE p126 molad moment"]
[{"$f":"2+8/24+429/25920","v":2.3498842592592593},"to radix",null,"from radix","Molad of Shevat 5782 = Yom Sheini at 8 hours 429 parts"]
["[ ; 24, 60, 18 ]",{"$f":"ToRadix(A433,A432)","v":"{2; 8, 23, 15}"},null,{"$f":"FromRadix(A433,B433)","v":2.3498842592592593},"with weekday and fractional hours minutes and parts"]
["Show Bases",{"$f":"ToRadix(A433,A432,TRUE)","v":"{2; 8/24, 23/60, 15/18}"},null,{"$f":"FromRadix(A433,B434)","v":2.3498842592592593}]
["Units to Apply","# weekday; hour, minute, part"]
["Apply Units",{"$f":"ApplyUnits(B433,B435)","v":"2 nd weekday ; 8 hours, 23 minutes, 15 parts"}]
["[ ; 24, 1080 ]",{"$f":"ToRadix(A437,A432)","v":"{2; 8, 429}"},null,{"$f":"FromRadix(A437,B437)","v":2.3498842592592593},"with weekday and fractional hours and parts"]
["Show Bases",{"$f":"ToRadix(A437,A432,TRUE)","v":"{2; 8/24, 429/1080}"},null,{"$f":"FromRadix(A437,B438)","v":2.3498842592592593}]
["Units to Apply","# weekday; hour, part"]
["Apply Units",{"$f":"ApplyUnits(B437,B439)","v":"2 nd weekday ; 8 hours, 429 parts"}]
[]
[]
["Arbitrary traditional molad moments:"]
["Hebrew Year:",5782]
["Hebrew Month (1-13):",13,"Nisan=1, Iyar=2, Sivan=3, Tammuz=4, Av=5, Elul=6, Tishrei=7, Cheshvan=8, Kislev=9, Tevet=10, Shevat=11, Adar=12, Adar2=13"]
["Lunation:",{"$f":"ElapsedMonths19(B444,B445)","v":71507},"calculated automatically"]
["Traditional molad:",{"$f":"TraditionalMolad(B446)","v":{"v":738217.4110725309,"d":"738217.41107253"}},null,"Hebrew Yom"]
["{7; 24, 60, 18}",{"$f":"ToRadix(A448,B447)","v":"{105459, 4; 9, 51, 17}"},null,{"$f":"Nth(B448,2)+1","v":5}]
["Show Bases",{"$f":"ToRadix(A448,B447,TRUE)","v":"{105459, 4/7; 9/24, 51/60, 17/18}"}]
["Units to Apply","week, # weekday; hour, minute, part"]
["Apply Units",{"$f":"ApplyUnits(B448,B450)","v":"105459 weeks, 4 th weekday ; 9 hours, 51 minutes, 17 parts"}]
["Back to Number",{"$f":"FromRadix(A448,B451)","v":{"v":738217.4110725309,"d":"738217.41107253"}},null,null,"In Excel, units don't interfere with FromRadix( )"]
["יחידות ליישום","שבוע/שבועות, יום/ימים; שעה/שעות, דקה/דקות, חלק/חלקים"]
["להחיל יחידות",{"$f":"ApplyUnits(B448,B453)","v":"105459 שבועות, 4 ימים ; 9 שעות, 51 דקות, 17 חלקים"},{"$nulls":3},"LibreOffice Calc incorrectly displays the number of weeks leftmost."]
["{7; 24, 1080}",{"$f":"ToRadix(A455,B447)","v":"{105459, 4; 9, 935}"},null,{"$f":"Nth(B455,2)+1","v":5},"Hebrew weekday in the radix list (2nd item): Yom Rishon=0 .. Shabbat=6"]
["Show Bases",{"$f":"ToRadix(A455,B447,TRUE)","v":"{105459, 4/7; 9/24, 935/1080}"},null,null,"+1 for Hebrew Yom column: Yom Rishon=1 .. Shabbat=7"]
["Units to Apply","week, # weekday; hour, part"]
["Apply Units",{"$f":"ApplyUnits(B455,B457)","v":"105459 weeks, 4 th weekday ; 9 hours, 935 parts"}]
["יחידות ליישום","שבוע/שבועות, יום/ימים; שעה/שעות, חלק/חלקים"]
["להחיל יחידות",{"$f":"ApplyUnits(B455,B459)","v":"105459 שבועות, 4 ימים ; 9 שעות, 935 חלקים"},{"$nulls":3},"LibreOffice Calc incorrectly displays the number of weeks leftmost."]
[]
[]
["100-Saros Lunisolar Calendar Leap Cycle with 22300 months per cycle = 100 x 223-month saros eclipse interval"]
["Months per cycle",12,"per year times",1803,"years =",{"$f":"B464*D464","v":21636},"common months per cycle"]
["Months in 100-Saros",{"$f":"100*223","v":22300},"minus common months per cycle = ",null,null,{"$f":"B465-F464","v":664},"leap months per cycle"]
["One saros interval =",6585.3211,"days, times 100 =",{"$f":"100*B466","v":658532.11},"days per cycle, rounded:",{"$f":"ROUND(D466,0)","v":658532},"days per cycle"]
["Calendar Mean Year = Days Per Cycle / Years Per Cycle =",{"$nulls":3},{"$f":"F466/D464","v":{"v":365.24237382140876,"d":"365.242373821409"}},"days"]
["Mean Year + Fraction",{"$f":"ReduceFraction(F466,D464)","v":"365+437/1803"},null,"days"]
["Bases for Time","[ ; 24, 60, 60, 1803 ]",null,"Start with least base = years per cycle, and try reducing the least radix value"]
["Mean Year from Fraction",{"$f":"ToRadix(B469,B468,TRUE)","v":"{365; 5/24, 49/60, 1/60, 177/1803}"}]
["Mean Year from Decimal",{"$f":"ToRadix(B469,E$467,TRUE)","v":"{365; 5/24, 49/60, 1/60, 177/1803}"},{"$nulls":3},"Least radix value reduces =",{"$f":"ReduceFraction(Nth(B471,-1),Nth(B469,-1))","v":"59/601"}]
["Reduced Bases","[ ; 24, 60, 60, 601 ]",null,null,"Change least base to match reduced denominator"]
["Mixed Radix with bases",{"$f":"ToRadix(B472,E$467,TRUE)","v":"{365; 5/24, 49/60, 1/60, 59/601}"},null,null,"Less than a second more than the mean northward equinoctial year."]
["Long Form",{"$f":"Nth(B473,1)&\" days, \"&Nth(B473,2)& \" hours, \"&Nth(B473,3)&\" minutes, and \"&Nth(B473,4)&\"+\"&Nth(B473,-1)&\"/\"&Nth(B472,-1)&\" seconds\"","v":"365 days, 5 hours, 49 minutes, and 1+59/601 seconds"}]
["Lunar Cycle",25,"saros times",223,"months per saros =",{"$f":"B475*223","v":5575},"months (1/4 of 100-saros cycle)"]
["Mean Month = Days Per Cycle / Months Per Cycle =",{"$nulls":3},{"$f":"F466/B465","v":{"v":29.530582959641254,"d":"29.5305829596413"}},"days =",{"$f":"ReduceFraction(F466,B465)","v":"29+2958/5575"},null,null,"days"]
["Bases for Fraction","[ ; 5575 ]"]
["Mixed Radix with bases",{"$f":"ToRadix(B477,E476,TRUE)","v":"{29; 2958/5575}"},null,"Thus the mean month is exactly ",null,{"$f":"Nth(B478,1)&\"+\"&Nth(B478,-1)&\"/\"&Nth(B477,-1)","v":"29+2958/5575"},"days, as above"]
["Bases for Time","[ ; 24, 60, 60, 5575 ]",null,"Start with least base = months per cycle, and try reducing the least radix value"]
["Mean Month from Fraction",{"$f":"ToRadix(B479,G$476,TRUE)","v":"{29; 12/24, 44/60, 2/60, 2050/5575}"}]
["Mean Month as Time",{"$f":"ToRadix(B479,E$476,TRUE)","v":"{29; 12/24, 44/60, 2/60, 2050/5575}"},{"$nulls":3},"Least radix value reduces =",{"$f":"ReduceFraction(Nth(B481,-1),Nth(B479,-1))","v":"82/223"}]
["Reduced Bases","[ ; 24, 60, 60, 223 ]",null,"Change least base to match reduced denominator = months per saros interval"]
["Mean Month as Time",{"$f":"ToRadix(B482,E$476,TRUE)","v":"{29; 12/24, 44/60, 2/60, 82/223}"},null,null,"Exact mean month in time units"]
["Long Form",{"$f":"Nth(B483,1)&\" days, \"&Nth(B483,2)& \" hours, \"&Nth(B483,3)&\" minutes, and \"&Nth(B483,4)&\"+\"&Nth(B483,-1)&\"/\"&Nth(B482,-1)&\" seconds\"","v":"29 days, 12 hours, 44 minutes, and 2+82/223 seconds"}]
[]
["A solar leap day calendar with 1803-year cycle and having the same mean year:"]
["Common Days Per Cycle = Years Per Cycle x 365",null,null,{"$f":"D464*365","v":658095},"common days"]
["Leap Days = Days Per Cycle - Non-Leap Days",null,null,{"$f":"F466-D487","v":437},"leap days"]
["So calendar mean year as an exact fraction =",null,null,{"$f":"ReduceFraction(D487+D488,D464)","v":"365+437/1803"},"days, as above"]
[]
["A solar leap week calendar with 1803-year cycle and having the same mean year:"]
["Common Days Per Cycle = Years Per Cycle x 364",null,null,{"$f":"D464*364","v":656292},"common days"]
["Leap Days = Days Per Cycle - Non-Leap Days",null,null,{"$f":"F466-D492","v":2240},"leap days"]
["Leap Weeks = Leap Days / 7",null,null,{"$f":"D493/7","v":320},"leap weeks"]
[]
[]
["Triple Saros interval",{"$f":"3*B466","v":19755.9633},null,"days","(Exeligmos)"]
["3 x 223 months =",{"$f":"3*223","v":669},null,"lunations"]
["Mean Lunation",{"$f":"B497/B498","v":{"v":29.53058789237668,"d":"29.5305878923767"}},null,"days"]
["Bases for Time","[ ; 24, 60, 60, 669 ]",null,"Try setting least base to months per cycle"]
["Mixed Radix",{"$f":"ToRadix(B500,B499)","v":"{29; 12, 44, 2, 531}"},null,null,"Least radix value reduces =",{"$f":"ReduceFraction(Nth(B501,-1),Nth(B500,-1))","v":"177/223"}]
["Bases for Time","[ ; 24, 60, 60, 223 ]",null,"Change least base to reduced denominator"]
["Mixed Radix",{"$f":"ToRadix(B502,B499)","v":"{29; 12, 44, 2, 177}"}]
["Long Form",{"$f":"Nth(B503,1)&\" days, \"&Nth(B503,2)& \" hours, \"&Nth(B503,3)&\" minutes, and \"&Nth(B503,4)&\"+\"&Nth(B503,-1)&\"/\"&Nth(B502,-1)&\" seconds\"","v":"29 days, 12 hours, 44 minutes, and 2+177/223 seconds"}]
[]
[]
["Ptolemy's Cycle of About 345 Years:"]
["Days Elapsed",{"$f":"126007+1/24","v":{"v":126007.04166666667,"d":"126007.0417"}},null,"days (decimal fraction is one hour)"]
["Lunar months",4267,null,"lunations"]
["Mean Lunation",{"$f":"B508/B509","v":{"v":29.530593313022422,"d":"29.5305933130224"}},null,"days"]
["Mean Lunation Fract",{"$f":"B508/B509-29","v":{"v":0.5305933130224219,"d":"0.530593313022422"}},null,"of a day"]
["Convert to Fraction",{"$f":"ContinuedFraction(B511,,12)","v":"54337/102408"},null,"12th convergent, denominator exactly 24 x the number of lunations"]
["Confirm Fraction",{"$f":"Numerator(B512)/Denominator(B512)","v":{"v":0.5305933130224201,"d":"0.530593313022420"}},null,"identical to above decimal points except the last (15th) digit"]
["Convert to Fraction",{"$f":"ContinuedFraction(B511,,10)","v":"13155/24793"},null,"10th convergent, denominator is about 5.8 x the number of lunations"]
["Confirm Fraction",{"$f":"Numerator(B514)/Denominator(B514)","v":{"v":0.5305933126285645,"d":"0.530593312628564"}},null,"first 8 decimal points match"]
["Bases for Time","[ ; 24, 60, 60, 4267 ]",null,"Try setting least base to months per cycle"]
["Mixed Radix 29 days +",{"$f":"ToRadix(B516,B511,,\"R\")","v":"{; 12, 44, 3, 1119/4267}"},null,"Least radix value doesn't reduce"]
["Long Form 29 days +",{"$f":"Nth(B517,1)&\" hours, \"&Nth(B517,2)& \" minutes, and \"&Nth(B517,3)&\"+\"&Least(B517,FALSE)&\" seconds\"","v":"12 hours, 44 minutes, and 3+1119/4267 seconds"}]
["Bases for Time","[ ; 24, 60, 60 ]"]
["Mixed Radix 29 days +",{"$f":"ToRadix(B519,B511,,6)","v":"{; 12, 44, 3.262245}"},null,"Least radix value (seconds) rounded to 6 decimal points"]
[]
[]
["years (364 days = 4 quarters), quarters (13 weeks), weeks, days, hours, minutes, seconds, milliseconds"]
["bases","[4, 13, 7; 24, 60, 60, 1000]"]
["calculated number",{"$f":"4*52*7+49*7+5+17/24+49/1440+(41.111605)/86400","v":1804.7428369398726},{"$nulls":3},"back to number"]
["truncate",{"$f":"ToRadix(B524,B525,,\"T\")","v":"{4, 3, 10, 5; 17, 49, 41, 111}"},{"$nulls":3},{"$f":"FromRadix(B$524,B526)","v":1804.7428369328704}]
["round least to 3 decimals",{"$f":"ToRadix(B524,B525,,3)","v":"{4, 3, 10, 5; 17, 49, 41, 111.605}"},{"$nulls":3},{"$f":"FromRadix(B$524,B527)","v":1804.7428369398726}]
["show bases",{"$f":"ToRadix(B524,B525,TRUE,3)","v":"{4, 3/4, 10/13, 5/7; 17/24, 49/60, 41/60, 111.605/1000}"},{"$nulls":3},{"$f":"FromRadix(B$524,B528)","v":1804.7428369398726}]
["round least to 0 decimals",{"$f":"ToRadix(B524,B525,,0)","v":"{4, 3, 10, 5; 17, 49, 41, 112}"},{"$nulls":3},{"$f":"FromRadix(B$524,B529)","v":1804.7428369444444}]
["show bases",{"$f":"ToRadix(B524,B525,TRUE,0)","v":"{4, 3/4, 10/13, 5/7; 17/24, 49/60, 41/60, 112/1000}"},{"$nulls":3},{"$f":"FromRadix(B$524,B530)","v":1804.7428369444444}]
["Units to Apply","year, quarter, week, day; hour, minute, second, millisecond, microsecond"]
["Apply Units",{"$f":"ApplyUnits(B527,B531)","v":"4 years, 3 quarters, 10 weeks, 5 days ; 17 hours, 49 minutes, 41 seconds, 111.605 milliseconds"}]
[]
[]
["bases","[4, 13, 7; 24, 60]"]
["calculated number",{"$f":"4*52*7+49*7+5+17/24+49/1440+(41.111)/86400","v":1804.7428369328702},{"$nulls":3},"back to number"]
["raw",{"$f":"ToRadix(B$535,B$536,,\"U\")","v":"{4, 3, 10, 5; 17, 49.685183333233}"},null,null,"too many decimals",{"$f":"FromRadix(B$535,B537)","v":1804.7428369328702}]
["truncate",{"$f":"ToRadix(B$535,B$536,,\"T\")","v":"{4, 3, 10, 5; 17, 49}"},{"$nulls":3},1804.7423611111112]
["round least to integer",{"$f":"ToRadix(B$535,B$536,,0)","v":"{4, 3, 10, 5; 17, 50}"},null,null,"this is default rounding",1804.7430555555557]
["show bases",{"$f":"ToRadix(B535,B536,TRUE,0)","v":"{4, 3/4, 10/13, 5/7; 17/24, 50/60}"},{"$nulls":3},1804.7430555555557]
["round least to 3 decimals",{"$f":"ToRadix(B$535,B$536,,3)","v":"{4, 3, 10, 5; 17, 49.685}"},{"$nulls":3},1804.7428368055555]
["show bases",{"$f":"ToRadix(B$535,B$536,TRUE,3)","v":"{4, 3/4, 10/13, 5/7; 17/24, 49.685/60}"},{"$nulls":3},1804.7428368055555]
["Apply Units",{"$f":"ApplyUnits(B539,B531)","v":"4 years, 3 quarters, 10 weeks, 5 days ; 17 hours, 50 minutes"},{"$nulls":3},"(extra post-radix names ignored)"]
[]
[]
["bases","[ 4, 13, 7; 24, 60, 60 ]"]
