Talk:Sexagesimal

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 Manual of Style (self-references to avoid). Rather then just revert I wanted to discuss the issue. The exact quote is as follows: "In this article, all sexagesimal digits are represented as decimal numbers..."How is that line not a self-references? meshach (talk) 23:36, 10 November 2010 (UTC)
 * Did you read the section "Neutral self-references are acceptable" in the style guide you linked to? Because this example seems almost exactly like the ones in that section to me. —David Eppstein (talk) 06:31, 11 November 2010 (UTC)
 * Yes I read that section but I did not think it applied here. But I guess I see how it is possible.  Cheers, meshach (talk) 18:44, 12 November 2010 (UTC)

Just say base 10 decimal numbers and it will not be a self reference. —Preceding unsigned comment added by 164.106.234.127 (talk) 17:21, 3 March 2011 (UTC)

Sexagesimal in modern French
The paragraph about modern French is not really correct:

> A vestige of the sexagesimal system exists in the European and Canadian dialects of the French language, > where the numbers from 70 to 79 are rendered by adding a number to 60: 70, for example, renders as > soixante-dix (sixty-ten), and 75 is called soixante-quinze (sixty-fifteen).

This is a residue of a vigesimal (base 20 system), in which there are words for 60 and 80, but 70-79 and 90-99, are made by adding the numbers 10 to 19 to 60 or 80. 80 is also 'quatre-vingt' or 4-20, showing more evidence of a vigesimal system.

In short, this is nothing to do with sexagesimal and I suggest that this paragraph be removed. — Preceding unsigned comment added by Komaba (talk • contribs) 12:17, 14 February 2011 (UTC)


 * Done. I'm not sure you can see from the modern words whether there is a historical connection to sexagesimal system, but it would require a source - and I'm pretty sure there IS no such connection, where as the vigesimal roots of French (and Danish) number words is undisputable.--Nø (talk) 15:07, 14 February 2011 (UTC)
 * I agree with removing this. —David Eppstein (talk) 17:04, 14 February 2011 (UTC)


 * One should check out Old English where the numbers from 60 to 120 are of the form hundsixty, hundseventy, ... hundelefty. Even though there is a base 20 substrate in french, the change at sixty should be noted.  --Wendy.krieger (talk) 07:33, 15 February 2011 (UTC)
 * Linguistic remnants of sexagesimal in Old English or French should be noted if, and only if, they can can be properly sourced as being accepted by experts in the subject. —David Eppstein (talk) 07:44, 15 February 2011 (UTC)
 * Who would ye like? Sweet? Clark Hall? Onions? A list of numbers in OE does show the change at sixty.--Wendy.krieger (talk) 07:14, 16 February 2011 (UTC)

Modern Usage
Does the numbering system used by digital microwave ovens fit within the definition of sexagesimal as a modern usage? I ask because it appears to be a hybrid of sexagesimal and decimal, allowing values >59 in the seconds columns. — Preceding unsigned comment added by TimWooley (talk • contribs) 15:08, 26 April 2011 (UTC)


 * It's actually mmss, but ye are permitted to enter 90 seconds as such, not as 130. Nothing restricts you to using digits in the normal range in any number system: it's just that the normal name for the number is in the reduced range.--Wendy.krieger (talk) 07:39, 28 April 2011 (UTC)


 * There is actually a hybrid system used by some ancient astronomers of recording the whole number part of a number in decimal notation and the fractional part in sexagesimal notation. --SteveMcCluskey (talk) 22:47, 25 October 2012 (UTC)

