# Talk:Sexagesimal

WikiProject Mathematics (Rated C-class, Mid-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 C Class
 Mid Importance
Field: History of mathematics (historical)

## Symbols used

What symbols were used historically for digits 11-59 in sexagesimal? If those used today (obviously in very narrow practice) are different, what are those, as well?

Today the symbol ':' is used for times (HH:MM:SS) in ISO 8601 and so is a de facto delimiter for sexagesimal digits. So we could have for example

 Sexagesimal  Decimal
15         15
01:03         63
05:00        300
16:41       1001
02:05:00       7500


I also put the fractions in the article into this notation, keeping the '.' as the sexagesimal point.

Karl 24 March 2004, 21 July 2004

I think some sort of example like this should go into the article, unless it contravenes some style guideline. What are the rules concerning examples in articles? ais523 11:26, 16 June 2006 (UTC)

-I am curios as to how the Sumarians were able to build a transmitter to transmit thier mathamatics 4000 years ago? —Preceding unsigned comment added by 72.89.188.197 (talk) 05:53, 28 January 2008 (UTC)

## Sexagesimals in Ancient India

The Ancient Indians had a sexagesimal system as well; as clearly explained in the Surya Siddhanta. I think there needs to be a paragraph on that. In fact, the two cultures, Sumeria & India, developed the sexagesimal system independently. —Preceding unsigned comment added by 67.180.39.64 (talk) 16:50, 22 April 2008 (UTC)

Supply appropriate sources (both for their use in the Surya Siddhanta and, if you wish it stated in the article, for the independence of their development from the Sumerians) and we can add it easily enough. —David Eppstein (talk) 16:55, 22 April 2008 (UTC)
The Surya Siddhanta is usually dated to the 3rd century CE based on the position of its vernal equinox, which is 'frozen' sidereally, now near April 14. This is many centuries after the Babylonians, and even after the Hellenistic astronomers, Hipparchus and Ptolemy, had used a sexagesimal system using Greek numerals. The supposed version dating to the 3rd century BCE is probably the Vedanga Jyotisha of Lagadha. Even that is still about 2500 years after the Sumerians are known to have used a fully formed sexagesimal system (except for zero) during the 3rd millennium BCE, based on surviving cuneiform tablets. I can't remember whether the Surya Siddhanta itself used sexagesimal notation. — Joe Kress (talk) 20:47, 22 April 2008 (UTC)
Regardless of the (lack of) merit of any priority claims, or the pointlessness of claiming priority for sexagesimal when they have a much stronger and more important claim for decimal, if some ancient Indians actually used sexagesimal then we should mention it in the article. —David Eppstein (talk) 22:36, 22 April 2008 (UTC)
Sixty was indeed used as a fraction system in india, especially since we have the day divided into 60 ghurries, each of 60 pali, of 60 vipali. India acquired the zero by way of the arabs, who got it from the greeks. --Wendy.krieger (talk) 11:02, 2 September 2009 (UTC)

## Sixty as a division-system

One should note that for the greater time, sixty-wise numbers are intended to be a division system, where the unit column is at the left, and more right-places are more precision. For example, 15 hours, 15:00 hours and 15:00:00 hours are all the same thing, going to minutes and seconds respectively.

Zeros in the sumerian system reflect the division system, so they have leading zeros and medial zeros, but not trailing ones: we see eg 0:0:1 for 1 second, and 1:0:1 for 1 hour 1 second, but not 1:0 for 1 hour 0 minutes. One could shift the lead column by changing the unit, eg 1:4 shock is 1.03333 shocks (of 60 in number), giving 64.

The ancient sumerian use of this system is a division-system (means of writing fractions), designed to avoid the arithmetic division. We note that one of the common tables that come down to us is the table of ordered recriprocals, eg 3 -- 20 3.20 -- 18 etc, one doing general division by way of interpolating this table.

Neugebauer gives a reference to Sachs having 'recently' found a tablet dealing with the evaluation of 1/7, 1/11, etc, in the sense that :08:34:16:59 < 1/7 < :08:34:18, when a division give the correct value of :08:34:17:08...