["calculated number",{"$f":"4*52*7+49*7+5+17/24+49/1440+(41.111671)/86400","v":1804.7428369406364},{"$nulls":3},"back to number"]
["raw",{"$f":"ToRadix(B546,B547,,\"U\")","v":"{4, 3, 10, 5; 17, 49, 41.1116709709167}"},null,null," too many decimals",{"$f":"FromRadix(B$546,B548)","v":1804.7428369406362}]
["truncate",{"$f":"ToRadix(B546,B547,,\"T\")","v":"{4, 3, 10, 5; 17, 49, 41}"},{"$nulls":3},1804.7428356481482]
["round least to integer",{"$f":"ToRadix(B546,B547,,0)","v":"{4, 3, 10, 5; 17, 49, 41}"},{"$nulls":3},1804.7428356481482]
["show bases format",{"$f":"ToRadix(B546,B547,TRUE,0)","v":"{4, 3/4, 10/13, 5/7; 17/24, 49/60, 41/60}"},{"$nulls":3},1804.7428356481482]
["round least to 3 decimals",{"$f":"ToRadix(B546,B547,,3)","v":"{4, 3, 10, 5; 17, 49, 41.112}"},{"$nulls":3},1804.7428369444444]
["round least to 6 decimals",{"$f":"ToRadix(B546,B547,,6)","v":"{4, 3, 10, 5; 17, 49, 41.111671}"},null,null,"exact",1804.7428369406366]
["show bases format",{"$f":"ToRadix(B546,B547,TRUE,6)","v":"{4, 3/4, 10/13, 5/7; 17/24, 49/60, 41.111671/60}"},{"$nulls":3},1804.7428369406366]
["Apply Units",{"$f":"ApplyUnits(B553,B531)","v":"4 years, 3 quarters, 10 weeks, 5 days ; 17 hours, 49 minutes, 41.111671 seconds"}]
[]
[]
["bases","[ 4, 13, 7; 24, 60, 60,1000,1000 ]",null,null,"even with microseconds","back to number"]
["raw",{"$f":"ToRadix(B558,B547,,\"U\")","v":"{4, 3, 10, 5; 17, 49, 41, 111, 670.96875}"},null,null,"too many decimal points",{"$f":"FromRadix(B$558,B559)","v":1804.7428369406362}]
["round least to integer",{"$f":"ToRadix(B558,B547,,0)","v":"{4, 3, 10, 5; 17, 49, 41, 111, 671}"},null,null,"rounded to microsecond",{"$f":"FromRadix(B$558,B560)","v":1804.7428369406366}]
["show bases format",{"$f":"ToRadix(B558,B547,TRUE,0)","v":"{4, 3/4, 10/13, 5/7; 17/24, 49/60, 41/60, 111/1000, 671/1000}"},{"$nulls":3},{"$f":"FromRadix(B$558,B561)","v":1804.7428369406366}]
["Apply Units",{"$f":"ApplyUnits(B560,B531)","v":"4 years, 3 quarters, 10 weeks, 5 days ; 17 hours, 49 minutes, 41 seconds, 111 milliseconds, 671 microseconds"}]
[]
[]
["Bases","[ 52, 7; 24, 60, 60, 1000 ]",null,null,"bases resolving to the millisecond"]
["Round to Millisecond",{"$f":"roundtoleast(B565,17/24+49/1440+(41.111671)/86400)","v":0.7428369444444445},null,null,"just the fractional part"]
["Mixed Radix / Bases",{"$f":"ToRadix(B565,B566,TRUE,\"U\")","v":"{0/7; 17/24, 49/60, 41/60, 112/1000}"},{"$nulls":3},"raw value already rounded"]
["back to fraction of day",{"$f":"FromRadix(B565,B567)","v":0.7428369444444445}]
["Excel 24h time format",{"$f":"B568","v":"17:49:41.112"},null,"with milliseconds decimal points on seconds"]
["Abbreviated Units","y,w,d;h,m,s,ms"]
["Apply Units",{"$f":"ApplyUnits(B567,B570,FALSE)","v":"0 d ; 17 h, 49 m, 41 s, 112 ms"},null,null,"no plurals"]
["Trim Zeros",{"$f":"TrimZeros(B567)","v":"{0/7; 17/24, 49/60, 41/60, 112/1000}"},null,null,"leaves at least one zero left of radix point"]
["Apply Units",{"$f":"ApplyUnits(B572,B571)","v":"0 d ; 17 h, 49 m, 41 s, 112 ms"}]
[]
[]
["Lunisolar Cycle"]
["Alternate Mean Years",{"$f":"365+437/1803","v":{"v":365.24237382140876,"d":"365.242373821409"}},null,null,"1803-year 100-saros solar cycle",null,1803]
["(copy/paste below)",{"$f":"365+127/524","v":{"v":365.24236641221376,"d":"365.242366412214"}},null,null,"524-year solar cycle with 127 leap days",null,524]
[]
["Alternate Mean Months",{"$f":"29+2958/5575","v":{"v":29.530582959641254,"d":"29.5305829596413"}},null,null,"25-saros 5575-month lunar cycle",null,5575]
["(copy/paste below)",{"$f":"29+451/850","v":{"v":29.53058823529412,"d":"29.5305882352941"}},null,null,"52-yerm 850-month lunar cycle",null,850]
[null,{"$f":"29+13753/25920","v":{"v":29.53059413580247,"d":"29.5305941358025"}},null,null,"traditional molad interval",null,25920]
[null,{"$f":"29+425/801","v":{"v":29.530586766541823,"d":"29.5305867665418"}},null,null,"49-yerm 801-month lunar cycle",null,801]
[]
["Target Mean Month",{"$f":"29+425/801","v":{"v":29.530586766541823,"d":"29.5305867665418"}},null,"days"]
["Target Mean Year",{"$f":"365+127/524","v":{"v":365.24236641221376,"d":"365.242366412214"}},null,"days"]
["Mean Ratio Month/Year",{"$f":"B585/B586","v":{"v":0.08085203000030261,"d":"0.0808520300003026"}}]
["Max Convergent",7,null,"Remove extraneous characters if LibreOffice CALC autocompletes here."]
["Continued Fraction",{"$f":"ContinuedFraction(B587,,B588)","v":"353/4366"},null,{"$f":"Numerator(B589)&\" years containing \"&Denominator(B589)&\" months\"","v":"353 years containing 4366 months"}]
["Confirm Fraction",{"$f":"Numerator(B589)/Denominator(B589)","v":{"v":0.08085203847915712,"d":"0.0808520384791571"}},null,"for 7th convergent, first 8 decimal points match"]
["Years",{"$f":"Numerator(B589)","v":353}]
["Months",{"$f":"Denominator(B589)","v":4366}]
["Mean Year",{"$f":"B592*B585/B591","v":{"v":365.2423281096929,"d":"365.242328109693"}},null,"days"]
["Bases","[ ;24, 60, 60 ]"]
["Lunisolar Mixed Radix",{"$f":"ToRadix(B594,B593,,3)","v":"{365; 5, 48, 57.149}"}]
["Target Mixed Radix",{"$f":"ToRadix(B594,B586,,3)","v":"{365; 5, 49, 0.458}"}]
["Absolute Difference",{"$f":"ABS(B586-B593)","v":{"v":3.8302520863453537E-5,"d":"0.000038302521"}},null,"days"]
["Units to Apply",";hour,minute,second"]
["Mixed Radix",{"$f":"ToRadix(B594,B597,,3)","v":"{; 0, 0, 3.309}"},null,{"$f":"IF(B593<B586,\"too short\",IF(B586<B593,\"too long\",\"\"))","v":"too short"}]
["Apply Units",{"$f":"ApplyUnits(B599,B598)","v":"; 0 hours, 0 minutes, 3.309 seconds"},null,null,{"$f":"IF(B593<B586,\"too short\",IF(B586<B593,\"too long\",\"\"))","v":"too short"}]
["Absolute Difference",{"$f":"1440*B597","v":0.055155630043373094},null,"minutes"]
["Bases","[ 60; 60 ]",null,"shifted radix point 2 positions to the right, dropped hours"]
["Mixed Radix",{"$f":"ToRadix(B602,B601,,3)","v":"{0; 3.309}"},null,{"$f":"IF(B593<B586,\"too short\",IF(B586<B593,\"too long\",\"\"))","v":"too short"}]
["Units to Apply","day,minute;second",null,"shifted radix point 2 positions to the right"]
["Apply Units",{"$f":"ApplyUnits(B603,B604)","v":"0 minutes ; 3.309 seconds"},null,null,{"$f":"IF(B593<B586,\"too short\",IF(B586<B593,\"too long\",\"\"))","v":"too short"}]
["Absolute Difference",{"$f":"60*B601","v":3.3093378026023856},null,"seconds","(don't need mixed radix conversion if less than one minute)"]
[]
[]
["Following example from https://stackoverflow.com/questions/759296/converting-a-decimal-to-a-mixed-radix-base-number"]
[]
["Bases","[52,7,24,60,60,1000]"]
["Units","week,day,hour,minute,second,millisecond"]
["Milliseconds",662321]
["Mixed Radix",{"$f":"ToRadix(B611,B613)","v":"{11, 2, 321}"}]
["Show Bases",{"$f":"ToRadix(B611,B613,TRUE)","v":"{11/60, 2/60, 321/1000}"}]
["Back to Milliseconds",{"$f":"FromRadix(B611,B614)","v":662321}]
["Apply Units",{"$f":"ApplyUnits(B614,B612)","v":"11 minutes, 2 seconds, 321 milliseconds"}]
["Back to Milliseconds",{"$f":"FromRadix(B611,B614)","v":662321}]
[]
[]
["Mixed Radix (given)","{32,5,7,45,15,500}"]
["Apply Units",{"$f":"ApplyUnits(B621,B612)","v":"32 weeks, 5 days, 7 hours, 45 minutes, 15 seconds, 500 milliseconds"}]
["Milliseconds",{"$f":"FromRadix(B611,B621)","v":19813515500}]
["Show Bases",{"$f":"ToRadix(B611,B623,TRUE)","v":"{32/52, 5/7, 7/24, 45/60, 15/60, 500/1000}"}]
[]
[]
["Bases","[365, 24, 60, 60, 1000]"]
["Units","year,day,hour,minute,second,millisecond"]
["Milliseconds",313732097]
["Mixed Radix",{"$f":"ToRadix(B627,B629)","v":"{3, 15, 8, 52, 97}"}]
["Apply Units",{"$f":"ApplyUnits(B630,B628)","v":"3 days, 15 hours, 8 minutes, 52 seconds, 97 milliseconds"}]
["Back to Milliseconds",{"$f":"FromRadix(B627,B629)","v":313732097}]
["Show Bases",{"$f":"ToRadix(B627,B629,TRUE)","v":"{3/365, 15/24, 8/60, 52/60, 97/1000}"}]
["Back to Milliseconds",{"$f":"FromRadix(B627,B633)","v":313732097}]
[]
[]
["bases","[7,24,60,60]",null,"[7;24,60,60]","make time fractional"]
["seconds in one week",{"$f":"7*86400","v":604800},null,7,"days"]
["exactly one week",{"$f":"ToRadix(B637,B638)","v":"{1, 0, 0, 0, 0}"},null,{"$f":"ToRadix(D637,D638)","v":"{1, 0; 0, 0, 0}"}]
["back to seconds",{"$f":"FromRadix(B637,B639)","v":604800},null,{"$f":"FromRadix(D637,D639)","v":7},"back to days"]
[{"$nulls":3},{"$f":"TrimZeros(D639)","v":"{1, 0}"},"trim zeros removes all trailing zeros and the radix point"]
[{"$nulls":3},{"$f":"FromRadix(D637,D641)","v":7},"back to days"]
[]
[]
["Bases","[24, 60, 60]"]
["Seconds",59263]
["Mixed Radix",{"$f":"ToRadix(B645,B646)","v":"{16, 27, 43}"}]
["Back to Seconds",{"$f":"FromRadix(B645,B647)","v":59263}]
["Show Bases",{"$f":"ToRadix(B645,B646,TRUE)","v":"{16/24, 27/60, 43/60}"}]
["Back to Seconds",{"$f":"FromRadix(B645,B649)","v":59263}]
["Units to Apply","day,hour,minute,second"]
["Apply Units",{"$f":"ApplyUnits(B647,B651)","v":"16 hours, 27 minutes, 43 seconds"}]
["Elapsed Fraction of Day",{"$f":"B646/86400","v":0.6859143518518519}]
["Excel 24h Time Format",{"$f":"B653","v":"16:27:43"}]
[]
[]
["degrees to circles, degrees ( ° ), arcminutes ( ' ), arcseconds ( \" )"]
["Bases","[ 360 ; 60, 60 ]"]
["Angle",633.4515,"degrees"]
["Mixed Radix",{"$f":"ToRadix(B658,B659)","v":"{1, 273; 27, 5}"},null,"rounded to the arcsecond"]
["Back to Number",{"$f":"FromRadix(B658,B660)","v":633.4513888888889},null,"changed slightly due to rounding"]
["Show Bases",{"$f":"ToRadix(B658,B659,TRUE)","v":"{1, 273/360; 27/60, 5/60}"}]
["Units to Apply","circle, degree; arcminute, arcsecond"]
["Apply Units",{"$f":"ApplyUnits(B660,B663)","v":"1 circle, 273 degrees ; 27 arcminutes, 5 arcseconds"}]
["Abbreviated Units","circle,°; ', \""]
["Apply Abbrev Units",{"$f":"ApplyUnits(B660,B665,FALSE)","v":"1 circle, 273 ° ; 27 ', 5 \""}]
[]
[]
["28 Day x 13 Month Calendar: Interconvert Elapsed Days with Elapsed Months, Weeks, Days"]
["Bases","[ 13, 4, 7 ]",null,null,"elapsed counts because each starts at zero"]
["Units","year, month, week, day"]
["Confirm Limit / Base",{"$f":"ApplyUnits(B670,B671)","v":"13 months, 4 weeks, 7 days"},null,null,"extra day and leap day at end of year"]
["Days Elapsed in Year",363,null,null,"meaningless if >= 364"]
["Mixed Radix",{"$f":"ToRadix(B670,B673)","v":"{12, 3, 6}"}]
["Show Bases",{"$f":"ToRadix(B670,B673,TRUE)","v":"{12/13, 3/4, 6/7}"}]
["Apply Units",{"$f":"ApplyUnits(B674,B671)","v":"12 months, 3 weeks, 6 days"},null,null,"elapsed months, weeks within month, days within week"]
["Back to Elapsed Days",{"$f":"FromRadix(B670,B674)","v":363},null,null,"this will recover days elapsed entry even if >=364 days"]
[]
["Month, week, day","7, 3, 6",null,"if you enter a number that is >= the limit for any base …"]
["Apply Units",{"$f":"ApplyUnits(B679,B671)","v":"7 months, 3 weeks, 6 days"}]
["Convert to Elapsed Days",{"$f":"FromRadix(B670,B679)","v":223}]
["Back to Mixed Radix",{"$f":"ToRadix(B670,B681)","v":"{7, 3, 6}"},null,"… then this won't recover your entry"]
[]
[]
["Year to Millennia, Centuries, Decades, Years"]
["Bases","[ 10, 10, 10 ]",null,"10 years per decade, 10 decades per century, 10 centuries per millenium"]
["Units","millennium/millennia, century/centuries, decade, year"]
["Confirm Limit / Base",{"$f":"ApplyUnits(B686,B687)","v":"10 centuries, 10 decades, 10 years"}]
["Year Number",2022]
["Radix Conversion",{"$f":"ToRadix(B686,B689)","v":"{2, 0, 2, 2}"}]
["Apply Units",{"$f":"ApplyUnits(B690,B687)","v":"2 millennia, 0 centuries, 2 decades, 2 years"}]
["This example is arguably trivial, because anybody can see the digits of the decimal year number,"]
["but consider that the computer did this calculation in binary! Also the most significant radix value"]
["has no limit. SpaceEvery( 1, Year ) will similarly separate the digits (without commas):",{"$nulls":4},{"$f":"SpaceEvery(1,B689)","v":"2 0 2 2"}]
[]
[]
["The following examples for the Tibetan Calendar are taken from \"Tibetan Calendar Mathematics\" by Svante Janson"]
["https://arxiv.org/abs/1401.6285v1",null,null,"(Janson is a Swedish mathematician at Uppsala University)"]
["http://www2.math.uu.se/~svante/papers/calendars/tibet.pdf",{"$nulls":3},"(revised 2022 Feb 13 to correct typographical error)"]
["https://katalog.uu.se/profile/?id=XX2949",null,null,"(Janson's academic web site at Uppsala University, in English)"]
["Janson omitted most radix points (semicolon), but here they are mandatory for obtaining the correct results."]