About the two digit digits...
Perhaps it would be easier to reprensent digits above 9 with the standard lettering aproach used in other number system articles, that is to follow up the digits 1-9 with new digits A-Z and then, in this case, the lowercase letters a-x? I understand that the decimal digit approach is used to represent time but that, to me, looks a bit visually overwhelming when you get past three digits and isn't using true sexagesimal reprensentation, and after all this article is called 'Sexagesimal' not 'Deca-sexagesimal'. Maybe save the decimal for articles linked to time measurment, instead of pure mathematics? Also as it's a multiple of a primorial it means it's one of the best at representing fractions, especially as it's also a highly composite number, so I'm all for a table like the one on Base 30 to show this, though, again I think it would look overwhelming with deciamal digits, though that might just be me. Robo37 (talk) 17:34, 30 November 2011 (UTC)
 * Oh, sorry, I forgot I already made a section about this. Still, my point on having a propper table still stands. Robo37 (talk) 17:45, 30 November 2011 (UTC)
 * It may not be using true sexagesimal but it's what both the Babylonians and our modern notation for times and angles uses. I'd prefer not to make up new original or not-well-used notations when we have a perfectly good standard notation to use. —David Eppstein (talk) 19:17, 30 November 2011 (UTC)


 * I don't know. The babylon use of two digits to represent numbers base 60, is probably no different to the romans using two digits I, V.  In practice, if one wants to experiment with the problems and such of the sumerian notation, it is best to do direct transliteration of the digits, viz A=10, B=20, C=30, D=40, E=50.  A number like 53 is written as E3.  The number 1 3 is 63.
 * Sexagesimal is best thought of as a division-system to avoid division. In sumerian and later, the Most significant digit is the units, the remainder is fraction.  One notes in modern parlance, 15 hours and 1500 hours are the same thing.   One finds among the tables of the reckoner, the multiples of various small numbers, like 1, 2, 3,  a table of recriprocals for numbers that 'come out', eg 1B1 = D4B6D  (ie 1.21 ~ 44.26.40), and tables of these recriprocals, eg 44.26.40×17 etc.  Other giveaways is the presence of things like papers on the 'seven brothers' (the result concludes 0 8C4A6 < 1/7 < 0 8C4A8).  (all of these are in Neugebauer, but he does not draw this conclusion).
 * I've seen in a number of references a calculation of sixty-number, laid out with the digits evenly spaced, like 3 . 8 3 for 38C (pi) with little regard to the alternation. While this looks strange to people who grew up on a diet of ten-like bases, for people who regularly use alternating bases, it is quite natural in regards to say, criss-cross multiplication. Wendy.krieger (talk) 07:50, 1 December 2011 (UTC)

Ptolemy
The current version of our article writes that Ptolemy used the value 3;8,30 for π. But this would be simpler written as 6;17 for 2π. Do we have a source indicating whether Ptolemy considered π or 2π as primary? Currently our statement about his value is unsourced. —David Eppstein (talk) 22:07, 2 December 2012 (UTC)


 * Sir Thomas Heath (1921, Dover reprint) 'a history of greek mathematics' vol 1, p233, gives an approximation, given in greekish runes, of 21,1875 : 6,7441 (greek demotic), and later on observes a sextent occupies 1;2.50, which makes the value of pi 3;8.30. The value 6;17 is not given anywhere in the discussion.  Wendy.krieger (talk) 06:51, 4 December 2012 (UTC)
 * So π/3 was primary? That would certainly explain why it's 360 degrees in a circle. Thanks for the reference. —David Eppstein (talk) 07:41, 4 December 2012 (UTC)
 * Ptolemy's Almagest, VI,7 (p. 302 of the Toomer translation) has "we assumed that the ratio of the circumference to the diameter [i.e., π] is 3;8,30 : 1, since this ratio is about half-way between 3 1/7 : 1 and 3 10/71 : 1, which Archimedes used as rough bounds."  Note that Ptolemy is treating π as a ratio.
 * As to the 360 degrees in a circle, that goes back to Babylonian usage and AFAIK, the origins of that are not clear. --SteveMcCluskey (talk) 20:55, 4 December 2012 (UTC)
 * The sumerian circle for tangable circles, is given in Heath (vol 2, p216), as 180 ells of 24 digits, which gives a circle diameter 60, pi=3. Circles by diameters is how one finds them in the real world (eg circular inch = area of circle 1 inch diam).  The division of the circle to 360 degrees comes only when one is standing in the centre thereof (eg astronomy), with r=60, pi=3.  Compare this with the present examples of the mill (r=1000, pi=3.2 or 3.15).  Wendy.krieger (talk) 07:18, 5 December 2012 (UTC)

Equation formatting question.
There have been two different versions of some mathematical equations in this article. In version one using the tag the Wikipedia default display makes the equation appear in a varying size font which (on a large screen) is much larger than the text:


 * The square root of 2, the length of the diagonal of a unit square, was approximated by the Babylonians of the Old Babylonian Period (1900 BC – 1650 BC) as
 * $$1;24,51,10=1+\frac{24}{60}+\frac{51}{60^2}+\frac{10}{60^3}=\frac{30547}{21600}\approx 1.414212\ldots$$
 * Because $$\sqrt{2}$$ is an irrational number, it cannot be expressed exactly in sexagesimal numbers, but its sexagesimal expansion does begin 1;24,51,10,7,46,6,4,44 ...

Version two, using the

template, the font is closer to that in the text although, as has been commented in a recent edit, there is a misalignment of the vertical fractions.


 * The square root of 2, the length of the diagonal of a unit square, was approximated by the Babylonians of the Old Babylonian Period (1900 BC – 1650 BC) as
 * Because √2 is an irrational number, it cannot be expressed exactly in sexagesimal numbers, but its sexagesimal expansion does begin 1;24,51,10,7,46,6,4,44 ...
 * Because √2 is an irrational number, it cannot be expressed exactly in sexagesimal numbers, but its sexagesimal expansion does begin 1;24,51,10,7,46,6,4,44 ...

I'm presenting the two versions here so editors can compare how they appear on their system(s) and discuss their preferences between the two versions. SteveMcCluskey (talk) 22:28, 14 April 2013 (UTC)


 * On my desktop computer (using generic Mozilla Firefox under Windows 7 and with no additional mathematical rendering tools), the equation font in version one is twice the height of the text font and appears quite jarring; in version two there is a slight misalignment between the vertical fractions (apparently because of the lack of a superscript marking an exponent in one of them).
 * On my iPhone the disproportion between the font sizes is not quite as bad, with the equation in version one about 1 1/2 times as large as the text font. Considering the fonts used, the font in version two is the same sans serif font as used in body of the text, while the font in version one is a serif font that looks like Times Roman.
 * In balance, version two looks better to me on both machines. SteveMcCluskey (talk) 22:42, 14 April 2013 (UTC)

On my machine, with MathJax turned on in preferences, version 1 looks much better than version 2. But even with the bitmap default formatting, I greatly prefer version 1, despite the font size issue, because I find the vertical misalignment severely jarring. Additionally, for a displayed (rather than inline) equation, version 2 has too small fonts for the fractions, whereas version 1 makes them all equal to the text size (in the MathJax rendering), and the horizontal spacing in version 2 is also bad (too tight in the sexagesimal part and too little space before the third plus sign and the equal sign). And as a general matter of principle, converting something that works with MathJax to something that doesn't is moving in the wrong direction. —David Eppstein (talk) 23:10, 14 April 2013 (UTC)
 * We haven't had any other comments on this, so I guess it's up to us to resolve this ourselves. You haven't convinced me that version one, using the  tag, is preferable.  Your strongest argument is that if you set the preferences to a non-standard setting, you can get an acceptable rendering of this equation using version one.  But we are writing this encyclopedia for general readers who use standard settings.  One could argue that the MathJax setting which you use should be made Wikipedia's default, but this is certainly not the place to debate that.
 * With the standard setting, version one renders the equation in such a differently sized font that it is glaringly unacceptable (it has been bothering me since I first edited this article, but I only recently discovered the

template I used in version two). Version two, on the other hand, makes a minor misalignment which requires careful attention to detail to even notice. Unless you present a convincing argument otherwise, I will revert to version two. SteveMcCluskey (talk) 21:01, 17 April 2013 (UTC)
 * No, I believe you are misrepresenting my opinion. The argument that I believe is strongest is that your version looks even uglier than the default bitmap rendering. I find the misalignment much worse than the font size issue. It is not minor. If you want a third opinion, you could try asking at WT:WPM. In the meantime, as for how to resolve an impasse with too few editors to declare a consensus: the default Wikipedia convention (e.g. in WP:RETAIN) is to not fix things that aren't broken: don't change formats without a clear consensus to do so. —David Eppstein (talk) 22:29, 17 April 2013 (UTC)
 * I very much prefer version 2, not only because in my opinion the misalignment is much less noticeable than the overly huge and thick font used in version 1. Version 2 also has other advantages: It actually displays text, therefore it will produce useful display content also with text-only browsers (for example with LYNX or DOSLYNX). Screen-readers for blind or visually impaired people should have an easier job to make sense of the version 2 display as well. Finally, you can easily copy and paste the version 2 numbers using your keyboard or mouse, which is not possible with the graphics displayed by version 1. --Matthiaspaul (talk) 23:21, 17 April 2013 (UTC)
 * To get additional input, I've added a note at Wikipedia:Third opinion. Hope this helps bring things to resolution.  SteveMcCluskey (talk) 15:29, 18 April 2013 (UTC)
 * I think you'll get a more informed opinion on mathematics formatting from WT:WPM. I've added a note there. —David Eppstein (talk) 15:36, 18 April 2013 (UTC)
 * Fine. SteveMcCluskey (talk) 15:49, 18 April 2013 (UTC)
 * Certainly, if you tuned your account for the use of MathJax, but neglect to set up a good appearance of &#123;{math}}, then you will obtain better results on &lt;math>. Note that for unregistered users both are unavailable. Incnis Mrsi (talk) 17:45, 18 April 2013 (UTC)

The misalignment in Version 2 is incredibly jarring and makes the equation hard to read. The size difference between the equations and the surrounding text in Version 1 is a little odd, but the equation is readable. I would take Version 1 any day. (I am using Chrome 26 on Linux; checking in Firefox 17 on Linux, Version 2 looks slightly less bad but is still worse than Version 1. I am using whatever the defaults are for how Wikipedia displays equations, i.e., I think it should be the same for me and for unregistered users.)  --JBL (talk) 17:53, 18 April 2013 (UTC)


 * I don't know if my opinion should be taken seriously, as I've been writing mathematical articles for over 40 years, but I think the misalignment in version 2 looks more jarring than the boldface in version 1. (As an aside, compare additional alternatives:
 * (1)
 * $$1;24,51,10=1+\frac{24}{60}+\frac{51}{60^2}+\frac{10}{60^3}=\frac{30547}{21600}\approx 1.414212\ldots$$
 * (1A)
 * $$1;24,51,10=1+\tfrac{24}{60}+\tfrac{51}{60^2}+\tfrac{10}{60^3}=\tfrac{30547}{21600}\approx 1.414212\ldots$$
 * (2)
 * (2A)
 * Sorry about 2A; I thought there is some HTML way to render text but make it invisble; Anyone is welcome to edit the HTML to fix the problem, but I think it looks better with the explicit 601 than without.
 * Version 1A is closer to version 2 in character size, and version 2A almost solves the alignment problem. I, personally, still prefer version 1, but 1A is a close second.  — Arthur Rubin  (talk) 18:27, 19 April 2013 (UTC)
 * Sorry about 2A; I thought there is some HTML way to render text but make it invisble; Anyone is welcome to edit the HTML to fix the problem, but I think it looks better with the explicit 601 than without.
 * Version 1A is closer to version 2 in character size, and version 2A almost solves the alignment problem. I, personally, still prefer version 1, but 1A is a close second.  — Arthur Rubin  (talk) 18:27, 19 April 2013 (UTC)

Which raises a question: What is the current state of the use of MathJax in Wikipedia pages? If that's what they use on math.stackexchange.com, it works really well. Is it still not the default here? Michael Hardy (talk) 22:36, 19 April 2013 (UTC)

I don't know the numbers, but I gave up on MathJax as being too incredibly slow to render, for instance, using the built in browser on my iPad 3. Some of the meatier math articles were taking 15-20 seconds to render. --Mark viking (talk) 15:53, 25 April 2013 (UTC)

This code will do the job:

83.28.151.17 (talk) 08:10, 4 July 2014 (UTC)


 * Thanks for the suggestion; it looked fine on my desktop machine but when I checked on my mobile device (an android bssed Nexus tablet) the presentation was -- how shall I say it -- strange. Looks like we still need to hunt an optimum format. --SteveMcCluskey (talk) 15:23, 4 July 2014 (UTC)


 * It looks fine in both Windows desktop and mobile android. 85.193.197.7 (talk) 21:06, 12 July 2014 (UTC)

Quick Question
Reading this article and some of the posts that other readers have placed on its talk page made me curious as to whether one could call the Babylonian number system a 'decimal-encoded sexagesimal system.' Would this idea have any merit either in this article or within the greater scientific community?

—&thinsp;RandomDSdevel (talk) 21:32, 5 September 2013 (UTC)
 * Because values of sexagesimal digits are encoded in decimal, aren’t they? In a fashion virtually identical to binary-coded decimal. Incnis Mrsi (talk) 15:01, 9 September 2013 (UTC)


 * "Decimal-encoded" doesn't seem to describe the Babylonian system (although it may describe the modern form in which Babylonian numbers are written using Arabic numerals). In Babylonian notation the units 1 to 9, and the tens 10 through 50, are each represented as simple tallies using different markers for the units (Y) and tens (<) places.  However, the resulting combined numbers in the range 1 to 59 are used in a sexagesimal place value system.  I don't know if there is a simple term to describe such a hybrid system of notation and I'll leave it to the experts on mathematical terminology to sort that out. SteveMcCluskey (talk) 19:14, 9 September 2013 (UTC)
 * The modern form is more like a mixed radix system that alternates between bases 6 and 10. In half of the positions, only digits 0-5 are used, rather than the full system of digits 0-9. —David Eppstein (talk) 19:34, 9 September 2013 (UTC)
 * Not exactly. First, see leap second. Second, do you feel “72h 00m 00s” is more abnormal than “48h 00m 00s”? Incnis Mrsi (talk) 17:28, 12 September 2013 (UTC)
 * In an ideal world, ΔT would always be zero, and leap seconds would never be necessary. They are a kludge to fit things in, just like leap days, except that they are so small that practically nobody really cares in real life. So I claim the first is a non issue.
 * The only reason why I don't feel the former is more abnormal is that the hours place isn't sexagesimal, it's quadrovigesimal. Time is not a pure sexagesimal system. So I think both are weird and would write 3 d and 2 d respectively. If there were 60 hours in a day, then I would surely find 72 h weird and 48 h not.
 * When adding and subtracting in sexagesimal, I think of it as a mixed 6-on-10 radix system. When multiplying and dividing, I tend to instead think of it as pure base 60 with two-piece digits, and use Michael DeVlieger's reciprocal divisor method. Both have their respective merits, although I suspect only 6-on-10 encoded sexagesimal à la Babylon could ever be a general-purpose base: pure sexagesimal would probably never be able to work as a base for general society. Double sharp (talk) 16:01, 10 November 2013 (UTC)

An error
When the article describes the factors of 60, it says it has 2 prime factors, but there is a third one: 2.71.204.170.66 (talk) 14:52, 11 October 2013 (UTC)
 * It says that "two, three, and five" are the prime numbers. That does not mean "two prime factors, the numbers three and five" — for that meaning, the punctuation would be different (see oxford comma). It means "the prime factors are the numbers two, three, and five". —David Eppstein (talk) 16:15, 11 October 2013 (UTC)

Table of sexagesimal reciprocals
To put in the article. Really it should have the decimal representations too for comparison, but I don't have time to add those just yet. The top half of the table (until 1/30) has been checked; the bottom half has not.

Double sharp (talk) 13:28, 11 November 2013 (UTC)


 * Why? Are these useful for something nowadays? Do you have a source for the table? —David Eppstein (talk) 22:58, 11 November 2013 (UTC)
 * In duodecimal we have such a table. It illustrates the points in the article about the large number of terminating fractions due to the three prime factors {2, 3, 5} and the fact that short repeating periods are only given by {59, 61}. And I think it counts as routine calculation?
 * About "useful for something" – not really, unless for some reason you want to find out how many minutes and seconds 1/7 of an hour is, since we don't really use sexagesimal any longer (we may use it for time, but I would say we're not fluent in it, or things like 45 sec × 7 = 5 min 15 sec wouldn't be problematic). Double sharp (talk) 02:47, 12 November 2013 (UTC)
 * (Prime factors were added on 3 July 2014 by User:83.28.151.17, not by me.) Double sharp (talk) 06:37, 22 February 2015 (UTC)

A table of reciprocals of the 5-smooth sexagesimal numbers (the ones that have finite representations) is already in the article, and such tables can be sourced and notable (because the Babylonians made and used them, and because it's useful to know how long an even fraction of an hour or minute is). But the reciprocals of all numbers as you list it above is, I think, going to be hard to source (so, likely original research, even though there is little danger of getting a simple calculation like that wrong), not as useful, and above all, long and cluttered. I think it violates WP:BALANCE: the size of the table is far out of proportion to its significance as a part of the story of sexagesimal numbers. —David Eppstein (talk) 07:46, 22 February 2015 (UTC)
 * Maybe only up to 1/20 or at the most 1/30 then? The reason is that duodecimal and senary have such tables, showing the simple representation of fractions: and since sexagesimal gets chosen for this quality even today a brief depiction of everything up to a small limit, and then the 5-smooth ones, should be enough for a brief overview. Double sharp (talk) 14:25, 24 February 2015 (UTC)
 * OK, I added the reciprocals of all numbers up till 20 (the recurring ones separately). That ought to be enough to get the idea, without bloating the page excessively. Double sharp (talk) 04:40, 19 March 2015 (UTC)

Roman numeral superscripts?
I have this belief that somewhere along the line, sexagesimal fractions were written by some scholars using Roman numeral superscripts for the parts, with an invented "0", like this:


 * 120 23I 34II 45III 56IV

Furthermore, this notation either influenced or was influenced by the familiar degree sign and single, double, triple prime notation, due to the obvious visual similarity. Did I make this up, or did I read it somewhere? —Steve Summit (talk) 17:55, 18 December 2015 (UTC)


 * Ah. I didn't make this up, and it's mentioned at Degree (angle). —Steve Summit (talk) 21:09, 18 December 2015 (UTC)


 * You may be right; I vaguely recall seeing it in some discussions of 17th c. astronomy (Newton? Kepler?  Halley?) but it does need a citation.  There is a citation needed flag in the Degree article.  --SteveMcCluskey (talk) 23:31, 18 December 2015 (UTC)


 * It is definitely a known notational style, it is used throughout the English language translation of the Almagest. I am viewing a version translated by R. Catesby Taliaferro.  The only question I have us whether this is the precedent for the ' and '' notations for minutes and seconds.  I am sure it is, but I am searching for confirmation (which is how I stumbled on this discussion BTW). LaurentianShield (talk) 23:58, 24 June 2017 (UTC)


 * I suspect it's a variant of the notation that Cajori attributes to Wallis's Mathesis universalis of 1657 (see the reference in the article). It makes sense that someone using Wallis's notation might change it to describe minuta quarta with a superscripted iv instead of  .  That would place the introduction of that notation somewhere around the 17th century.  --SteveMcCluskey (talk) 01:39, 25 June 2017 (UTC)


 * Found an excellent reference: Samuel Jeake A Compleat Body of Arithmetic. He explains the Roman Numeral superscript notation explicitly. I am still searching for better general explanations, but the term to search on is "Astronomicals". LaurentianShield (talk) 02:00, 25 June 2017 (UTC)

lowest common multiple 30?
it's 30 isn't it? "60 is the smallest number that is divisible by every number from 1 to 6; that is, it is the lowest common multiple of 1, 2, 3, 4, 5, and 6" — Preceding unsigned comment added by 122.108.108.108 (talk) 14:35, 25 February 2017 (UTC)
 * 30 is not divisible by 4. Double sharp (talk) 15:36, 25 February 2017 (UTC)

I had a comment on the same sentence, so I'll just add it to this section: my thinking is, "60 is the smallest number that is divisible by every number from 1 to 6" does not get to "the point"; 60 is the smallest number divisible by 3, 4 & 5 is what is important. divisibility by 6 is a "nice to have" but it just comes for free with the 5 and the two 2's in the 4. If we didn't want 5, we could go with 24 instead of 60. The point is, in understanding why 60 is an admirable base, focus on the important points. It's not that it's divisible by 6. 68.175.11.48 (talk) 18:39, 25 July 2017 (UTC)
 * If you didn't want 5, 12 would already suffice. Indeed the important ones are just 3, 4, and 5, but the others do also come as a nice bonus for free. Halves and sixths are still very useful fractions. Double sharp (talk) 23:42, 25 July 2017 (UTC)
 * you want as many as possible, and we have 5 fingers so you want 5. but wanting as many as possible is tempered by "but the base can get too big", so that's why 7 is not included (420 anyone?). But since 6 comes for free with 3 4 5, it doesn't need to be a reason. 3 4 5 is a tighter explanation. — Preceding unsigned comment added by 68.175.11.48 (talk) 03:56, 2 August 2017 (UTC)
 * We need reliable sources, not the opinions of random anonymous people on the internet. According to some sources I've found, the argument that 60 was chosen because of its divisibility goes back to Theon of Alexandria, but beyond that the details get muddy. One popularized book reports that it was because 60 is the LCM of {1,2,3,4,5} (note: includes the numbers 1 and 2 which also comes free with the others), but another says it's because 60 is a highly composite number (more divisors than anything else so small), and other more scholarly sources that I've found are also more vague. Perhaps Theon was himself vague. —David Eppstein (talk) 04:53, 2 August 2017 (UTC)
 * To the extent that this issue implies a rational mathematical origin for the sexagesimal system, Neugebauer, History of Ancient Mathematical Astronomy, vol. 2, p. 589, n.1 has the following on the origins of the sexagesimal system:
 * "Contrary to a widespread belief the sexagesimal system did not originate from any astronomical concept. Its beginnings go back to the earliest Mesopotamian civilization, more than a millennium before any computational astronomy existed.  Its origin can be found in the norms for weights and measures in combination with palaeographical processes which led to the place value notation which is the most characteristic element of this number system."
 * Of course, this says nothing about why 60 was chosen as the norm for weights and measures.--SteveMcCluskey (talk) 22:29, 4 August 2017 (UTC)

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Base 60
It's interesting to see the base 60 numerals: 0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWX FreddieBimble384 (talk) 07:32, 11 September 2021 (UTC)

iPhone incorrect photo (English)
On the iPhone app for the English Sexagesimal article, the main photo displayed doesn't relate to the topic. However when clicked on, a different photo, related to the article and displayed later on is displayed. This photo doesn't appear in the web version, and I'm not sure if this is an isolated issue to my device, but it was a jarring experience to open the article and see what appears to be a medical or surgical photo I wasn't prepared for. 68.186.87.217 (talk) 19:05, 13 September 2023 (UTC)


 * There was some template vandalism a week ago that might have caused this, quickly removed (within two minutes of its occurrence). I have bumped the protection level of the template in hope of preventing repeat incidents. —David Eppstein (talk) 20:32, 13 September 2023 (UTC)
 * Yeah so this is still the case. It’s weird. If I on the first time upon viewing the article tap and hold on the bad image (it only works the first time), I could see which one it is: https://commons.m.wikimedia.org/wiki/File:Lipoma_04.jpg Please be advised the image may not be for the squiemish. It’s a surgical/medical image. I’m not sure if this is a case of data corruption somewhere because the image is not in this article. Northgrove (talk) 21:27, 27 September 2023 (UTC)
 * It never was in the article itself. The vandalism was to a template used by the article. Anyway, if you're still seeing it, weeks later, it's probably in your browser cache. We can't do anything about that from this side. Maybe explicitly asking your browser to reload will help. —David Eppstein (talk) 21:30, 27 September 2023 (UTC)
 * @Northgrove If you go to Special:Preferences and click to the "gadgets" tab, there is a checkbox for "Add a 'Purge' option to the top of the page, which purges the page's cache". Ticking this will add a "*" link at the top of pages which will clear cached images, which can sometimes help fix this kind of issue. (Sometimes you may need to clear your local browser cache instead.) –jacobolus (t) 02:07, 28 September 2023 (UTC)