Neugebauer also gives the number of the sumerians for the multiplication scale. It's a mixture of units, such as using i (one), x (10), I (60 = big 1) and X (big 10 = 100), along with U (120 = 2*60). A date consistently refered to in the table as 3:12, would elsewhere be written as XIxxii (ie hundred+sixty+thirty+two).

I have yet to see a practical application of sixty as a multiple-system, in the sense of other bases.

Ref: O Neugebauer "the exact sciences in antiquity" --Wendy.krieger (talk) 12:07, 27 August 2009 (UTC)

## Symbols

The Base 62 article uses the 26 uppercase letters and then the 26 lowercase letters to represent numbers greater then 9, why doesn't this article follow along with the pattern by using A-x? It will look a lot better this way, at the moments it’s hard to tell the difference between numbers that are in decimal and the ones that are in sexagesimal. If we included letters we will be able to show repeating decimals more easily. Robo37 (talk) 17:58, 27 August 2009 (UTC)

Because sexagesimal is actually in standard use today (for instance in showing times as hours:minutes:seconds) and that standard use represents each base-60 digit as a pair of decimal digits. We should be following standard conventions here, not trying to make up new and more logical conventions: see WP:OR. —David Eppstein (talk) 20:59, 27 August 2009 (UTC)
But the standard convention is to use the 10 numerical digits first, then the 26 uppercase letters, and then finally the 26 lowercase letters; why should the 60th base be the only one that doesn't fit in with this pattern? Yes, we do express time under this format, but this article isn't about time; it's about numbers. 24 is also often used to express time but letters are still used in the article about the number's respective base. Robo37 (talk) 21:37, 27 August 2009 (UTC)
There is no record of the sumerians using a system like this: it's always been alternating symbols from the set 1-9, and A-F (for 10,20,30,40,50). Many of the things that i see written of this system is exactly what one would expect of an alternating-base division system. I use an ordinary alternating base of 12*10, so these things occur in ordinary life.--Wendy.krieger (talk) 07:40, 1 September 2009 (UTC)
See here for an example of what goes wrong when one treats it as a mixed-radious system with alternating bases 6 and 10. They're not the same, and the differences show up primarily for sexagesimal digits that are either less than 10 or a multiple of 10. As for using a representation different than the one we use for time, degrees, etc., I think that would seriously impair the readability of the artcle for a large fraction of its audience. —David Eppstein (talk) 14:03, 1 September 2009 (UTC)
The use of zero to show sixtyone is wrong. One should remember that any notation is to write the position of stones on the abacus, and that one has either full-value tokens (like C = 100 or $1 1c for$1.01, or some kind of spacing empty column-marker, like zero. The egyptians had a zero too, but it was used to show there are no stones on the abacus.
One can represent sumerian numbers in a notation that matches the written runes: 0, 1-9, and A-F for 10-50. Semicolons are used to indicate columns, are not in the source. There is evidently no confusion between 2 (II) and 1:1 (I I). A zero 0 is written as a full stop (that's the usual meaning of that symbol), is written either leading or medially (so 1 second might be written as 001, or 0:0:1, assuming the unit degree, or 0:0:0:1 (the sextant). One could write 61 as 11 and 3601 as 101. In the first example, there is a missing 10, so this is skipped (the instruction is to put 1 stone in a column unit, and the next in the next column unit). There is no symbol for a semimedial zero (ie D 1 vs D1, ie :D:1: = 40.01 vs :D1: = 41, but this is no great miss.
I have use an alternating base for many years. Alternating bases behave like regular bases when the full scope of the column is taken in one place. So grouping pairs of alternating digits like 60, is no different to grouping threes of digits in binary->octal, or 10>1000, or 18>5832. What makes me think that it is an alternating base, is that one sees calculations where the digits are evenly spaced, like 3 D 5 (for 1/16 = :03:45), where the digits are presented without punctuation. I have seen seven or eight digits of 60 thus represented. It's usually a marker that criss-cross multiplication is under way.
One must also note that there are many representations of sixty, especially after the greeks (who used decimal numbers and had access to egyptian and sumerian fractions, along with their home grown one (eg x parts where y is ...) Euclid has lines representing a ratio of integers.--Wendy.krieger (talk) 10:58, 2 September 2009 (UTC)