[]
["Strangely, despite introducing the use of mixed radix conversion, the authors of CCUE wrote in their"]
["chapter about the Tibetan Calendar: \"Though traditional calendarists use mixed radix fractions in their calculations,"]
["we convert everything to ordinary rational numbers, as we have done for other calendars in this book.\""]
["Perhaps they did that because their \"from-radix( )\" function only returns decimal fractions."]
["Also note that they cited Janson's manuscript as reference 4 of their Tibetan Calendar chapter!"]
[]
["I modified my FromRadix( ) function so that it has the option to return fractions, and, as can be seen below,"]
["my functions are able to reproduce all of Janson's mixed radix calculations, which validates that mine are"]
["working properly and also confirms that Janson didn't make any mixed radix calculation mistakes"]
["(except that he intentionally omitted most of the radix points and stubbornly refuses to put them in)."]
[]
["Meanings of a few of the base values and other significant numbers (as best as I've managed to glean so far):"]
[5,"elements (\"male\" and \"female\") and their associated colours, used in naming years"]
[7,"days per week (Saturday=0 … Friday=6)"]
[7,"moving celestial bodies (Saturn=0, Sun=1, Moon=2, Mars=3, Mercury=4, Jupiter=5, Venus=6)"]
[11,"karana names (lunar half-days)"]
[12,"month names"]
[12,"animal totems used in naming years"]
[13,"?"]
[27,"0 to 26 lunar mansions used for planets other than Mercury"]
[null,"used to calculate longitude by dividing the ecliptic into 27 lunar mansions"]
[27,"0 to 26 yogas (combined motions of Sun and Moon)"]
[28,"lunar mansions (stations, naksatras) in a full circle"]
[28,"anomaly is calculated modulo 28"]
[30,"degrees per zodiac sign"]
[30,"New Moon taken as 30th day of the month"]
[60,"sexagenary cycle of Tibetan years (there are two: rabjyung, Vṛhaspati = Drukchu kor)"]
[null,"also arcminutes, arcseconds"]
[67,"?"]
[126,"?"]
[480,"each month the karana mean weekday falls behind the siddhanta by 480 of the smallest 707 sub-units"]
[707,"traditional denominator for least radix value of mean synodic month = 480"]
[null,"707 x 8 = mean month denominator 5656"]
[null,"mystical: traditional subdivision of each breath into 707 sub-units"]
[null,"(seems quite intentional since 707 is a divisor of 5656)"]
[6811,"?"]
[149209,"?"]
[]
["p21 (7.2 to 7.4)","m1",null,null,"m2 = m1/30","m0"]
["Bases","( ; 60, 60, 6, 707 )",null,null,{"$f":"B742","v":"( ; 60, 60, 6, 707 )"},{"$f":"B742","v":"( ; 60, 60, 6, 707 )"}]
["Radices","29; 31, 50, 0, 480",null,null,"0; 59, 3, 4, 16","0; 50, 44, 2, 38"]
["Mean Lunar Month",{"$f":"FromRadix(B742,B743)","v":{"v":29.530586987270155,"d":"29.5305869872702"}},null,"Mean Lunar Day",{"$f":"FromRadix(E742,E743)","v":0.9843528995756718},{"$f":"FromRadix(F742,F743)","v":0.8456506364922207}]
["as Mixed Fraction",{"$f":"FromRadix(B742,B743,\"M\")","v":"29+3001/5656"},null,null,{"$f":"FromRadix(E742,E743,\"M\")","v":"11135/11312"},{"$f":"FromRadix(F742,F743,\"M\")","v":"4783/5656"}]
["as Improper Fraction",{"$f":"FromRadix(B742,B743,\"I\")","v":"167025/5656"}]
["Divisors of 5656",{"$f":"Divisors(5656)","v":"2', 4, 7', 8, 14, 28, 56, 101', 202, 404, 707, 808, 1414, 2828, 5656"}]
["see also p34 (12.1)",{"$f":"\"note that sub-units base 707 is 1/\"&5656/707&\" of denominator 5656 (months per lunar cycle)\"","v":"note that sub-units base 707 is 1/8 of denominator 5656 (months per lunar cycle)"}]
["convert to time units","[;24,60,60,1000,1000]"]
["Radices",{"$f":"ToRadix(B749,B744)","v":"{29; 12, 44, 2, 715, 700}"}]
["Units to Apply","day;hour,minute,second,millisecond,microsecond"]
["Apply Units",{"$f":"ApplyUnits(B750,B751)","v":"29 days ; 12 hours, 44 minutes, 2 seconds, 715 milliseconds, 700 microseconds"}]
["Abbrev Units to Apply","d;h,m,s,ms,us"]
["Apply Abbrev Units",{"$f":"ApplyUnits(B750,B753,FALSE)","v":"29 d ; 12 h, 44 m, 2 s, 715 ms, 700 us"},null,null,"plurals suppressed"]
["Bases for Exact Time","[;24,60,60,5656]",null,null,"set least base to months per cycle"]
["Convert to time units",{"$f":"ToRadix(B755,B745)","v":"{29; 12, 44, 2, 4048}"}]
["Same, Reduce Least",{"$f":"ToRadix(B755,B745,,\"R\")","v":"{29; 12, 44, 2, 506/707}"},null,null,"least fraction is nearly 5/7 second"]
["Least Fraction",{"$f":"Least(B757,FALSE)","v":"506/707"},"of a second"]
["Least Fraction Decimal",{"$f":"Numerator(B758)/Denominator(B758)","v":{"v":0.7157001414427157,"d":"0.715700141442716"}},null,"of a second","excellent for the present era"]
[]
["Compare m1 with 52-yerm lunar cycle mean month = 29+451/850 days (accurate in the present era):"]
["Fraction to Decimal",{"$f":"29+451/850","v":{"v":29.53058823529412,"d":"29.5305882352941"}}]
["Bases for time units","[;24,60,60,850]",null,null,"set least base to months per cycle"]
["Convert to time units",{"$f":"ToRadix(B763,B762)","v":"{29; 12, 44, 2, 700}"}]
["Same, Reduce Least",{"$f":"ToRadix(B763,B762,,\"R\")","v":"{29; 12, 44, 2, 14/17}"}]
["Least Fraction",{"$f":"Least(B765,FALSE)","v":"14/17"},"of a second"]
["Least Fraction Decimal",{"$f":"Numerator(B766)/Denominator(B766)","v":{"v":0.8235294117647058,"d":"0.823529411764706"}},null,"of a second","excellent for the present era"]
[]
["Compare m1 with 25-saros lunar cycle mean month = 29+2958/(25x223) = 29+2958/5575 days (slightly short for present era):"]
["Fraction to Decimal",{"$f":"29+2958/5575","v":{"v":29.530582959641254,"d":"29.5305829596413"}}]
["Bases for time units","[;24,60,60,5575]",null,null,"set least base to months per cycle"]
["Convert to time units",{"$f":"ToRadix(B771,B770)","v":"{29; 12, 44, 2, 2050}"}]
["Same, Reduce Least",{"$f":"ToRadix(B771,B770,,\"R\")","v":"{29; 12, 44, 2, 82/223}"}]
["Least Fraction",{"$f":"Least(B773,FALSE)","v":"82/223"},"of a second"]
["Least Fraction Decimal",{"$f":"Numerator(B774)/Denominator(B774)","v":{"v":0.36771300448430494,"d":"0.367713004484305"}},null,"of a second","good for the near future"]
[]
["p22 middle"]
["Radices","6; 57, 53, 2, 20"]
["Decimal Number",{"$f":"FromRadix($B742,B778)","v":6.96481612446959}]
["as Mixed Fraction",{"$f":"FromRadix($B742,B778,\"M\")","v":"6+5457/5656"}]
["as Improper Fraction",{"$f":"FromRadix($B742,B778,\"I\")","v":"39393/5656"}]
[]
["p22 bottom"]
["Bases (True Weekday)",{"$f":"B742","v":"( ; 60, 60, 6, 707 )"}]
["Radices","3; 11, 27, 2, 332"]
["Decimal Number",{"$f":"FromRadix($B784,B785)","v":3.190947666195191}]
["as Mixed Fraction",{"$f":"FromRadix($B784,B785,\"M\")","v":"3+135/707"}]
["as Improper Fraction",{"$f":"FromRadix($B784,B785,\"I\")","v":"2256/707"}]
[]
["p23 (7.6 to 7.8)","s1",null,"s2 = s1/30",null,"s0"]
["Bases (Lunar Mansions)","(;27, 60, 60, 6, 67)",null,{"$f":"B791","v":"(;27, 60, 60, 6, 67)"},null,{"$f":"B791","v":"(;27, 60, 60, 6, 67)"}]
["Radices",";2, 10, 58, 1, 17",null,";0, 4, 21, 5, 43",null,";24, 57, 5, 2, 16"]
["Decimal Number",{"$f":"FromRadix(B791,B792)","v":0.0808457711442786},null,{"$f":"FromRadix(D791,D792)","v":0.00269485903814262},null,{"$f":"FromRadix(F791,F792)","v":0.9241293532338308}]
["Fraction",{"$f":"FromRadix(B791,B792,\"P\")","v":"65/804"},null,{"$f":"FromRadix(D791,D792,\"P\")","v":"13/4824"},null,{"$f":"FromRadix(F791,F792,\"P\")","v":"743/804"}]
["Mean Year = m1/s1",{"$f":"B744/B793","v":365.2706451963878}]
["convert to time units","[;24,60,60,1000,1000]"]
["Radices",{"$f":"ToRadix(B796,B795)","v":"{365; 6, 29, 43, 744, 968}"}]
["Units to Apply","day;hour,minute,second,millisecond,microsecond"]
["Apply Units",{"$f":"ApplyUnits(B797,B798)","v":"365 days ; 6 hours, 29 minutes, 43 seconds, 744 milliseconds, 968 microseconds"}]
["Abbrev Units to Apply","d;h,m,s,ms,us"]
["Apply Abbrev Units",{"$f":"ApplyUnits(B797,B800,FALSE)","v":"365 d ; 6 h, 29 m, 43 s, 744 ms, 968 us"},null,null,"this is appreciably longer than the sidereal year!"]
[]
["p23","(7.9)",null,"(7.12)","(7.13)","(7.14)","(7.15)",null,"(7.16)"]
["Bases",{"$f":"B791","v":"(;27, 60, 60, 6, 67)"},null,"(;28, 126)",{"$f":"D804","v":"(;28, 126)"},{"$f":"E804","v":"(;28, 126)"},{"$f":"F804","v":"(;28, 126)"},null,{"$f":"G804","v":"(;28, 126)"}]
["Radices",";25, 9, 10, 4, 32",null,";2, 1",";1, 0",";3, 97",";13, 103",null,";21, 90"]
["Decimal Number",{"$f":"FromRadix(B804,B805)","v":0.931592039800995},null,{"$f":"FromRadix(D804,D805)","v":0.07171201814058957},{"$f":"FromRadix(E804,E805)","v":0.03571428571428571},{"$f":"FromRadix(F804,F805)","v":0.13463718820861678},{"$f":"FromRadix(G804,G805)","v":0.4934807256235828},null,{"$f":"FromRadix(I804,I805)","v":0.7755102040816326}]
["Fraction",{"$f":"FromRadix(B804,B805,\"P\")","v":"749/804"},null,{"$f":"FromRadix(D804,D805,\"P\")","v":"253/3528"},{"$f":"FromRadix(E804,E805,\"P\")","v":"1/28"},{"$f":"FromRadix(F804,F805,\"P\")","v":"475/3528"},{"$f":"FromRadix(G804,G805,\"P\")","v":"1741/3528"},null,{"$f":"FromRadix(I804,I805,\"P\")","v":"38/49"}]
[]
[null,"p24 (7.19) 1/4",null,"p25 (7.24)"]
["Bases","(;27, 60)",null,"(;28, 126, 30)",{"$f":"D810","v":"(;28, 126, 30)"},{"$f":"E810","v":"(;28, 126, 30)"}]
["Radices",";6, 45",null,";1, 0, 1",";1, 0, 0",";0,0,1"]
["Decimal Number",{"$f":"FromRadix(B810,B811)","v":0.25},null,{"$f":"FromRadix(D810,D811)","v":0.03572373393801965},{"$f":"FromRadix(E810,E811)","v":0.03571428571428571},{"$f":"FromRadix(F810,F811)","v":{"v":9.44822373393802e-06,"d":"0.00000944822373393802"}}]
["Fraction",{"$f":"FromRadix(B810,B811,\"P\")","v":"1/4"},null,{"$f":"FromRadix(D810,D811,\"P\")","v":"3781/105840"},{"$f":"FromRadix(E810,E811,\"P\")","v":"1/28"},{"$f":"FromRadix(F810,F811,\"P\")","v":"1/105840"}]
[]
["p31"]
["Mean solar longitude in Signs, Degrees, Minutes"]
["Bases","(12,30,60)"]
[]
["p39","(A.1)",null,"(A.2)",null,"(A.3)"]
["Bases","(;60, 60, 6, 13, 707)",null,"(;27, 60, 60, 6, 13, 67)",null,"(;60, 60, 6, 13, 707)"]
["Radices","29; 31, 50, 0, 8, 584",null,";2, 10, 58, 1, 3, 20",null,"4; 14, 6, 2, 2, 666"]
["Decimal Number",{"$f":"FromRadix(B820,B821)","v":29.530586987270155},null,{"$f":"FromRadix(D820,D821)","v":0.08084577114427861},null,{"$f":"FromRadix(F820,F821)","v":4.235103069830792}]
["Improper Fraction",{"$f":"FromRadix(B820,B821,\"I\")","v":"167025/5656"},null,{"$f":"FromRadix(D820,D821,\"I\")","v":"65/804"},null,{"$f":"FromRadix(F820,F821,\"I\")","v":"32337553/7635600"}]
["Mixed Fraction",{"$f":"FromRadix(B820,B821,\"M\")","v":"29+3001/5656"},{"$nulls":3},{"$f":"FromRadix(F820,F821,\"M\")","v":"4+1795153/7635600"}]
[]
["Compare A.1 with 25-saros lunar cycle mean month = 29+2958/(25x223) = 29+2958/5575 days"]
[]
["p39","(A.4)",null,"(A.5)","(A.6)","(A.7)","(A.8)"]
["Bases","(;27, 60, 60, 6, 13, 67)",null,"(;28, 126)","(;60, 60, 6, 13, 707)","(;27, 60, 60, 6, 13, 67)","(;28, 126)"]
["Radices",";-1, -29, -17, -5, -6, -1",null,";14, 99","2; 9, 24, 2, 5, 417",";0, 1, 22, 2, 4, 18",";0, 72"]
["Decimal Number",{"$f":"FromRadix(B829,B830)","v":-0.055122535470794175},null,{"$f":"FromRadix(D829,D830)","v":0.5280612244897959},{"$f":"FromRadix(E829,E830)","v":2.156779166011839},{"$f":"FromRadix(F829,F830)","v":0.000847613782937166},{"$f":"FromRadix(G829,G830)","v":0.02040816326530612}]
["Fraction",{"$f":"FromRadix(B829,B830,\"I\")","v":"–5983/108540"},null,{"$f":"FromRadix(D829,D830,\"I\")","v":"207/392"},{"$f":"FromRadix(E829,E830,\"I\")","v":"16468303/7635600"},{"$f":"FromRadix(F829,F830,\"I\")","v":"23/27135"},{"$f":"FromRadix(G829,G830,\"I\")","v":"1/49"}]
["Mixed Fraction",{"$nulls":3},{"$f":"FromRadix(E829,E830,\"M\")","v":"2+1197103/7635600"}]
[]
["p42","(A.11)",null,"(A.12)",null,"(A.13)"]
["Bases","(;60, 60, 6, 67, 707)",null,"(;27, 60, 60, 6, 67)",null,"(;28, 126)"]
["Radices","1; 55, 13, 3, 31, 394",null,";26, 39, 51, 0, 18",null,";24, 22"]
["Decimal Number",{"$f":"FromRadix(B836,B837)","v":1.9204384724186705},null,{"$f":"FromRadix(D836,D837)","v":0.9875621890547263},null,{"$f":"FromRadix(F836,F837)","v":0.8633786848072562}]
["Fraction",{"$f":"FromRadix(B836,B837,\"I\")","v":"5431/2828"},null,{"$f":"FromRadix(D836,D837,\"I\")","v":"397/402"},null,{"$f":"FromRadix(F836,F837,\"I\")","v":"1523/1764"}]
["Mixed Fraction",{"$f":"FromRadix(B836,B837,\"M\")","v":"1+2603/2828"}]
[]
["p42 Remark 23","alternative m0",null,"alternative m1",null,"standard m1"]
["Bases","(;60, 60, 6, 707)",null,"(;60, 60, 6, 67, 707)",null,"(;60, 60, 6, 707)"]
["Radices","1; 55, 13, 3, 333",null,"1; 31, 50, 0, 45, 345",null,"1; 31, 50, 0, 480"]
["Decimal Number",{"$f":"FromRadix(B843,B844)","v":1.9204384724186703},null,{"$f":"FromRadix(D843,D844)","v":1.5305869872701554},null,{"$f":"FromRadix(F843,F844)","v":1.5305869872701556}]
["Fraction",{"$f":"FromRadix(B843,B844,\"I\")","v":"5431/2828"},null,{"$f":"FromRadix(D843,D844,\"I\")","v":"8657/5656"},null,{"$f":"FromRadix(F843,F844,\"I\")","v":"8657/5656"}]
["Mixed Fraction",{"$f":"FromRadix(B843,B844,\"M\")","v":"1+2603/2828"},null,{"$f":"FromRadix(D843,D844,\"M\")","v":"1+3001/5656"},null,{"$f":"FromRadix(F843,F844,\"M\")","v":"1+3001/5656"}]
["Janson adds 28 to each of the results obtained in Remark 23 (obtained by changing \"1;\" to \"29;\" above)"]
[]
["p42","(A.14)",null,"(A.15)",null,"(A.16)"]
["Bases","(;60, 60, 6, 707)",null,"(;27, 60, 60, 6, 67)",null,"(;28, 126)"]
["Radices","1; 52, 41, 2, 524",null,";26, 9, 37, 3, 45",null,";24, 19"]
["Decimal Number",{"$f":"FromRadix(B851,B852)","v":1.8781824611032532},null,{"$f":"FromRadix(D851,D852)","v":0.9689054726368159},null,{"$f":"FromRadix(F851,F852)","v":0.8625283446712018}]
["Fraction",{"$f":"FromRadix(B851,B852,\"I\")","v":"10623/5656"},null,{"$f":"FromRadix(D851,D852,\"I\")","v":"779/804"},null,{"$f":"FromRadix(F851,F852,\"I\")","v":"3043/3528"}]
["Mixed Fraction",{"$f":"FromRadix(B851,B852,\"M\")","v":"1+4967/5656"}]
[]
["p45","(A.19)",null,"(A.20)",null,"(A.21)"]
["Bases","(;60, 60, 707)",null,"(;27, 60, 60, 67)",null,"(;28, 126)"]
["Radices","2; 4, 24, 552",null,";0, 24, 10, 50",null,";3, 30"]
["Decimal Number",{"$f":"FromRadix(B858,B859)","v":2.0735502121640734},null,{"$f":"FromRadix(D858,D859)","v":0.014925373134328358},null,{"$f":"FromRadix(F858,F859)","v":0.11564625850340135}]
["Fraction",{"$f":"FromRadix(B858,B859,\"I\")","v":"1466/707"},null,{"$f":"FromRadix(D858,D859,\"I\")","v":"1/67"},null,{"$f":"FromRadix(F858,F859,\"I\")","v":"17/147"}]
["Mixed Fraction",{"$f":"FromRadix(B858,B859,\"M\")","v":"2+52/707"}]
[]
["p46 Kalacakra Tantra","(A.25)",null,"(A.26)",null,"(A.27)"]
["Bases","(;60, 60)",null,"(;27, 60, 60, 6, 13)",null,"(;28, 126)"]
["Radices","29; 31, 50",null,";2, 10, 58, 2, 10",null,";2, 1"]
["Decimal Number",{"$f":"FromRadix(B865,B866)","v":29.530555555555555},null,{"$f":"FromRadix(D865,D866)","v":0.08084836973725862},null,{"$f":"FromRadix(F865,F866)","v":0.07171201814058957}]
["Fraction",{"$f":"FromRadix(B865,B866,\"I\")","v":"10631/360"},null,{"$f":"FromRadix(D865,D866,\"I\")","v":"1277/15795"},null,{"$f":"FromRadix(F865,F866,\"I\")","v":"253/3528"}]
["Mixed Fraction",{"$f":"FromRadix(B865,B866,\"M\")","v":"29+191/360"}]
["Bases for Time","[;24,60,60]"]
["Mixed Radices",{"$f":"ToRadix(B870,B867)","v":"{29; 12, 44, 0}"}]
["Units to Apply","day;hour,minute,second"]
["Apply Units",{"$f":"ApplyUnits(B871,B872)","v":"29 days ; 12 hours, 44 minutes, 0 seconds"}]
["Thus (A.25) yields a rather short mean lunar month having zero seconds. "]
[]
["p46-47 top","(A.29)","(A.30)","(A.31)","(A.32)","(A.33)"]
[null,"m0","s0","a0","m2","s2"]
["Bases","(;60, 60)","(;27, 60)","(;28, 126)","(;60)","(;27, 60, 60)"]
["Radices","2; 30, 0","; 26, 58",";5, 112","0; 59",";0, 4, 20"]
["Decimal Number",{"$f":"FromRadix(B878,B879)","v":2.5},{"$f":"FromRadix(C878,C879)","v":0.9987654320987653},{"$f":"FromRadix(D878,D879)","v":0.21031746031746032},{"$f":"FromRadix(E878,E879)","v":0.9833333333333333},{"$f":"FromRadix(F878,F879)","v":0.002674897119341564}]
["Fraction",{"$f":"FromRadix(B878,B879,\"I\")","v":"5/2"},{"$f":"FromRadix(C878,C879,\"I\")","v":"809/810"},"53/252","59/60","13/4860"]
["Mixed Fraction",{"$f":"FromRadix(B878,B879,\"M\")","v":"2+1/2"}]
[]
["p47 middle","(A.34)","(A.35)","(A.36)"]
[null,"m'1","s'1","a'1"]
["Bases","(;60, 60)","(;27, 60)","(;28, 126)"]
["Radices","29; 32, 0","; 2, 11",";2,0"]
["Decimal Number",{"$f":"FromRadix(B886,B887)","v":29.533333333333335},{"$f":"FromRadix(C886,C887)","v":0.0808641975308642},{"$f":"FromRadix(D886,D887)","v":0.07142857142857142}]
["Fraction",{"$f":"FromRadix(B886,B887,\"I\")","v":"443/15"},{"$f":"FromRadix(C886,C887,\"I\")","v":"131/1620"},"1/14"]
["Mixed Fraction",{"$f":"FromRadix(B886,B887,\"M\")","v":"29+8/15"}]
[]
["p49 Sherab Ling","(A.40)"]
["Bases","(;27, 60, 60, 6, 707, 6811)"]
["Radices",";2, 10, 58, 2, 564, 5546"]
["Decimal Number",{"$f":"FromRadix(B893,B894)","v":0.08084842059095268}]
["Fraction",{"$f":"FromRadix(B893,B894,\"I\")","v":"3114525/38523016"}]
[]
["p50 Table 6 epoch variants (decimal numbers formatted to 6 decimals to match Janson manuscript)"]
["(;60, 60, 6, 707)","m0","Phugpa",null,"Tsurphu","Mongolia","Bhutan"]
["Radices",null,"2; 22, 34, 2, 518",null,"2; 25, 20, 2, 352","2; 25, 6, 3, 327","2; 24, 37, 5, 431"]
["Decimal Number",null,{"$f":"FromRadix($A899,C900)","v":{"v":2.376237623762376,"d":"2.376238"}},null,{"$f":"FromRadix($A899,E900)","v":{"v":2.422337864738855,"d":"2.422338"}},{"$f":"FromRadix($A899,F900)","v":{"v":2.4184936350777937,"d":"2.418494"}},{"$f":"FromRadix($A899,G900)","v":{"v":2.4105374823196604,"d":"2.410537"}}]
["Fraction",null,{"$f":"FromRadix($A899,C900,\"I\")","v":"240/101"},null,{"$f":"FromRadix($A899,E900,\"I\")","v":"18496003/7635600"},{"$f":"FromRadix($A899,F900,\"I\")","v":"13679/5656"},{"$f":"FromRadix($A899,G900,\"I\")","v":"6817/2828"}]
["Mixed Fraction",null,{"$f":"FromRadix($A899,C900,\"M\")","v":"2+38/101"},null,{"$f":"FromRadix($A899,E900,\"M\")","v":"2+3224803/7635600"},{"$f":"FromRadix($A899,F900,\"M\")","v":"2+2367/5656"},{"$f":"FromRadix($A899,G900,\"M\")","v":"2+1161/2828"}]
[]
["(;27, 60, 60, 6, 67)","s0"]
["Radices",null,";0, 8, 3, 3, 33",null,"; 0, 29, 34, 5, 37","; 0, 38, 17, 0, 6",";0, 28, 12, 3, 15"]
["Decimal Number",null,{"$f":"FromRadix($A905,C906)","v":{"v":0.004975124378109453,"d":"0.004975"}},null,{"$f":"FromRadix($A905,E906)","v":{"v":0.01826054910632025,"d":"0.018261"}},{"$f":"FromRadix($A905,F906)","v":{"v":0.023631840796019904,"d":"0.023632"}},{"$f":"FromRadix($A905,G906)","v":{"v":0.01741293532338308,"d":"0.017413"}}]
["Fraction",null,{"$f":"FromRadix($A905,C906,\"I\")","v":"1/201"},null,{"$f":"FromRadix($A905,E906,\"I\")","v":"991/54270"},{"$f":"FromRadix($A905,F906,\"I\")","v":"19/804"},{"$f":"FromRadix($A905,G906,\"I\")","v":"7/402"}]
[]
["(;28, 126)","a0"]
["Radices",null,";5, 98",null,";5, 112",";5, 101",";6, 22"]
["Decimal Number",null,{"$f":"FromRadix($A910,C911)","v":{"v":0.20634920634920637,"d":"0.206349"}},null,{"$f":"FromRadix($A910,E911)","v":{"v":0.21031746031746032,"d":"0.210317"}},{"$f":"FromRadix($A910,F911)","v":{"v":0.2071995464852608,"d":"0.207200"}},{"$f":"FromRadix($A910,G911)","v":{"v":0.22052154195011336,"d":"0.220522"}}]
["Fraction",null,{"$f":"FromRadix($A910,C911,\"I\")","v":"13/63"},null,{"$f":"FromRadix($A910,E911,\"I\")","v":"53/252"},{"$f":"FromRadix($A910,F911,\"I\")","v":"731/3528"},{"$f":"FromRadix($A910,G911,\"I\")","v":"389/1764"}]
[]
["p61 Phugpas definition points",null,"(C.28) p1",null,"(C.29) p0"]
["Bases",null,"(;27, 60)"]
["Radices",null," ;23, 6"]
["Decimal Number",null,{"$f":"FromRadix(C916,C917)","v":0.8555555555555555},null,{"$f":"C918-1/12","v":0.7722222222222221}]
["Fraction",null,{"$f":"FromRadix(C916,C917,\"I\")","v":"77/90"},"-1/12",{"$f":"ReduceFraction(Numerator(C919)*Denominator(D919)+Denominator(C919)*Numerator(D919),Denominator(C919)*Denominator(D919))","v":"139/180"}]
["Degrees (above x 360°)",null,{"$f":"360*C918","v":308},"°",{"$f":"360*E918","v":278},"°"]
[]
[]
["p62 Tsurphu definition points",null,"(C.40) p1",null,"(C.41) p0","(C.43) s0"]
["Bases",null,"(;27, 60, 60)",null,null,"(;27, 60, 60, 6, 13, 67)"]
["Radices",null,"; 23, 1, 30",null,null,";0, 1, 22, 2, 4, 18"]
["Decimal Number",null,{"$f":"FromRadix(C924,C925)","v":0.8527777777777779},null,{"$f":"Numerator(E927)/Denominator(E927)","v":0.7694444444444445},{"$f":"FromRadix(F924,F925)","v":0.000847613782937166}]
["Fraction",null,{"$f":"FromRadix(C924,C925,\"I\")","v":"307/360"},null,"277/360",{"$f":"FromRadix(F924,F925,\"I\")","v":"23/27135"}]
["Degrees (above x 360°)",null,{"$f":"360*C926","v":307},"°",{"$f":"360*E926","v":277},"°"]
[]
["p63 Mongolian definition points",null,"(C.47) p1",null,"(C.48) p0"]
["Bases",null,"(;27, 60)"]
["Radices",null,";23, 9"]
["Decimal Number",null,{"$f":"FromRadix(C931,C932)","v":0.8574074074074074},null,{"$f":"Numerator(E934)/Denominator(E934)","v":0.774074074074074}]
["Fraction",null,{"$f":"FromRadix(C931,C932,\"I\")","v":"463/540"},null,"209/270"]
["Degrees (above x 360°)",null,{"$f":"360*C933","v":308.6666666666667},"°",{"$f":"360*E933","v":278.66666666666663},"°"]
["Degrees with fraction",null,{"$f":"ReduceFraction(360*Numerator(C934),Denominator(C934))","v":"308+2/3"},"°",{"$f":"ReduceFraction(360*Numerator(E934),Denominator(E934))","v":"278+2/3"},"°"]
[]
["p65 Bhutanese definition points",null,"(C.63) p1",null,"(C.64) p0","(C.65) s0"]
["Bases",null,"(;27, 60, 60)",null,"(;27, 60, 60)","(;27, 60, 60, 67)"]
["Radices",null,";23, 10, 30",null,";20, 55, 30",";0, 24, 10, 50"]
["Decimal Number",null,{"$f":"FromRadix(C939,C940)","v":0.8583333333333334},null,{"$f":"FromRadix(E939,E940)","v":0.775},{"$f":"FromRadix(F939,F940)","v":0.014925373134328358}]
["Fraction",null,{"$f":"FromRadix(C939,C940,\"I\")","v":"103/120"},null,{"$f":"FromRadix(E939,E940,\"I\")","v":"31/40"},{"$f":"FromRadix(F939,F940,\"I\")","v":"1/67"}]
["Degrees (above x 360°)",null,{"$f":"360*C941","v":309},"°",{"$f":"360*E941","v":279},"°"]
[]
["p66 original Lhawang Lodro",null,"(C.67) p1",null,"(C.68) p0"]
["Bases",null,"(;27, 60)",null,"(;27, 60)"]
["Radices",null,";23, 15",null,";21, 0"]
["Decimal Number",null,{"$f":"FromRadix(C946,C947)","v":0.8611111111111112},null,{"$f":"FromRadix(E946,E947)","v":0.7777777777777778}]
["Fraction",null,{"$f":"FromRadix(C946,C947,\"I\")","v":"31/36"},null,{"$f":"FromRadix(E946,E947,\"I\")","v":"7/9"}]
["Degrees (above x 360°)",null,{"$f":"360*C948","v":310},"°",{"$f":"360*E948","v":280},"°"]
[]
["p66 Karana definition points",null,"(C.70) p1",null,"(C.71) p0"]
["Bases",null,"(;27, 60)"]
["Radices",null,";22, 30"]
["Decimal Number",null,{"$f":"FromRadix(C953,C954)","v":0.8333333333333333},null,{"$f":"Numerator(E956)/Denominator(E956)","v":0.75}]
["Fraction",null,{"$f":"FromRadix(C953,C954,\"I\")","v":"5/6"},null,"9/12"]
["Degrees (above x 360°)",null,{"$f":"360*C955","v":300},"°",{"$f":"360*Numerator(E956)/Denominator(E956)","v":270},"°"]
[]
["p66 footnote 74",{"$nulls":3},"p5"]
["Bases",null,"(;27, 60, 60, 67)",null,"(;27, 60)"]
["Radices",null,";5, 14, 19, 47",null,";5, 15"]
["Decimal Number",null,{"$f":"FromRadix(C960,C961)","v":0.19402985074626863},null,{"$f":"FromRadix(E960,E961)","v":0.19444444444444445}]
["Fraction",null,{"$f":"FromRadix(C960,C961,\"I\")","v":"13/67"},null,{"$f":"FromRadix(E960,E961,\"I\")","v":"7/36"}]
["Degrees (above x 360°)",null,{"$f":"360*C962","v":{"v":69.8507462686567,"d":"69.851"}},"°",{"$f":"360*E962","v":70},"°"]
[]
["p70","(D.8) s'1",null,"(D.9) s'0"]
["Bases","(;27, 60, 60, 6, 149209)",null,"(;27, 60, 60, 6, 149209)"]
["Radices",";0, 4, 26, 0, 93156",null,";25, 9, 20, 0, 97440"]
["Decimal Number",{"$f":"FromRadix(B967,B968)","v":2.7376960430596604E-3},null,{"$f":"FromRadix(D967,D968)","v":0.9316883625578142}]
["Fraction",{"$f":"FromRadix(B967,B968,\"I\")","v":"18382/6714405"},null,{"$f":"FromRadix(D967,D968,\"I\")","v":"6255733/6714405"}]
[{"$nulls":3},"1 minus above ="]
[{"$nulls":3},{"$f":"ReduceFraction(1*Denominator(D970)-1*Numerator(D970),Denominator(D970))","v":"458672/6714405"}]
[]
[]
["The following information and arithmetic examples are from Edward Henning's book, \"Kālacakra and the Tibetan Calendar\", 2007"]
["https://b-ok.cc/dl/11341578/382dfe"]
[]
["Tibetan solar days:",1,"week"]
["each week contains",7,"days"]
["each day contains",60,"nadi","each is about 24 minutes"]
["each nadi contains",60,"pala","each is about 24 seconds"]
["each pala contains",6,"breaths","each is about 4 seconds"]
["each breath contains",707,"subunits","707 x 8 = mean month denominator 5656 (months per lunar cycle)"]
["Bases for Solar Days","[ 7 ; 60, 60, 6, 707 ]"]
["Units to Apply","week, # weekday; nadi/nadi, pala/pala, breath, subunit"]
[]
[null,1,"sidereal year"]
[null,12,"zodiacal signs (each 30 degrees)"]
["each zodiacal sign",30,"zodiacal days (each about one degree)"]
[null,null,"(actual time taken for the mean Sun to move through 1/30 of a zodiacal sign"]
["Bases for Zodiacal Signs","[ 12, 30 ]"]
["Units to Apply","sidereal year, zodiac sign, zodiacal day"]
[]
["Each of the 12 zodiacal signs has 135 nadi x 12 = 1620 nadi"]
["Bases for Zodiacal Signs","[ 12, 135 ]"]
["Units to Apply","zodiac sign, nadi/nadi"]
[]
["Tibetan lunar mansions:",null,null,"mansions are numbered 0 to 26"]
[null,27,"mansions","each is the approximate sidereal distance that Moon travels in one day"]
[null,60,"nadi"]
["each nadi contains",60,"pala"]
["each pala contains",6,"breaths"]
[null,null,"one or more other fractional units in a longitude, varrying according to the quantity being described"]
[]
["Lunar Bases","[ ; 27, 60, 60, 6 ]",null,"and others?"]
["Units to Apply","mansion, nadi/nadi, pala/pala, breath"]
[]
["p20"]
["In Tibetan Calendar arithmetic, 65 solar years is exactly equal to 804 lunar months"]
["Thus the cycle of different intercalary months repeats itself:"]
["In every 65 years there are 780 normal months and 24 intercalary, which totals 804 months."]
[]
["p18, result middle of p19"]
["Starting number",840]
["Bases",{"$f":"SolarBases","v":"[ 7 ; 60, 60, 6, 707 ]"}]
["Radices","1;31,50,0,480"]
["Convert to Number",{"$f":"FromRadix(B1015,B1016)","v":1.5305869872701556}]
["Multiply",{"$f":"B1014*B1017","v":1285.6930693069307}]
["Radices to Add","6;57,53,2,20"]
["Convert to Number",{"$f":"FromRadix(B1015,B1019)","v":6.96481612446959}]
["Sum",{"$f":"B1018+B1020","v":1292.6578854314002}]
["Mixed Radix Values",{"$f":"ToRadix(B1015,B1021)","v":"{184, 4; 39, 28, 2, 230}"}]
["Apply Units",{"$f":"ApplyUnits(B1022,SolarDayUnits)","v":"184 weeks, 4 th weekday ; 39 nadi, 28 pala, 2 breaths, 230 subunits"}]
["Drop Days from Bases","(;60, 60 , 6, 707)",null,"ignore extra weeks"]
["Back to Number",{"$f":"FromRadix(B1024,B1022)","v":4.657885431400283}]
["Back to Radices",{"$f":"ToRadix(B1024,B1025)","v":"{4; 39, 28, 2, 230}"},null,"final value for the mean weekday"]
["Apply Units",{"$f":"ApplyUnits(B1026,SolarDayUnits)","v":"4 th weekday ; 39 nadi, 28 pala, 2 breaths, 230 subunits"}]
["Weekday",{"$f":"Nth(B1027,1)","v":4},{"$f":"TibetanWeekday(B1028)","v":"Wednesday"}]
[]
["p21"]
["Mean Synodic Month","29;31,50,0,480",null,"solar days"]
["Solar Days",{"$f":"FromRadix(SolarBases,B1031,\"I\")","v":"167025/5656"}]
["/30 = Lunar Day",{"$f":"Numerator(B1032)/30","v":5567.5}]
["Exact Fraction",{"$f":"ReduceFraction(2*B1033,2*Denominator(B1032))","v":"11135/11312"},null,"solar days"]
["Thus",{"$f":"Denominator(B1034)","v":11312},"equals",{"$f":"Numerator(B1034)","v":11135},"solar days"]
[]
["p21 Anomaly"]
["Starting Number",840]
["Bases","(28; 126)"]
["Multiplier","2;1"]
["Convert to Number",{"$f":"FromRadix(B1039,B1040)","v":2.007936507936508}]
["Multiply",{"$f":"B1038*B1041","v":1686.6666666666667}]
["Radices to Add","13;103"]
["Convert to Number",{"$f":"FromRadix(B1039,B1043)","v":13.817460317460318}]
["Sum",{"$f":"B1042+B1044","v":1700.484126984127}]
["Back to Radices",{"$f":"ToRadix(B1039,B1045)","v":"{60, 20; 61}"}]
["Modulo 28",{"$f":"modulus(B1045,28)","v":20.484126984126988},null,"anomaly is calculated modulo 28"]
["Back to Radices",{"$f":"ToRadix(B1039,B1047)","v":"{20; 61}"}]
[]
["p22 Mean Solar Longitude"]
["Starting Number",840]
["Bases","(27;60,60,6,67)"]
["Multipler Radices","2;10,58,1,17"]
["Convert to Number",{"$f":"FromRadix(B1052,B1053)","v":2.1828358208955225}]
["Multiply",{"$f":"B1051*B1054","v":1833.5820895522388}]
["Radices to Add","25;9,10,4,32"]
["Convert to Number",{"$f":"FromRadix(B1052,B1056)","v":25.152985074626866}]
["Sum",{"$f":"B1055+B1057","v":1858.7350746268658}]
["Back to Radices",{"$f":"ToRadix(B1052,B1058)","v":"{68, 22; 44, 6, 1, 41}"}]
["Modulo 27",{"$f":"modulus(B1058,27)","v":22.735074626865753},null,"ecliptic is divided into 27 lunar mansions, only want remainder"]
["Back to Radices",{"$f":"ToRadix(B1052,B1060)","v":"{22; 44, 6, 1, 41}"}]
[]
["p22 middle"]
["Degrees in Circle",360,"°"]
["Lunar Mansions",27]
["Exact Fraction",{"$f":"ReduceFraction(B1064,B1065)","v":"13+1/3"},"degrees per lunar mansion"]
["Sexagesimal Base","[60]","can't input as (60) because Excel considers that to be -60"]
["Degrees and Minutes",{"$f":"ToRadix(B1067,B1066)","v":"{13; 20}"},"the radix point position is forced by presence of fractional value"]
["Abbreviated Units","°; ', \""]
["Apply Abbrev Units",{"$f":"ApplyUnits(B1068,B1069,FALSE)","v":"13 ° ; 20 '"}]
[]
["p22 bottom"]
["Mean Solar Longitude",{"$f":"B1061","v":"{22; 44, 6, 1, 41}"},null,"as above"]
["… as decimal number",{"$f":"B1060","v":22.735074626865753},null,"as above"]
["Multiply by °/mansion",{"$f":"B1074*B1064/B1065","v":303.13432835821004},null,"Don't know why Henning did a much more complicated calculation"]
["Zodiac Bases","(12,30;60,60,60)",null,"to obtain almost the same result (he ignored units smaller than arcminutes)."]
["Convert to Mixed Radices",{"$f":"ToRadix(B1076,B1075)","v":"{10, 3; 8, 3, 35}"}]
["Units to Apply","# sign,°; ', \",'''"]
["Apply Units",{"$f":"ApplyUnits(B1077,B1078,FALSE)","v":"10 th sign, 3 ° ; 8 ', 3 \", 35 '''"}]
["Zodiac Sign",{"$f":"ZodiacSign(B1075)","v":"Aquarius"}]
[]
["p23","Weekday:",null,null,"Sun:"]
["Constant","0;59,3,4,16",null,null,"0;4,21,5,43"]
["Solar Bases",{"$f":"SolarBases","v":"[ 7 ; 60, 60, 6, 707 ]"},null,null,{"$f":"B1052","v":"(27;60,60,6,67)"},"as above"]
["Convert to Decimal",{"$f":"FromRadix(SolarBases,B1083)","v":0.9843528995756718},null,null,{"$f":"FromRadix(SolarBases,E1083)","v":0.07273429723924774}]
["Weekday",{"$f":"B1026","v":"{4; 39, 28, 2, 230}"},null,"as above",{"$f":"B1073","v":"{22; 44, 6, 1, 41}"},"mean solar longitude as above"]
["Weekday in Decimal",{"$f":"B1025","v":4.657885431400283},null,"as above",{"$f":"B1074","v":22.735074626865753},"as above"]
["Nth Lunar Day",3,{"$f":"ReturnNumberSuffix(B1088)","v":"rd"}]
["Multiply",{"$f":"B1088*B1085","v":2.9530586987270153},null,null,{"$f":"B1088*E1085","v":0.2182028917177432},"product"]
["Sum",{"$f":"B1087+B1089","v":7.610944130127298},null,null,{"$f":"E1087+E1089","v":22.953277518583498}]
["Back to Radices",{"$f":"ToRadix(B1084,B1090)","v":"{1, 0; 36, 39, 2, 278}"},null,null,{"$f":"ToRadix(E1084,E1090)","v":"{22; 57, 11, 4, 53}"},"results not given by Henning"]
[]
["p24"]
["Bases",{"$f":"B1039","v":"(28; 126)"},null,"as above, Anomaly"]
["Anomaly",{"$f":"B1048","v":"{20; 61}"},null,"as above, Anomaly"]
["Decimal Value",{"$f":"B1047","v":20.484126984126988},null,"as above, Anomaly"]
["Divide by 14",{"$f":"B1096/14","v":1.4631519274376419}]
["Back to Radices",{"$f":"ToRadix(B1094,B1097)","v":"{1; 58}"},null,"Henning gives result as quotient of 1 and remainder 6"]
[]
["p26"]
["Solar Bases",{"$f":"SolarBases","v":"[ 7 ; 60, 60, 6, 707 ]"}]
["Radices","4;39,28,2,230"]
["Convert to Number",{"$f":"FromRadix(B1101,B1102)","v":4.657885431400283}]
["Radices","0;24,29,0,202"]
["Convert to Number",{"$f":"FromRadix(B1101,B1104)","v":0.4080687830687831}]
["Subtract",{"$f":"B1103-B1105","v":4.2498166483315}]
["Back to Radices",{"$f":"ToRadix(B1101,B1106)","v":"{4; 14, 59, 2, 28}"}]
[]
["p31"]
["Result from p26",{"$f":"B1107","v":"{4; 14, 59, 2, 28}"}]
["Decimal Number",{"$f":"B1106","v":4.2498166483315}]
["New Bases","(7;60,60,6,67,707)"]
["New Radices",{"$f":"ToRadix(B1112,B1111)","v":"{4; 14, 59, 2, 2, 462}"}]
[]
["Mean Solar Long Bases",{"$f":"B1052","v":"(27;60,60,6,67)"},null,"from p22"]
["Radices result p22",{"$f":"B1061","v":"{22; 44, 6, 1, 41}"}]
["Decimal Number",{"$f":"B1060","v":22.735074626865753},null,"from above"]
["Radices to Subtract","{6;45,0,0,0}"]
["Decimal Number",{"$f":"FromRadix(B1115,B1118)","v":6.75},"x 360/27 =",{"$f":"B1119*360/27","v":90},{"$f":"ZodiacSign(D1119)","v":"Cancer"}]
["Subtract",{"$f":"B1117-B1119","v":15.985074626865753},null,null,"Birth-Sign of Sun"]
["Back to Radices",{"$f":"ToRadix(B1115,B1120)","v":"{15; 59, 6, 1, 41}"}]
[]
["p32"]
["Bases","(27;60,60,6,67)"]
["Radices","15;59,6,1,41"]
["Decimal Number",{"$f":"FromRadix(B1124,B1125)","v":15.985074626865671}]
["Radices to Subtract","13;30,0,0,0",null,"half-circle (180°) = value x (13+1/3)"]
["Decimal Number",{"$f":"FromRadix(B1124,B1127)","v":13.5}]
["Subtract",{"$f":"B1126-B1128","v":2.4850746268656714}]
["Back to Radices",{"$f":"ToRadix(B1124,B1129)","v":"{2; 29, 6, 1, 41}"}]
[]
["p35"]
["Bases","(7;60,60,6,67)"]
["Radices","4;14,59,2,2"]
["Decimal Number",{"$f":"FromRadix(B1133,B1134)","v":4.249816196793809}]
["Radices to Add","0;6,25,0,30"]
["Decimal Number",{"$f":"FromRadix(B1133,B1136)","v":0.10696517412935325}]
["Add",{"$f":"B1135+B1137","v":4.356781370923162}]
["Back to Radices",{"$f":"ToRadix(B1133,B1138)","v":"{4; 21, 24, 2, 32}"}]
[]
["Bases","(27;60,60,6,67)"]
["Radices","22;44,6,1,41"]
["Decimal Number",{"$f":"FromRadix(B1141,B1142)","v":22.73507462686567}]
["Radices to Add","0;6,25,0,30"]
["Decimal Number",{"$f":"FromRadix(B1141,B1144)","v":0.10696517412935325}]
["Add",{"$f":"B1143+B1145","v":22.842039800995025}]
["Back to Radices",{"$f":"ToRadix(B1141,B1146)","v":"{22; 50, 31, 2, 4}"}]
[]
["p38"]
["Bases","(27;60,60,6,67)"]
["Radices","22;9,51,0,18",null,null,"Radices","1;12,32,1,29"]
["Decimal Number",{"$f":"FromRadix(B1150,B1151)","v":22.16417910447761},null,null,"Decimal Number",{"$f":"FromRadix(B1150,F1151)","v":1.208955223880597}]
["Multiply by 13+1/3",{"$f":"13+1/3","v":13.333333333333334},null,null,"Multiply by 13+1/3",{"$f":"13+1/3","v":13.333333333333334}]
["Tibetan Apogee° 1603",{"$f":"B1152*B1153","v":{"v":295.5223880597015,"d":"295.522"}},"formatted to 3 decimals",null,"Tibetan Apogee° 1888",{"$f":"F1152*F1153","v":{"v":16.119402985074625,"d":"16.119"}},"3 decimals"]
[]
["p43"]
["Bases","(27;60,60,6,67)"]
["Radices","23;35,28,2,10",null,"solar longitude"]
["Decimal Number",{"$f":"FromRadix(B1157,B1158)","v":23.591210613598673}]
["Radices to Add","9;0,0,0,0"]
["Decimal Number",{"$f":"FromRadix(B1157,B1160)","v":9}]
["Add",{"$f":"B1159+B1161","v":32.591210613598676}]
["Modulo 27",{"$f":"modulus(B1162,27)","v":5.591210613598676}]
["Back to Radices",{"$f":"ToRadix(B1157,B1163)","v":"{5; 35, 28, 2, 10}"}]
[]
["Bases","(27;60,60,6,67)"]
["Radices","5;35,28,2,10"]
["Decimal Number",{"$f":"FromRadix(B1166,B1167)","v":5.591210613598673}]
["Radices to Subtract","0;50,13,4,57"]
["Decimal Number",{"$f":"FromRadix(B1166,B1169)","v":0.8371690160309564}]
["Subtract",{"$f":"B1168-B1170","v":4.754041597567717}]
["Back to Radices",{"$f":"ToRadix(B1166,B1171)","v":"{4; 45, 14, 3, 20}"},null,"lunar longitude"]
[]
["p44"]
["Bases","(27;60,60,6,67)"]
["Radices",{"$f":"B1158","v":"23;35,28,2,10"}]
["Decimal Number",{"$f":"FromRadix(B1175,B1176)","v":23.591210613598673}]
["Radices to Add",{"$f":"B1172","v":"{4; 45, 14, 3, 20}"}]
["Decimal Number",{"$f":"FromRadix(B1175,B1178)","v":4.754041597567717}]
["Add",{"$f":"B1177+B1179","v":28.34525221116639}]
["Modulo 27",{"$f":"modulus(B1180,27)","v":1.3452522111663896}]
["Back to Radices",{"$f":"ToRadix(B1175,B1181)","v":"{1; 20, 42, 5, 30}"}]
[]
["p45"]
["Bases","(27;60,60,6,67)"]
["Whole Day","1;0,0,0,0"]
["Decimal Number",{"$f":"FromRadix(B1185,B1186)","v":1}]
["Radices to Subtract","0;46,2,1,53"]
["Decimal Number",{"$f":"FromRadix(B1185,B1188)","v":0.7673051409618574}]
["Subtract",{"$f":"B1187-B1189","v":0.23269485903814258}]
["Back to Radices",{"$f":"ToRadix(B1185,B1190)","v":"{0; 13, 57, 4, 14}"}]
[]
["From this point onward Henning used an abbreviated format for bases, or omitted them entirely,"]
["making it difficult to reproduce his calculations. Perhaps a more careful read of the book would help,"]
["but it has a total of 385 enumerated pages!"]

{"sheet":"Mixed","index":2,"n_rows":268,"n_cols":6,"rows_kept":268,"merged":[[37,1,37,2],[39,1,39,2],[41,1,41,2],[44,1,44,2],[46,1,46,2],[195,1,195,2],[196,1,196,2],[199,1,199,2],[222,1,222,2],[223,1,223,2],[228,1,228,2],[235,1,235,2],[239,1,239,2]]}
["Examples and Validation of Mixed Radix Conversions"]
["http://individual.utoronto.ca/intervalmath/mixed/radix.html"]
["Note: The abbreviation \"CCUE\" refers to the book \"Calendrical Calculations: The Ultimate Edition\" (4th, 2018) by Reingold and Dershowitz,"]
["which introduced Mixed Radix conversions on page 27."]
[]
["Cells marked with a","yellow","background and bold test contain numbers that the user can change."]
[]
["Tactical units in a Roman army"]
["Bases","[10,6,10,8]",null,null,"legion","unlimited"]
["Given Number",16894,null,null,"cohort",10]
["Mixed Radix",{"$f":"ToRadix(B9,B10)","v":"{3, 5, 1, 1, 6}"},null,null,"century",6]
["Back to Number",{"$f":"FromRadix(B9,B11)","v":16894},null,null,"contubernia",10]
["Show Bases",{"$f":"ToRadix(B9,B10,TRUE)","v":"{3, 5/10, 1/6, 1/10, 6/8}"},null,null,"soldier",8]
["Back to Number",{"$f":"FromRadix(B9,B13)","v":16894}]
["Units to Apply","legion, cohort, century/centuries, contubernia/contuberniae, soldier"]
["Apply Units",{"$f":"ApplyUnits(B11,B15)","v":"3 legions, 5 cohorts, 1 century, 1 contubernia, 6 soldiers"}]
[]
[]
["Roman Coins: Augustan Values (27 BC to 301 AD)"]
["Contributed by Stephan Lenk, of Germany, March 2022"]
[]
["Name of Coin","Metal","Sestertii","Radix Base"]
["Aureus","gold",100]
["Quinarius Aureus","gold",50,null,"no ancient authority, 12+1/2 denarii, half-aureus, omit"]
["Denarius","silver",4,25,"1/25 of Aureus"]
["Quinarius Argenteus","silver",2,2,"shorten to Quinarius since we must omit the gold version"]
["Sestertius","silver",1,"2;","base denomination, radix point to right"]
["Dupondius","brass","1/2",2]
["As or Assarius","bronze or copper","1/4",2,"use the longer name, even though rarely used by Romans"]
["Semis","copper","1/8",2]
["Quadrans","bronze","1/16",2]
[]
["Names to Apply/Plurals","aureus/aurei, denarius/dēnāriī, quinarius/quinarii, sestertius/sestertii; dupondius/dupondii, assarius/assarii, semis/semises, quadrans/quadrantes"]
[null,"any coin that has a radix base of 2 will never be seen in more than 1 because 2 carries to the next higher denomination"]
[null,"nevertheless the plural form of the coin name is applied to zeros"]
[]
["Bases","[25, 2, 2; 2, 2, 2, 2]"]
["Total Value",{"$f":"559+1/16+1/8+1/4+1/2","v":559.9375},null,"arithmetic expression used to intentionally get at least one for each coin"]
["Mixed Radices",{"$f":"ToRadix(B37,B38)","v":"{5, 14, 1, 1; 1, 1, 1, 1}"}]
["Total Value from Radices",{"$f":"FromRadix(B37,B39)","v":559.9375}]
["Apply Names",{"$f":"ApplyUnits(B39,B33)","v":"5 aurei, 14 dēnāriī, 1 quinarius, 1 sestertius ; 1 dupondius, 1 assarius, 1 semis, 1 quadrans"}]
["Total Value from Names",{"$f":"FromRadix(B37,B41)","v":559.9375}]
["Add 1 quadrans",{"$f":"B42+1/16","v":560},null,"adding just 1 quadrans = 1/16 leaves only 5 aurei and 15 "]
["Mixed Radices Again",{"$f":"ToRadix(B37,B43)","v":"{5, 15, 0, 0; 0, 0, 0, 0}"}]
["Total Value from Radices",{"$f":"FromRadix(B37,B44)","v":560}]
["Apply Names Again",{"$f":"ApplyUnits(B44,B33)","v":"5 aurei, 15 dēnāriī, 0 quinarii, 0 sestertii ; 0 dupondii, 0 assarii, 0 semises, 0 quadrantes"}]
["Total Value from Names",{"$f":"FromRadix(B37,B46)","v":560}]
["Names Without Zeros",{"$f":"ApplyUnits(B44,B33,,TRUE)","v":"5 aurei, 15 dēnāriī"},null,"used the optional SkipZeros flag, but can't go back to the total value:"]
["Total Value from Names",{"$f":"FromRadix(B37,B48)","v":25},null,"can't recover total after skipping radix values of zero (positions wrong)"]
[]
[]
["Factorial number system: each base is the next sequential integer value"]
["Bases","[ 9, 8, 7, 6, 5, 4, 3, 2, 1 ]"]
["Given Number",437]
["Mixed Radix",{"$f":"ToRadix(B53,B54)","v":"{3, 3, 0, 2, 1, 0}"}]
["Back to Number",{"$f":"FromRadix($B53,B55)","v":437}]
["Show Bases",{"$f":"ToRadix(B53,B54,TRUE)","v":"{3/6, 3/5, 0/4, 2/3, 1/2, 0/1}"}]
["Back to Number",{"$f":"FromRadix($B53,B57)","v":437}]
["Units to Apply","x 9!, x 8!, x 7!, x 6!, x 5!, x 4!, x 3!, x 2!, x 1!"]
["Apply Units",{"$f":"ApplyUnits(B55,B59,FALSE)","v":"3 x 6!, 3 x 5!, 0 x 4!, 2 x 3!, 1 x 2!, 0 x 1!"}]
[]
[]
["Fibonorial number system: each base is the next sequential Fibonacci number:"]
["Bases","[ 144, 89, 55, 34, 21, 13, 8, 5, 3, 2 ]"]
["Given Number",1784006]
["Mixed Radix",{"$f":"ToRadix(B$64,B$65)","v":"{27, 4, 10, 2, 4, 1, 0}"}]
["Back to Number",{"$f":"FromRadix(B$64,B66)","v":1784006}]
["Show Bases",{"$f":"ToRadix(B$64,B$65,TRUE)","v":"{27/34, 4/21, 10/13, 2/8, 4/5, 1/3, 0/2}"}]
["Back to Number",{"$f":"FromRadix(B$64,B68)","v":1784006}]
[]
[]
["Primorial number system: each base is the next sequential prime number"]
["Bases","[ 29,23,19,17,13,11,7,5,3,2 ]"]
["Given Number",3218147]
["Mixed Radix",{"$f":"ToRadix(B$73,B$74)","v":"{6, 5, 2, 1, 3, 2, 2, 1}"}]
["Back to Number",{"$f":"FromRadix(B$73,B75)","v":3218147}]
["Show Bases",{"$f":"ToRadix(B$73,B$74,TRUE)","v":"{6/19, 5/17, 2/13, 1/11, 3/7, 2/5, 2/3, 1/2}"}]
["Back to Number",{"$f":"FromRadix(B$73,B77)","v":3218147}]
[]
[]
["Bases","[ 15,10,5 ]"]
["Given Number",534]
["Mixed Radix",{"$f":"ToRadix(B$81,B$82)","v":"{10, 6, 4}"}]
["Back to Number",{"$f":"FromRadix(B$81,B83)","v":534}]
["Show bases",{"$f":"ToRadix(B$81,B$82,TRUE)","v":"{10/15, 6/10, 4/5}"}]
["Back to Number",{"$f":"FromRadix(B$81,B85)","v":534}]
[]
[]
["Bases","[ 12,9,6 ]"]
["Given Number",571]
["Mixed Radix",{"$f":"ToRadix(B$89,B$90)","v":"{10, 5, 1}"}]
["Back to Number",{"$f":"FromRadix(B$89,B91)","v":571}]
["Show bases",{"$f":"ToRadix(B$89,B$90,TRUE)","v":"{10/12, 5/9, 1/6}"}]
["Back to Number",{"$f":"FromRadix(B$89,B93)","v":571}]
[]
[]
["UK currency before decimalisation"]
["12 pence per shilling, 20 shillings per pound"]
[null,"pound",4]
[20,"shilling",13]
[12,"penny",7]
[]
["bases","[ 20, 12 ]"]
["mixed radix",{"$f":"\"{ \"&C99&\", \"&C100&\", \"&C101& \" }\"","v":"{ 4, 13, 7 }"},"assembled from list above"]
["Units to Apply","pound, shilling, penny/pence"]
["Apply Units",{"$f":"ApplyUnits(B104,B105)","v":"4 pounds, 13 shillings, 7 pence"}]
["money total",{"$f":"FromRadix(B103,B104)","v":1123},"pence"]
["pounds (total/(20x12))",{"$f":"B107/(20*12)","v":4.679166666666666},"pounds"]
[]
["Bases","[ 4, 3, 2 ]"]
["Number",40]
["Mixed Radix",{"$f":"ToRadix(B110,B111)","v":"{1, 2, 2, 0}"}]
["Back to number",{"$f":"FromRadix(B$110,B112)","v":40}]
["Show Bases",{"$f":"ToRadix(B110,B111,TRUE)","v":"{1, 2/4, 2/3, 0/2}"}]
["Back to number",{"$f":"FromRadix(B$110,B114)","v":40}]
[]
[]
["Bases","[16,16,16,16]"]
["Number",732847]
["Mixed Radix",{"$f":"ToRadix(B118,B119)","v":"{11, 2, 14, 10, 15}"},null,"numeric radix values if more than a single base specified"]
["Back to Number",{"$f":"FromRadix(B118,B120)","v":732847}]
["Show Bases",{"$f":"ToRadix(B118,B119,TRUE)","v":"{11, 2/16, 14/16, 10/16, 15/16}"}]
["Back to Number",{"$f":"FromRadix(B118,B122)","v":732847},null,"show bases not ignored but not useful"]
[]
[]
["{8,4;5,10}"]
[0.5]
[{"$f":"ToRadix(A126,A127)","v":"{0; 2, 5}"},"may have leading zeros"]
[{"$f":"FromRadix(A126,A128)","v":0.5},"back to number"]
[{"$f":"TrimZeros(A128)","v":"{0; 2, 5}"},"Trim zeros (removes 0, but not 0; or 0/base;)"]
[{"$f":"FromRadix(A126,A130)","v":0.5},"back to number"]
[{"$f":"ToRadix(A126,A127,TRUE)","v":"{0/4; 2/5, 5/10}"},"show bases"]
[{"$f":"FromRadix(A126,A132)","v":0.5},"back to number"]
[{"$f":"TrimZeros(A132)","v":"{0/4; 2/5, 5/10}"},"Trim zeros (removes 0, but not 0; or 0/base;)"]
[{"$f":"FromRadix(A126,A134)","v":0.5},"without leading zeros, back to number"]
[]
[]
["Canadian Cash","denom","base"]
["Currency bills",{"v":100,"d":"$100"},"leftmost radix will count hundreds of dollars"]
[null,{"v":50,"d":"$50"},"omit, uncommonly used, not divisible by $20"]
[null,{"v":20,"d":"$20"},"5 make $100, doesn't divide into $50 but very commonly used"]
[null,{"v":10,"d":"$10"},"2 make $20"]
[null,{"v":5,"d":"$5"},"2 make $10, but not divisible by $2, so omit"]
["Toonie bimetalic coin",{"v":2,"d":"$2"},"5 make $10, commonly used"]
["Loonie brass plate coin",{"v":1,"d":"$1"},"2 make $2, commonly used"]
["half-dollar",{"v":0.5,"d":"$0.50"},"2 make $1, but omit because rarely used"]
["quarter",{"v":0.25,"d":"$0.25"},"4 make $1"]
["dime",{"v":0.1,"d":"$0.10"},"omit, doesn't divide into 25 cents"]
["nickel",{"v":0.05,"d":"$0.05"},"5 make $0.25"]
["No pennies/cents --- rule in Canada for cash transactions is to round amount to nearest 5 cents."]
[]
["Bases","[ 5, 2, 5, 2 ; 4, 5 ]"]
["Amount",{"v":673.88,"d":"$673.88"}]
["Round to nickel",{"$f":"roundtoleast(B152,B153)","v":{"v":673.9,"d":"$673.90"}}]
["Money",{"$f":"ToRadix(B152,B154)","v":"{6, 3, 1, 1, 1; 3, 3}"}]
["Money with Bases",{"$f":"ToRadix(B152,B154,TRUE)","v":"{6, 3/5, 1/2, 1/5, 1/2; 3/4, 3/5}"}]
["Back to round number",{"$f":"FromRadix(B152,B156)","v":{"v":673.9,"d":"$673.90"}}]
["Units to Apply","x $100/x $100, x $20/x $20, x $10/x $10, toonie, loonie; quarter, nickel"]
["Apply Units",{"$f":"ApplyUnits(B155,B158)","v":"6 x $100, 3 x $20, 1 x $10, 1 toonie, 1 loonie ; 3 quarters, 3 nickels"}]
[]
[]
["Test Fraction with Bases for Trailing Zeros, and demo of calling the TrimZeros( ) function"]
["Bases","[ 2 ; 8, 6, 4, 2 ]"]
["Number",0.5]
["Mixed Radix",{"$f":"ToRadix(B$163,B$164)","v":"{0; 4, 0, 0, 0}"},null,"has leading and trailing zeros"]
["Trim Zeros",{"$f":"TrimZeros(B165)","v":"{0; 4}"},null,"leaves at least one zero to the left of the radix point, if it was present"]
["Show Bases",{"$f":"ToRadix(B$163,B$164,TRUE)","v":"{0/2; 4/8, 0/6, 0/4, 0/2}"}]
["Trim Zeros",{"$f":"TrimZeros(B167)","v":"{0/2; 4/8}"},null,"leaves at least one zero to the left of the radix point, if it was present"]
["Back to Number",{"$f":"FromRadix(B163,B168)","v":0.5}]
["Mixed Radix",{"$f":"TrimZeros(ToRadix(B$163,B$164))","v":"{0; 4}"},null,"Nested Trim Zeros call removes leading and trailing zeros"]
["Back to Number",{"$f":"FromRadix(B163,B170)","v":0.5}]
[]
[]
["United States customary units: Length"]
["Miles, Yards (1760), Feet (3), Inches (12); Pica (6), Points (12)"]
["Bases","[ 1760, 3, 12; 6, 12 ]"]
["Number",{"$f":"340.78-3+12","v":349.78}]
["Round To Least",{"$f":"roundtoleast(B176,B177)","v":349.77777777777777},"rounded to an integer number of Points"]
["Mixed Radix",{"$f":"ToRadix(B$176,B$177)","v":"{9, 2, 1; 4, 8}"}]
["Show Bases",{"$f":"ToRadix(B$176,B$177,TRUE)","v":"{9/1760, 2/3, 1/12; 4/6, 8/12}"}]
["Back to Rounded Number",{"$f":"FromRadix(B176,B179)","v":349.77777777777777}]
["Units to Apply","mile, yard, foot/feet, inch/inches; pica, point"]
["Apply Units",{"$f":"ApplyUnits(B179,B182)","v":"9 yards, 2 feet, 1 inch ; 4 picas, 8 points"}]
[]
[]
["United States customary units: Liquid Volume"]
["oil barrel, gallon (42), quart (4), pint (2), cup (2), ounce (8), tablespoon (2); teaspoon (3), minim (80)"]
["Note: excluded US fluid dram = 60 minims because not a divisor of the next higher unit = teaspoon = 80 minims"]
["Note: excluded US shot = 3 tablespoons because not a multiple of the next lower unit = ounce = 2 tablespoons"]
["Note: excluded gill = 4 ounces because rarely used, whereas cups above and ounces below are heavily used"]
["Note: excluded pottle even though valid base because rarely used, whereas gallons above and quarts below are heavily used"]
["Note: excluded liquid barrel because not an integer multiple of gallons = 31.5"]
["Note: excluded hogshead because not an integer multiple of oil barrels = 1.5"]
[]
["Bases","[ 42, 4, 2, 2, 8, 2, 3 ; 80 ]"]
["Number",{"$f":"45198+4+37.45/80","v":45202.468125}]
["Round To Least",{"$f":"roundtoleast(B195,B196)","v":45202.4625},null,"rounded to an integer number of minims"]
["Mixed Radix",{"$f":"ToRadix(B$195,B$196)","v":"{1, 16, 3, 0, 1, 5, 1, 1; 37}"}]
["Show Bases",{"$f":"ToRadix(B$195,B$196,TRUE)","v":"{1, 16/42, 3/4, 0/2, 1/2, 5/8, 1/2, 1/3; 37/80}"}]
["Back to Rounded Number",{"$f":"FromRadix(B195,B198)","v":45202.4625}]
["Units to Apply","oil barrel, gallon, quart, pint, cup, ounce, tablespoon, teaspoon ; minim"]
["Apply Units",{"$f":"ApplyUnits(B198,B201)","v":"1 oil barrel, 16 gallons, 3 quarts, 0 pints, 1 cup, 5 ounces, 1 tablespoon, 1 teaspoon ; 37 minims"}]
[]
[]
["United States customary units: Dry Volume"]
["Note: excluded cubic inches because not an integer number in a pint"]
["Note: excluded dry barrels because not an integer number of bushels"]
["Bases","[ 4, 2, 4, 2 ]"]
["Units to Apply","bushel, peck, dry gallon, dry quart, dry pint"]
["Confirm Limit / Unit",{"$f":"ApplyUnits(B208,B209)","v":"4 pecks, 2 dry gallons, 4 dry quarts, 2 dry pints"}]
["Number",173]
["Mixed Radix",{"$f":"ToRadix(B208,B211)","v":"{2, 2, 1, 2, 1}"}]
["Show Bases",{"$f":"ToRadix(B208,B211,TRUE)","v":"{2, 2/4, 1/2, 2/4, 1/2}"}]
["Apply Units",{"$f":"ApplyUnits(B212,B209)","v":"2 bushels, 2 pecks, 1 dry gallon, 2 dry quarts, 1 dry pint"}]
[]
[]
["United States customary units: Avoirdupois Weight"]
["Bases","[ 20, 100; 16, 16 ]",null,"1 dram = 27+11⁄32 grains, not an integer so exclude grains"]
["Units to Apply","ton, hundredweight, pound; ounce, dram"]
["Abbreviated Units","ton, cwt, lb; oz, dr"]
["Confirm Limit / Unit",{"$f":"ApplyUnits(B218,B219)","v":"20 hundredweights, 100 pounds ; 16 ounces, 16 drams"}]
["Abbrev Limit / Unit",{"$f":"ApplyUnits(B218,B220,FALSE)","v":"20 cwt, 100 lb ; 16 oz, 16 dr"},null,null,"plurals suppressed"]
["Number",6517.425]
["Round To Least",{"$f":"roundtoleast(B218,B223)","v":6517.42578125}]
["Mixed Radix",{"$f":"ToRadix(B218,B224)","v":"{3, 5, 17; 6, 13}"}]
["Show Bases",{"$f":"ToRadix(B218,B224,TRUE)","v":"{3, 5/20, 17/100; 6/16, 13/16}"}]
["Apply Units",{"$f":"ApplyUnits(B225,B219)","v":"3 tons, 5 hundredweights, 17 pounds ; 6 ounces, 13 drams"}]
["Apply Abbrev Units",{"$f":"ApplyUnits(B226,B220,FALSE)","v":"3 ton, 5 cwt, 17 lb ; 6 oz, 13 dr"},null,null,"plurals suppressed"]
["Back to Number",{"$f":"FromRadix(B218,B228)","v":6517.42578125}]
[]
[]
["United States FDA Cooking Measures"]
["Bases","[ 8, 2, 3 ]"]
["Units to Apply","cup, fluid ounce, tablespoon, teaspoon"]
["Confirm Limit / Unit",{"$f":"ApplyUnits(B233,B234)","v":"8 fluid ounces, 2 tablespoons, 3 teaspoons"}]
["Number",155]
["Mixed Radix",{"$f":"ToRadix(B233,B236)","v":"{3, 1, 1, 2}"}]
["Show Bases",{"$f":"ToRadix(B233,B236,TRUE)","v":"{3, 1/8, 1/2, 2/3}"}]
["Apply Units",{"$f":"ApplyUnits(B237,B234)","v":"3 cups, 1 fluid ounce, 1 tablespoon, 2 teaspoons"}]
["Back to Number",{"$f":"FromRadix(B233,B239)","v":155}]
[]
[]
["Israeli Bank Notes","₪ = Shekel (שֶׁקֶל)",null,"=","100 Agarot (אגרות)"]
["₪200"]
["₪100","x 2 = 200"]
["₪50","x 2 = 100","but excluded because not a multiple of 20"]
["₪20","x 5 = 100"]
["₪10","rarely used because of same-denomination coin"]
["₪5","rarely used because of same-denomination coin"]
["Israeli Coins"]
["₪10","x 2 = 20"]
["₪5","x 2 = 10","but excluded because not a multiple of 2"]
["₪2","x 5 = 10"]
["₪1","1 shekel","x 2 = 2"]
["₪1/2","1/2 shekel","x 2 = 1"]
["10 agorot","x 5 = 1/2"]
["5 agorot","removed from circulation in 2008"]
["1 agora","removed from circulation in 1991"]
[]
["Israeli Money"]
["Bases","[ 2, 5, 2, 5, 2; 2, 5 ]"]
["Units to Apply","x ₪200, x ₪100, x ₪20, x ₪10, x ₪2, x ₪1; x ₪0.5, x 10 אגרות"]
["Confirm Max / Unit",{"$f":"ApplyUnits(B261,B262,FALSE)","v":"2 x ₪100, 5 x ₪20, 2 x ₪10, 5 x ₪2, 2 x ₪1 ; 2 x ₪0.5, 5 x 10 אגרות"}]
["Shekels",972.5,"₪"]
["Mixed Radix",{"$f":"ToRadix(B261,B264)","v":"{4, 1, 3, 1, 1, 0; 1, 0}"}]
["Show Bases",{"$f":"ToRadix(B261,B264,TRUE)","v":"{4, 1/2, 3/5, 1/2, 1/5, 0/2; 1/2, 0/5}"}]
["Trim Zeros",{"$f":"TrimZeros(B265)","v":"{4, 1, 3, 1, 1, 0; 1}"}]
["Apply Units",{"$f":"ApplyUnits(B267,B262)","v":"4 x ₪200s, 1 x ₪100, 3 x ₪20s, 1 x ₪10, 1 x ₪2, 0 x ₪1s ; 1 x ₪0.5"}]

{"sheet":"Single","index":3,"n_rows":184,"n_cols":5,"rows_kept":184,"merged":[[150,1,150,2],[153,1,153,2],[154,1,154,2],[163,1,163,2]]}
["Examples and Validation of Single Base Radix Conversions"]
["http://individual.utoronto.ca/intervalmath/mixed/radix.html"]
[]
["Single Base Names:"]
["base","number system"]
[2,"binary",null,{"$f":"ToRadix(A6,D$14)","v":"101100001"}]
[3,"ternary",null,"111002"]
[4,"quaternary",null,"11201"]
[5,"quinary",null,"2403"]
[6,"senary or heximal",null,"1345"]
[7,"septenary",null,"1013"]
[8,"octal",null,"541"]
[9,"nonary",null,"432"]
[10,"decimal",null,353,"change this to watch the other bases values update"]
[11,"undenary",null,{"$f":"ToRadix(A15,D$14)","v":"2A1"}]
[12,"duodecimal or dozenal",null,{"$f":"ToRadix(A16,D$14)","v":"255"}]
[14,"quadroodecimal ",null,{"$f":"ToRadix(A17,D$14)","v":"1B3"}]
[13,"tridecimal",null,{"$f":"ToRadix(A18,D$14)","v":"212"}]
[14,"tetradecimal",null,"1B3"]
[15,"pentadecimal",null,"188"]
[16,"hexadecimal",null,"161"]
[17,"heptadecimal",null,"13D"]
[18,"octodecimal",null,"11B"]
[19,"enneadecimal",null,"IB"]
[20,"vigesimal",null,"HD"]
[36,"hexatrigesimal",null,"9T"]
[60,"sexagesimal",null,"{5, 53}"]
[]
[]
["Octal",null,"Binary",null,"If a number greater than 511 is input then there"]
["{8,8,8,8,8,8,8,8}",null,"{2,2,2,2,2,2,2,2}",null,"won't be enough bases, and the leftmost radix value"]
[1278,null,72,null,"will become an unlimited base 10 number."]
[{"$f":"ToRadix(A31,A32)","v":"{2, 3, 7, 6}"},null,{"$f":"ToRadix(C31,C32)","v":"{1, 0, 0, 1, 0, 0, 0}"},null,"with leading zeros forced by specified bases"]
[{"$f":"TrimZeros(A33)","v":"{2, 3, 7, 6}"},null,{"$f":"TrimZeros(C33)","v":"{1, 0, 0, 1, 0, 0, 0}"},null,"trim zeros"]
[{"$f":"FromRadix(A31,A34)","v":1278},null,{"$f":"FromRadix(C31,C34)","v":72}]
[]
["Decimal"]
["{10,10,10,10,10,10}"]
[1278,"convert base 10 to base 10"]
[{"$f":"ToRadix(A38,A39)","v":"{1, 2, 7, 8}"},"with leading zeros"]
[{"$f":"FromRadix(A38,A40)","v":1278},"impressive, right?"]
[{"$f":"TrimZeros(A40)","v":"{1, 2, 7, 8}"},"trim zeros"]
[{"$f":"FromRadix(A38,A42)","v":1278},"back to number"]
[]
["Binary"]
["{2}","single base treated like radix conversion without delimiters or brackets"]
[94,"the leftmost bit after conversion will always be binary"]
[{"$f":"ToRadix(A46,A47)","v":"1011110"},"always without leading zeros"]
[{"$f":"ToRadix(A46,A47,TRUE)","v":"{1, 0, 1, 1, 1, 1, 0}"},"with single base, show bases option shows individual radices without /base"]
[]
[2,"single base for direct binary conversion"]
[54648,"base 10 number to convert"]
[{"$f":"ToRadix(A51,A52)","v":"1101010101111000"},"result has no brackets or delimiters"]
[{"$f":"FromRadix(A51,A53)","v":54648},"back to base 10 number"]
[{"$f":"ToRadix(A51,A52,TRUE)","v":"{1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0}"},null,null,"show bases forces number groups"]
[{"$f":"FromRadix(A51,A55)","v":54648},"back to number"]
["Optionally use SpaceEvery( ) function to insert spaces for readability:"]
[{"$f":"SpaceEvery(4,A53)","v":"1101 0101 0111 1000"},"inserted spaces to separate into nybbles"]
[{"$f":"FromRadix(A51,A58)","v":54648},"inserted spaces don't interfere with converting to number"]
[{"$f":"SpaceEvery(8,A53)","v":"11010101 01111000"},"inserted spaces to separate into bytes"]
[{"$f":"SpaceEvery(4,A55)","v":"{1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0}"},null,null,"inserting spaces has no effect on numeric radix groups"]
[]
[]
["Ternary (base 3)"]
["single base",3]
["number",371]
["converted to base 3",{"$f":"ToRadix(B65,B66)","v":"111202"}]
["back to number",{"$f":"FromRadix(B65,B67)","v":371}]
[]
[]
["Quaternary (base 4)"]
["single base",4]
["number",5718]
["converted to base 4",{"$f":"ToRadix(B72,B73)","v":"1121112"}]
["back to number",{"$f":"FromRadix(B72,B74)","v":5718}]
[]
[]
["Octal"]
["{8}","single base treated like radix conversion"]
[65535]
[{"$f":"ToRadix(A79,A80)","v":"177777"}]
[{"$f":"FromRadix(A79,A81)","v":65535},"back to number"]
[{"$f":"ToRadix(A79,A80,TRUE)","v":"{1, 7, 7, 7, 7, 7}"},"show bases forces number groups"]
[{"$f":"FromRadix(A79,A83)","v":65535},"back to number"]
[]
[]
["Single bases 11 thru 36 use letters A thru Z for digits 10 thru 35"]
[]
[]
["Hexadecimal"]
[16,"brackets not necessary"]
[{"$f":"2^19-1+1/7","v":524287.14285714284},"if number has fractional part then"]
["hexadecimal uses 0 thru 9 and letters A thru F"]
[{"$f":"ToRadix(A91,A92)","v":"7FFFF;2492492"},"converted radix has fractional part"]
[{"$f":"FromRadix(A91,A94)","v":524287.1428571418},"back to number"]
[{"$f":"ToRadix(A91,A92,TRUE)","v":"{7, 15, 15, 15, 15; 2, 4, 9, 2, 4, 9, 2}"},null,null,"show bases forces number groups"]
[{"$f":"FromRadix(A91,A96)","v":524287.1428571418},"back to number"]
["Optionally use SpaceEvery( ) function to insert spaces for readability:"]
[{"$f":"SpaceEvery(2,A94)","v":"7 FF FF ; 24 92 49 2"},"bytes relative to radix point"]
[{"$f":"SpaceEvery(4,A94)","v":"7 FFFF ; 2492 492"},"32-bit words relative to radix point"]
[165.9,"all fractions of 10 terminate in decimal"]
[{"$f":"ToRadix(A91,A101)","v":"A5;E666666"},"but repeat in hexadecimal"]
[165.5,"(except for 5/10 = 0.5 decimal"]
[{"$f":"ToRadix(A91,A103)","v":"A5;8"},"and 0.8 in hexadecimal)"]
[-241.2,"make it a negative number"]
[{"$f":"ToRadix(A91,A105)","v":"-F1;3333333"},"yields one minus sign in front of the converted value"]
[{"$f":"FromRadix(A91,A106)","v":240.80000000074506},"and when converting back to the number the minus sign is preserved"]
[]
[]
["Hexadecimal","[16,16,16,16]",null,"fractions ignored unless radix point and fractional bases provided"]
["Given Number",732847]
["Radix values",{"$f":"ToRadix(B110,B111)","v":"{11, 2, 14, 10, 15}"},null,null,"numeric radix values if more than a single base specified"]
[null,{"$f":"FromRadix(B110,B112)","v":732847}]
["show bases not ignored",{"$f":"ToRadix(B110,B111,TRUE)","v":"{11, 2/16, 14/16, 10/16, 15/16}"}]
["Back to Number",{"$f":"FromRadix(B110,B114)","v":732847}]
[]
[]
["Vigesimal (base 20), was used by the Aztecs and Mayans (not with the number and letter symbols that we use, of course)"]
["single base","[20]"]
["number",3471.58]
["converted to base 20",{"$f":"ToRadix(B119,B120)","v":"8DB;BC"},"uses numerals 0 thru 9, and letters A through J"]
["back to number",{"$f":"FromRadix(B119,B121)","v":3471.58}]
[]
[]
[36,"brackets not necessary"]
[87654321]
[{"$f":"ToRadix(A125,A126)","v":"1G6QI9"},"uses numerals 0 thru 9, and letters A through Z"]
[{"$f":"FromRadix(A125,A127)","v":87654321},"back to number"]
[{"$f":"ToRadix(A125,A126,TRUE)","v":"{1, 16, 6, 26, 18, 9}"},"show bases forces number groups"]
[{"$f":"FromRadix(A125,A129)","v":87654321},"back to number"]
[{"$f":"LOWER(A127)","v":"1g6qi9"},"lowercase of hex value above"]
[{"$f":"FromRadix(A125,A131)","v":87654321},"lettercase doesn't matter when converting back to number"]
[]
[]
["Sexigesimal (base 60)"]
["{60}"]
[5418356.0034]
[{"$f":"ToRadix(A136,A137)","v":"{25, 5, 5, 56; 0, 12, 14, 24}"}]
[{"$f":"FromRadix(A136,A138)","v":5418356.0034}]
[]
[]
["sexigesimal fraction"]
["{;60,60,60,60,60,60,60,60}",null,"1/3 is a single radix and 1/9 is a dual radix fraction in sexigesimal"]
[0.14285714285714285,null,"any fraction of 7 generates 3 repeating fractional radix groups in sexigesimal"]
[{"$f":"ToRadix(A143,A144)","v":"{; 8, 34, 17, 8, 34, 17, 8, 34}"},null,"any fraction of 11 generates 5 repeating fractional radix groups (2nd set limited by floating point imprecision)"]
[{"$f":"FromRadix(A143,A145)","v":0.14285714285714116},"back to number"]
[]
[]
["Mathmetica 5 book examples (pages 725-6) --- long before MixedRadix[{ list of bases }] was introduced to the language"]
["base = hexadecimal","{16}"]
["given number",1234135634]
["converted to hex",{"$f":"ToRadix(B150,B151)","v":"498F6A52"}]
["cvt to numeric groups",{"$f":"ToRadix(B150,B151,TRUE)","v":"{4, 9, 8, 15, 6, 10, 5, 2}"},null,"forced by ShowBases=True"]
["from hex string",{"$f":"FromRadix(B150,B152)","v":1234135634}]
["from number groups",{"$f":"FromRadix(B150,B153)","v":1234135634}]
[]
["base = binary","{2}"]
["two number examples",56,37]
["converted to binary",{"$f":"ToRadix(B157,B158)","v":"111000"},{"$f":"ToRadix(B157,C158)","v":"100101"}]
["back to numbers",{"$f":"FromRadix(B157,B159)","v":56},{"$f":"FromRadix(B157,C159)","v":37}]
[]
["base = hexadecimal","{16}"]
["lettercase doesn't matter","ffffaa00"]
["converted to base 10",{"$f":"FromRadix(B162,B163)","v":4294945280}]
[]
["hexadecimal numbers","converted to base 10"]
["fffaa2",{"$f":"FromRadix(B$162,A167)","v":16775842}]
["ff",{"$f":"FromRadix(B$162,A168)","v":255}]
[null,-1]
[null,{"$f":"SUM(B167:B169)","v":16776096},"sum"]
[]
["base = binary","{2}"]
["binary # with fraction","101;100101"]
["convert to base 10",{"$f":"FromRadix(B172,B173)","v":5.578125},"Mathematica 5 book rounds this to 5.57813"]
[]
[]
["Converting between numeric bases (simply go through base 10 as the intermediate)"]
["Starting base","[10]"]
["Number",1234]
["Hexadecimal","[16]"]
[null,{"$f":"ToRadix(B180,B179)","v":"4D2"}]
["Octal","[8]"]
[null,{"$f":"ToRadix(B182,B179)","v":"2322"}]
["Or nest the conversion",{"$f":"ToRadix(B182,FromRadix(B180,B181))","v":"2322"},"(base 10 intermediate is internal, not displayed)"]

{"sheet":"Web","index":4,"n_rows":28,"n_cols":1,"rows_kept":28}
["Mixed Radix Conversion Resources on the Web"]
[]
["Home page for this workbook:"]
["http://individual.utoronto.ca/intervalmath/mixed/radix.html"]
[]
["\"Mixed radix\" topic at Wikipedia:"]
["https://en.wikipedia.org/wiki/Mixed_radix"]
[]
["\"Factorial number system\" topic at Wikipedia:"]
["https://en.wikipedia.org/wiki/Factorial_number_system"]
[]
["\"Primorial\" topic at Wikipedia:"]
["https://en.wikipedia.org/wiki/Primorial"]
[]
["Mixed Radix Numeral System class and Counter  (open source Mixed Radix Calculator):"]
["https://csharpcodewhisperer.blogspot.com/2015/10/mixed-radix-numeral-system-class-and.html"]
[]
["Mixed Radix Number Representations at Wolfram --- web demo of up to 4 bases for converting a number:"]
["https://demonstrations.wolfram.com/MixedRadixNumberRepresentations/"]
[]
["Description of the term \"Base\" at Wolfram MathWorld:"]
["https://mathworld.wolfram.com/Base.html"]
[]
["\"Standard positional numeral systems\" subtopic at Wikipedia:"]
["https://en.wikipedia.org/wiki/List_of_numeral_systems#Standard_positional_numeral_systems"]
[]
["Henning's web site concerning the Kālacakra Calendar (Tibetan Calendar)"]
["http://www.kalacakra.org/calendar/kcal.htm"]