Using either the modern notation (ie columns of 60, with markers), or sixty separate runes for base 60 confuses the issues as presented in sumerian and other records. In practice, the thing is an alternating base, used mainly for division (fractions). A transliteration of the sumerian runes gives, eg symbols for 1-9, and 10,20,30,40,50, in the form of eg A,B,C,D,E,F. It's also the same form i use for all alternating bases, eg Mayan. In essence, the numerals stand for the lower row of the abacus, while the letters stand for the upper row. The zero rune reflects actual zero usage. The word UNIX is shuffled around, to represent U,I as the high row, and N,X as the low row. Mayan numbers are read in NUXI, so a number like 1957 becomes 4.17.17, or 4 2C 2C (the dashes, representing the 5's follow the dots. Digits are clearly separated. A quoted value for sqrt2 runs 1B4E1A, of equal spacing, but no head. We see this becomes in modern script as 1:24:51:10. The next digit is 7, in the form 1BE1A 7, where A 7 represents 10:07, not 17. This is not apparent had the digits been written with included zeros. --Wendy.krieger (talk) 08:26, 4 September 2010 (UTC)

### A standard notation exists

There is an accepted scholarly notation for sexagesimal numbers that I recently added to the article. The article's main text uses a method of separating orders of sexagesimal numbers by colons. I have never seen this notation before except in time reckoning. Is there a source for the extension of this method to a more general sexagesimal notation?
If there is no source for the article's current notation, I would recommend following the accepted practice used by Aaboe, Neugebauer, and others. --SteveMcCluskey (talk) 22:44, 25 October 2012 (UTC)
I did some further checking and found that until these changes, the article consistently used the accepted scholarly notation in which digits in sexagesimal numbers where separated by commas, while the fractional part was separated from the whole number part by a semicolon. Does anyone know of a rationale for this change? --SteveMcCluskey (talk) 02:15, 28 October 2012 (UTC)
The reason for this change is that this is the notation we use when we write hours:minutes:seconds. So it should be much more familiar to readers than some alternative notation involving commas. And it is very far from being unsourceable, because it is a standard notation taught to kids in elementary school and used by many people every day. As for the lack of distinction between integer and fractional parts of the numbers: that's because the Babylonians made no such distinction. —David Eppstein (talk) 03:40, 28 October 2012 (UTC)
Thanks for the reply. As I read it, you seem to be saying that there is no source for the use of this notation except in the limited field of expressing units of time. If that is so, one could equally well argue for extending the familiar angle notation for degrees, minutes, and seconds (° ' ") to apply generally to sexagesimal numbers.
Your comment that Babylonian notation didn't distinguish integer and fractional parts of the number may be true (although I'm not certain about later Babylonian texts) but it certainly isn't true for later astronomers using sexagesimal numbers in Greek, Arabic, and Latin. They, like we, wrote digits as integers in their various customary notations, sometimes using just spaces to mark separation of digits (see Aaboe's transcription of part of Ptolemy's Table of Chords, Episodes from the Early History of Mathematics, p. 103). Even if it were universally true, it isn't an argument against the article's use of a notation that does makes this distinction.
Lacking a source for the article's extension of modern time notation to sexagesimal numbers in general, I think that, as an encyclopedia, Wikipedia should use the comma and semicolon notation that is widely accepted in the scholarly literature, where it is applied to units of time, angle, length, and to pure numbers such as Pi. --SteveMcCluskey (talk) 20:07, 28 October 2012 (UTC)

## Self-reference in introduction

I removed a dew sentences in the opening paragraph because I feel they are self references. They are almost verbatim of the first item from Wikipedia:Manual of Style (self-references to avoid). Rather then just revert I wanted to discuss the issue. The exact quote is as follows: