Talk:Sexagesimal - Knowledge

548:

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. --

2357:"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. 95: 85: 64: 31: 510:

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.

3705:"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." 1541:

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. —

1072:

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).

22: 1000:

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

1602:(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. -- 3341:

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).

3303:

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

2411:

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

1540:

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

1067:

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,

647:

Lacking a source for the article's extension of modern time notation to sexagesimal numbers in general, I think that, as an encyclopedia, Knowledge 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

637:

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

618:

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

427:

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

411:

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

369:

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.

3957:

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

1168:

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

1116:

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

1071:

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

633:

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

547:

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,

505:

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

488:

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

377:

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

3620:

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

2415:

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

1075:

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

373:

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

3885:

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

1601:

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

1195:

I just revised the recent addition on "Facts on the Calculation of Degrees and Minutes" to make it accurately reflect the source, but after doing it, I'm uncertain whether it contributes much to the article. If anyone else feels it's superfluous, go ahead and delete the section and I won't be the

509:

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

283:

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.

1561:

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 {{math}} template I used in version two). Version two, on the other hand,

1557:

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

1515:

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

1509:

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

3650:

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.

1576:

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

517:

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

385:

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

1696:

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

2308:

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

999:

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

469:

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

1971: 1151:, VI,7 (p. 302 of the Toomer translation) has "we assumed that the ratio of the circumference to the diameter 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. 3934:

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.

3495:. 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). 1838: 1351: 315:

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.

521:

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

1100:

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.

830:

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

604:, 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? -- 572:

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

513:

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-:

3299:

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

2407:

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

3621:

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.

3380:

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.

798:

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.

232:

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.

2267:

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.

446:

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

1697:

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 Knowledge displays equations, i.e., I think it should be the same for me and for unregistered users.) --

288:

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

381:

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...

3680:, 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 952:

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

770:, since, as far as I can see, it is just a description of digits used in a fictional universe created by the author of the website? The “Community“ in the name of the website is just a fictional community established in 2057… 3901:

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.

3425:

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?

2483:

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.

342:

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.

2160:

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.

865:

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.

1849: 920:

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 :

2125:

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 60 than without.

3463:

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. --

2145:

Which raises a question: What is the current state of the use of MathJax in Knowledge pages? If that's what they use on math.stackexchange.com, it works really well. Is it still not the default here?

1724: 1237: 3636:

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.

3920:

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.

151: 2459:

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

975:

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. --

4004: 619:

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. —

3916:

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:

3417:

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:

1387:

Version two, using the {{math}} 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.

259:(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. — 1716:

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:

701:

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. —

675: 3676:, 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 1377: 880:

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. —

638:

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,

2416:

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.

1562:

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.

3994: 3395:

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.

173:

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?

3559:. He explains the Roman Numeral superscript notation explicitly. I am still searching for better general explanations, but the term to search on is "Astronomicals". 766:

was removed as spam (although it contains a description of digits of base 60 that should manifest mathematical logic). Is there any reason to keep the other link to

196:

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?

4009: 3808: 3804: 3790: 1556:

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 <math: -->

3583:

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"

1169:

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).

3346:, 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 35: 4019: 1041:

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

141: 1492:

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.

3989: 3886:

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.

3684:(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. — 3999: 2375:

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. —

117: 4014: 3590: 739: 674:

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

644:, 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. 1581:. In the meantime, as for how to resolve an impasse with too few editors to declare a consensus: the default Knowledge convention (e.g. in 1216:

tag the Knowledge default display makes the equation appear in a varying size font which (on a large screen) is much larger than the text:

640: 455:; 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. 3076:

01,16,35,44,40,51,03,49,47,14,02,33,11,29,21,42,07,39,34,28,05,06,22,58,43,24,15,19,08,56,10,12,45,57,26,48,30,38,17,52,20,25,31,54,53,37

1966:{\displaystyle 1;24,51,10=1+{\tfrac {24}{60}}+{\tfrac {51}{60^{2}}}+{\tfrac {10}{60^{3}}}={\tfrac {30547}{21600}}\approx 1.414212\ldots } 3887: 2445: 771: 213: 3658: 3622: 2287: 2253: 489:

use for time, degrees, etc., I think that would seriously impair the readability of the artcle for a large fraction of its audience. —

240: 1833:{\displaystyle 1;24,51,10=1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}={\frac {30547}{21600}}\approx 1.414212\ldots } 1346:{\displaystyle 1;24,51,10=1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}={\frac {30547}{21600}}\approx 1.414212\ldots } 3786:

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

108: 69: 3529:

with a superscripted instead of ''''. That would place the introduction of that notation somewhere around the 17th century. --

3984: 3756: 938: 1215:

There have been two different versions of some mathematical equations in this article. In version one using the <math: -->

184:

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

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of 1657 (see the reference in the article). It makes sense that someone using Wallis's notation might change it to describe

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decimal and the ones that are in sexagesimal. If we included letters we will be able to show repeating decimals more easily.

1488:, it cannot be expressed exactly in sexagesimal numbers, but its sexagesimal expansion does begin 1;24,51,10,7,46,6,4,44 ... 1383:, it cannot be expressed exactly in sexagesimal numbers, but its sexagesimal expansion does begin 1;24,51,10,7,46,6,4,44 ... 3766: 3851: 818: 44: 2995:

01,27,48,17,33,39,30,43,54,08,46,49,45,21,57,04,23,24,52,40,58,32,11,42,26,20,29,16,05,51,13,10,14,38,02,55,36,35,07,19

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As to the 360 degrees in a circle, that goes back to Babylonian usage and AFAIK, the origins of that are not clear. --

3350:: the size of the table is far out of proportion to its significance as a part of the story of sexagesimal numbers. — 206:-I am curios as to how the Sumarians were able to build a transmitter to transmit thier mathamatics 4000 years ago? 578:

If there is no source for the article's current notation, I would recommend following the accepted practice used by

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to show this, though, again I think it would look overwhelming with deciamal digits, though that might just be me.

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Oh, sorry, I forgot I already made a section about this. Still, my point on having a propper table still stands.

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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the

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A vestige of the sexagesimal system exists in the European and Canadian dialects of the French language, : -->

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So π/3 was primary? That would certainly explain why it's 360 degrees in a circle. Thanks for the reference. —

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To the extent that this issue implies a rational mathematical origin for the sexagesimal system, Neugebauer,

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setting which you use should be made Knowledge's default, but this is certainly not the place to debate that.

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Yes I read that section but I did not think it applied here. But I guess I see how it is possible. Cheers,

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But the reciprocals of all numbers as you list it above is, I think, going to be hard to source (so, likely

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It's interesting to see the base 60 numerals: 0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWX

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When the article describes the factors of 60, it says it has 2 prime factors, but there is a third one: 2.

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where the numbers from 70 to 79 are rendered by adding a number to 60: 70, for example, renders as : -->

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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with

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

50: 3747:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit 3319: 894:

Who would ye like? Sweet? Clark Hall? Onions? A list of numbers in OE does show the change at sixty.--

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

3925: 3654: 3586: 2336: 1585:) is to not fix things that aren't broken: don't change formats without a clear consensus to do so. — 926: 806: 735: 236: 209: 930: 21: 3962: 3954: 3936: 3903: 3711: 3685: 3677: 3530: 3464: 3367: 3351: 3282: 2464: 2376: 2358: 2269: 2132:

solves the alignment problem. I, personally, still prefer version 1, but 1A is a close second. —

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on Knowledge. If you would like to participate, please visit the project page, where you can join

2313: 2147: 1698: 1635: 1578: 1170: 1118: 1077: 954: 895: 867: 549: 523: 471: 393: 344: 100: 3811:

before doing mass systematic removals. This message is updated dynamically through the template

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In short, this is nothing to do with sexagesimal and I suggest that this paragraph be removed.

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https://web.archive.org/web/20090305190121/http://www.ethnomath.org/resources/bowers1977.pdf

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Of course, this says nothing about why 60 was chosen as the norm for weights and measures.--

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https://web.archive.org/web/20070928061154/http://www.uog.ac.pg/glec/thesis/ch4web/ch4.htm

1403:, was approximated by the Babylonians of the Old Babylonian Period (1900 BC – 1650 BC) as 678:. Rather then just revert I wanted to discuss the issue. The exact quote is as follows: 583: 279: 1232:, was approximated by the Babylonians of the Old Babylonian Period (1900 BC – 1650 BC) as 836: 3491:

a known notational style, it is used throughout the English language translation of the

3959: 3793:, "External links modified" talk page sections are no longer generated or monitored by 3442: 294: 3833:

If you found an error with any archives or the URLs themselves, you can fix them with

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https://web.archive.org/web/20070905170848/http://www.uog.ac.pg/glec/thesis/thesis.htm

3978: 3450: 3431: 1510:(apparently because of the lack of a superscript marking an exponent in one of them). 3760: 1045:

or not-well-used notations when we have a perfectly good standard notation to use. —

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and so is a de facto delimiter for sexagesimal digits. So we could have for example

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Why? Are these useful for something nowadays? Do you have a source for the table? —

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02,04,08,16,33,06,12,24,49,39,18,37,14,28,57,55,51,43,26,53,47,35,10,20,41,22,45,31

2839:

02,04,08,16,33,06,12,24,49,39,18,37,14,28,57,55,51,43,26,53,47,35,10,20,41,22,45,31

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it means it's one of the best at representing fractions, especially as it's also a

716: 687: 429: 256: 3780: 3702:, vol. 2, p. 589, n.1 has the following on the origins of the sexagesimal system: 3556: 3800: 3740: 2390: 2372: 1400: 1229: 1028: 1013: 810: 579: 568:

There is an accepted scholarly notation for sexagesimal numbers that I recently

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01,07,55,28,18,06,47,32,49,48,40,45,16,58,52,04,31,41,53,12,27,10,11,19,14,43

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In this article, all sexagesimal digits are represented as decimal numbers...

3521:

I suspect it's a variant of the notation that Cajori attributes to Wallis's

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Certainly, if you tuned your account for the use of MathJax, but neglect to

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C=30, D=40, E=50. A number like 53 is written as E3. The number 1 3 is 63.

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soixante-dix (sixty-ten), and 75 is called soixante-quinze (sixty-fifteen).

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I think you'll get a more informed opinion on mathematics formatting from

3446: 3427: 2393:. Second, do you feel “72h 00m 00s” is more abnormal than “48h 00m 00s”? 1396: 1225: 1147: 178: 759: 2463:). It means "the prime factors are the numbers two, three, and five". — 2332: 1537: 732:

Just say base 10 decimal numbers and it will not be a self reference.

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Version 1A is closer to version 2 in character size, and version 2A

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alternating base of 12*10, so these things occur in ordinary life.--

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Maybe only up to 1/20 or at the most 1/30 then? The reason is that

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I don't know if my opinion should be taken seriously, as I've been

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The supposed version dating to the 3rd century BCE is probably the

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18,15,39,07,49,33,54,46,57,23,28,41,44,20,52,10,26,05,13,02,36,31

2764:

02,36,31,18,15,39,07,49,33,54,46,57,23,28,41,44,20,52,10,26,05,13

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is quite natural in regards to say, criss-cross multiplication.

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01,13,28,09,47,45,18,22,02,26,56,19,35,30,36,44,04,53,52,39,11

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01,23,43,15,20,55,48,50,13,57,12,33,29,18,08,22,19,32,05,34,53

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In balance, version two looks better to me on both machines.

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for additional information. I made the following changes:

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It looks fine in both Windows desktop and mobile android.

789:

The paragraph about modern French is not really correct:

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it was used to show there are no stones on the abacus.

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Ref: O Neugebauer "the exact sciences in antiquity" --

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03,09,28,25,15,47,22,06,18,56,50,31,34,44,12,37,53,41

2956:

34,44,12,37,53,41,03,09,28,25,15,47,22,06,18,56,50,31

2711:

03,09,28,25,15,47,22,06,18,56,50,31,34,44,12,37,53,41

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Today the symbol ':' is used for times (HH:MM:SS) in

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Knowledge:Manual of Style (self-references to avoid)

112:, a collaborative effort to improve the coverage of 3803:using the archive tool instructions below. Editors 2335:, aren’t they? In a fashion virtually identical to 3918:https://commons.m.wikimedia.org/File:Lipoma_04.jpg 3441:Ah. I didn't make this up, and it's mentioned at 1965: 1832: 1371: 1345: 3761:http://www.ethnomath.org/resources/bowers1977.pdf 600:I did some further checking and found that until 1620:. Hope this helps bring things to resolution. 451:under this format, but this article isn't about 4005:Knowledge level-5 vital articles in Mathematics 3781:http://www.uog.ac.pg/glec/thesis/ch4web/ch4.htm 3789:This message was posted before February 2018. 1616:To get additional input, I've added a note at 641:Episodes from the Early History of Mathematics 8: 3318:(Prime factors were added on 3 July 2014 by 3771:http://www.uog.ac.pg/glec/thesis/thesis.htm 3652: 3584: 3557:Samuel Jeake A Compleat Body of Arithmetic 58: 3739:I have just modified 3 external links on 3700:History of Ancient Mathematical Astronomy 2327:Because values of sexagesimal digits are 1942: 1930: 1920: 1908: 1898: 1883: 1851: 1811: 1800: 1791: 1780: 1771: 1758: 1726: 1362: 1360: 1324: 1313: 1304: 1293: 1284: 1271: 1239: 1008:, so I'm all for a table like the one on 686:How is that line not a self-references? 2486: 518:criss-cross multiplication is under way. 3995:Knowledge vital articles in Mathematics 1516:serif font that looks like Times Roman. 634:apply generally to sexagesimal numbers. 60: 19: 4010:C-Class vital articles in Mathematics 374:shocks (of 60 in number), giving 64. 7: 3958:your local browser cache instead.) – 1674:set up a good appearance of {{math}} 293:itself used sexagesimal notation. — 106:This article is within the scope of 2942:01,37,17,50,16,12,58,22,42,09,43,47 1677:. Note that for unregistered users 49:It is of interest to the following 14: 4020:Mid-priority mathematics articles 3743:. Please take a moment to review 126:Knowledge:WikiProject Mathematics 3990:Knowledge level-5 vital articles 3881:iPhone incorrect photo (English) 2479:Table of sexagesimal reciprocals 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 2371:The modern form is more like a 146:This article has been rated as 4000:C-Class level-5 vital articles 3966:02:07, 28 September 2023 (UTC) 3945:21:30, 27 September 2023 (UTC) 3930:21:27, 27 September 2023 (UTC) 3912:20:32, 13 September 2023 (UTC) 3896:19:05, 13 September 2023 (UTC) 3876:07:32, 11 September 2021 (UTC) 3555:Found an excellent reference: 2403:17:28, 12 September 2013 (UTC) 768:Community Standard Sexagesimal 670:Self-reference in introduction 648:to pure numbers such as Pi. -- 1: 3615:15:36, 25 February 2017 (UTC) 3599:14:35, 25 February 2017 (UTC) 3473:23:31, 18 December 2015 (UTC) 3455:21:09, 18 December 2015 (UTC) 3436:17:55, 18 December 2015 (UTC) 3391:14:25, 24 February 2015 (UTC) 3360:07:46, 22 February 2015 (UTC) 3332:06:37, 22 February 2015 (UTC) 3314:02:47, 12 November 2013 (UTC) 3291:22:58, 11 November 2013 (UTC) 3276:13:28, 11 November 2013 (UTC) 2864:01,56,07,44,30,58,03,52,15,29 2426:16:01, 10 November 2013 (UTC) 2385:19:34, 9 September 2013 (UTC) 2367:19:14, 9 September 2013 (UTC) 2349:15:01, 9 September 2013 (UTC) 2322:21:32, 5 September 2013 (UTC) 1211:Equation formatting question. 1055:19:17, 30 November 2011 (UTC) 1037:17:45, 30 November 2011 (UTC) 1022:17:34, 30 November 2011 (UTC) 995:About the two digit digits... 904:07:14, 16 February 2011 (UTC) 890:07:44, 15 February 2011 (UTC) 876:07:33, 15 February 2011 (UTC) 855:17:04, 14 February 2011 (UTC) 845:I agree with removing this. — 841:15:07, 14 February 2011 (UTC) 823:12:17, 14 February 2011 (UTC) 780:12:56, 30 December 2010 (UTC) 725:18:44, 12 November 2010 (UTC) 711:06:31, 11 November 2010 (UTC) 696:23:36, 10 November 2010 (UTC) 558:08:26, 4 September 2010 (UTC) 532:10:58, 2 September 2009 (UTC) 499:14:03, 1 September 2009 (UTC) 480:07:40, 1 September 2009 (UTC) 386:hundred+sixty+thirty+two). 353:11:02, 2 September 2009 (UTC) 228:Sexagesimals in Ancient India 120:and see a list of open tasks. 4015:C-Class mathematics articles 3857:13:48, 6 December 2017 (UTC) 2473:16:15, 11 October 2013 (UTC) 2454:14:52, 11 October 2013 (UTC) 1638:. I've added a note there. — 1179:07:18, 5 December 2012 (UTC) 1164:20:55, 4 December 2012 (UTC) 1141:07:41, 4 December 2012 (UTC) 1127:06:51, 4 December 2012 (UTC) 1111:22:07, 2 December 2012 (UTC) 1086:07:50, 1 December 2011 (UTC) 985:22:47, 25 October 2012 (UTC) 922:59 in the seconds columns. 785:Sexagesimal in modern French 658:20:07, 28 October 2012 (UTC) 629:03:40, 28 October 2012 (UTC) 614:02:15, 28 October 2012 (UTC) 596:22:44, 25 October 2012 (UTC) 378:interpolating this table. 222:05:53, 28 January 2008 (UTC) 193:24 March 2004, 21 July 2004 3443:Degree (angle)#Subdivisions 3413:Roman numeral superscripts? 2496:Sexagesimal representation 2175:This code will do the job: 1372:{\displaystyle {\sqrt {2}}} 764:Kubaz's mathematical corner 465:21:37, 27 August 2009 (UTC) 442:20:59, 27 August 2009 (UTC) 422:17:58, 27 August 2009 (UTC) 402:12:07, 27 August 2009 (UTC) 4036: 3820:(last update: 5 June 2024) 3736:Hello fellow Wikipedians, 3720:22:29, 4 August 2017 (UTC) 3694:04:53, 2 August 2017 (UTC) 3667:03:56, 2 August 2017 (UTC) 3605:30 is not divisible by 4. 3579:lowest common multiple 30? 3405:04:40, 19 March 2015 (UTC) 2171:15:53, 25 April 2013 (UTC) 2156:22:36, 19 April 2013 (UTC) 2140:18:27, 19 April 2013 (UTC) 1707:17:53, 18 April 2013 (UTC) 1691:17:45, 18 April 2013 (UTC) 1662:15:49, 18 April 2013 (UTC) 1648:15:36, 18 April 2013 (UTC) 1630:15:29, 18 April 2013 (UTC) 1612:23:21, 17 April 2013 (UTC) 1595:22:29, 17 April 2013 (UTC) 1572:21:01, 17 April 2013 (UTC) 1551:23:10, 14 April 2013 (UTC) 1529:22:42, 14 April 2013 (UTC) 1502:22:28, 14 April 2013 (UTC) 1206:17:46, 31 March 2013 (UTC) 963:07:39, 28 April 2011 (UTC) 943:15:08, 26 April 2011 (UTC) 563:A standard notation exists 365:Sixty as a division-system 326:22:36, 22 April 2008 (UTC) 303:20:47, 22 April 2008 (UTC) 269:16:55, 22 April 2008 (UTC) 249:16:50, 22 April 2008 (UTC) 3646:23:42, 25 July 2017 (UTC) 3631:18:39, 25 July 2017 (UTC) 3569:02:00, 25 June 2017 (UTC) 3539:01:39, 25 June 2017 (UTC) 3505:23:58, 24 June 2017 (UTC) 2296:21:06, 12 July 2014 (UTC) 748:17:21, 3 March 2011 (UTC) 201:11:26, 16 June 2006 (UTC) 145: 78: 57: 2389:Not exactly. First, see 2278:15:23, 4 July 2014 (UTC) 2262:08:10, 4 July 2014 (UTC) 152:project's priority scale 3732:External links modified 3682:highly composite number 3126:10,35,17,38,49,24,42,21 2903:45,52,56,28,14,07,03,31 2686:03,31,45,52,56,28,14,07 1618:Knowledge:Third opinion 1006:highly composite number 109:WikiProject Mathematics 3985:C-Class vital articles 1967: 1834: 1373: 1347: 684: 1968: 1835: 1374: 1348: 758:The external link to 680: 36:level-5 vital article 3801:regular verification 3523:Mathesis universalis 2337:binary-coded decimal 1850: 1725: 1536:On my machine, with 1395:, the length of the 1359: 1238: 1224:, the length of the 132:mathematics articles 3955:Special:Preferences 3791:After February 2018 3678:Theon of Alexandria 255:Supply appropriate 3845:InternetArchiveBot 3796:InternetArchiveBot 1963: 1952: 1937: 1915: 1893: 1830: 1369: 1343: 101:Mathematics portal 45:content assessment 3821: 3669: 3657:comment added by 3601: 3589:comment added by 3344:original research 3320:User:83.28.151.17 3265: 3264: 2182:1;24,51,10 = 1 + 2055:1;24,51,10 = 1 + 1981:1;24,51,10 = 1 + 1951: 1936: 1914: 1892: 1819: 1806: 1786: 1766: 1681:are unavailable. 1486:irrational number 1407:1;24,51,10 = 1 + 1381:irrational number 1367: 1332: 1319: 1299: 1279: 1196:least offended. 946: 929:comment added by 826: 809:comment added by 738:comment added by 586:, and others. -- 515:octal, or 10: --> 251: 239:comment added by 224: 212:comment added by 166: 165: 162: 161: 158: 157: 4027: 3868:FreddieBimble384 3855: 3846: 3819: 3818: 3797: 3674:reliable sources 3561:LaurentianShield 3497:LaurentianShield 3371: 3261: 3236: 3222: 3208: 3194: 3180: 3155: 3141: 3127: 3102: 3077: 3063: 3038: 3024: 3010: 2996: 2971: 2957: 2943: 2918: 2904: 2890: 2865: 2840: 2826: 2801: 2765: 2751: 2737: 2712: 2687: 2651: 2637: 2612: 2565: 2487: 2247: 2245: 2243: 2242: 2239: 2236: 2229: 2227: 2226: 2223: 2220: 2213: 2211: 2210: 2207: 2204: 2197: 2195: 2194: 2191: 2188: 2120: 2118: 2116: 2115: 2112: 2109: 2102: 2100: 2099: 2096: 2093: 2086: 2084: 2083: 2080: 2077: 2070: 2068: 2067: 2064: 2061: 2046: 2044: 2042: 2041: 2038: 2035: 2028: 2026: 2025: 2022: 2019: 2012: 2010: 2009: 2006: 2003: 1996: 1994: 1993: 1990: 1987: 1972: 1970: 1969: 1964: 1953: 1944: 1938: 1935: 1934: 1922: 1916: 1913: 1912: 1900: 1894: 1885: 1839: 1837: 1836: 1831: 1820: 1812: 1807: 1805: 1804: 1792: 1787: 1785: 1784: 1772: 1767: 1759: 1483: 1482: 1472: 1470: 1468: 1467: 1464: 1461: 1454: 1452: 1451: 1448: 1445: 1438: 1436: 1435: 1432: 1429: 1422: 1420: 1419: 1416: 1413: 1393:square root of 2 1378: 1376: 1375: 1370: 1368: 1363: 1352: 1350: 1349: 1344: 1333: 1325: 1320: 1318: 1317: 1305: 1300: 1298: 1297: 1285: 1280: 1272: 1222:square root of 2 945: 923: 825: 803: 750: 516:1000, or 18: --> 286:Vedanga Jyotisha 234: 207: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 4035: 4034: 4030: 4029: 4028: 4026: 4025: 4024: 3975: 3974: 3883: 3864: 3849: 3844: 3812: 3805:have permission 3795: 3749:this simple FaQ 3734: 3591:122.108.108.108 3581: 3415: 3365: 3259: 3234: 3220: 3206: 3192: 3178: 3153: 3139: 3125: 3100: 3075: 3061: 3036: 3022: 3008: 2994: 2969: 2955: 2941: 2916: 2902: 2888: 2863: 2838: 2824: 2799: 2763: 2749: 2735: 2710: 2685: 2649: 2635: 2610: 2563: 2481: 2442: 2306: 2240: 2237: 2234: 2233: 2231: 2224: 2221: 2218: 2217: 2215: 2208: 2205: 2202: 2201: 2199: 2192: 2189: 2186: 2185: 2183: 2181: 2113: 2110: 2107: 2106: 2104: 2097: 2094: 2091: 2090: 2088: 2081: 2078: 2075: 2074: 2072: 2065: 2062: 2059: 2058: 2056: 2054: 2039: 2036: 2033: 2032: 2030: 2023: 2020: 2017: 2016: 2014: 2007: 2004: 2001: 2000: 1998: 1991: 1988: 1985: 1984: 1982: 1980: 1926: 1904: 1848: 1847: 1796: 1776: 1723: 1722: 1480: 1478: 1465: 1462: 1459: 1458: 1456: 1449: 1446: 1443: 1442: 1440: 1433: 1430: 1427: 1426: 1424: 1417: 1414: 1411: 1410: 1408: 1406: 1357: 1356: 1309: 1289: 1236: 1235: 1213: 1193: 1098: 997: 924: 918: 831:undisputable.-- 804: 787: 756: 740:164.106.234.127 733: 672: 565: 409: 367: 291:Surya Siddhanta 280:Surya Siddhanta 230: 185: 171: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 4033: 4031: 4023: 4022: 4017: 4012: 4007: 4002: 3997: 3992: 3987: 3977: 3976: 3973: 3972: 3971: 3970: 3969: 3968: 3937:David Eppstein 3914: 3904:David Eppstein 3882: 3879: 3863: 3860: 3839: 3838: 3831: 3784: 3783: 3775:Added archive 3773: 3765:Added archive 3763: 3755:Added archive 3733: 3730: 3729: 3728: 3727: 3726: 3725: 3724: 3723: 3722: 3712:SteveMcCluskey 3708: 3707: 3706: 3686:David Eppstein 3618: 3617: 3580: 3577: 3576: 3575: 3574: 3573: 3572: 3571: 3548: 3547: 3546: 3545: 3544: 3543: 3542: 3541: 3531:SteveMcCluskey 3512: 3511: 3510: 3509: 3508: 3507: 3478: 3477: 3476: 3475: 3465:SteveMcCluskey 3458: 3457: 3423: 3422: 3421:12 23 34 45 56 3414: 3411: 3410: 3409: 3408: 3407: 3368:David Eppstein 3352:David Eppstein 3339: 3338: 3337: 3336: 3335: 3334: 3322:, not by me.) 3304:problematic). 3301: 3283:David Eppstein 3263: 3262: 3256: 3253: 3249: 3248: 3245: 3242: 3238: 3237: 3231: 3228: 3224: 3223: 3217: 3214: 3210: 3209: 3203: 3200: 3196: 3195: 3189: 3186: 3182: 3181: 3179:05,27,16,21,49 3175: 3172: 3168: 3167: 3164: 3161: 3157: 3156: 3150: 3147: 3143: 3142: 3136: 3133: 3129: 3128: 3122: 3119: 3115: 3114: 3111: 3108: 3104: 3103: 3097: 3094: 3090: 3089: 3086: 3083: 3079: 3078: 3072: 3069: 3065: 3064: 3058: 3055: 3051: 3050: 3047: 3044: 3040: 3039: 3037:21,49,05,27,16 3033: 3030: 3026: 3025: 3019: 3016: 3012: 3011: 3005: 3002: 2998: 2997: 2991: 2988: 2984: 2983: 2980: 2977: 2973: 2972: 2966: 2963: 2959: 2958: 2952: 2949: 2945: 2944: 2938: 2935: 2931: 2930: 2927: 2924: 2920: 2919: 2913: 2910: 2906: 2905: 2899: 2896: 2892: 2891: 2889:49,05,27,16,21 2885: 2882: 2878: 2877: 2874: 2871: 2867: 2866: 2860: 2857: 2853: 2852: 2849: 2846: 2842: 2841: 2835: 2832: 2828: 2827: 2821: 2818: 2814: 2813: 2810: 2807: 2803: 2802: 2796: 2793: 2789: 2788: 2785: 2782: 2778: 2777: 2774: 2771: 2767: 2766: 2760: 2757: 2753: 2752: 2750:43,38,10,54,32 2746: 2743: 2739: 2738: 2732: 2729: 2725: 2724: 2721: 2718: 2714: 2713: 2707: 2704: 2700: 2699: 2696: 2693: 2689: 2688: 2682: 2679: 2675: 2674: 2671: 2668: 2664: 2663: 2660: 2657: 2653: 2652: 2646: 2643: 2639: 2638: 2632: 2629: 2625: 2624: 2621: 2618: 2614: 2613: 2611:05,27,16,21,49 2607: 2604: 2600: 2599: 2596: 2593: 2589: 2588: 2585: 2582: 2578: 2577: 2574: 2571: 2567: 2566: 2560: 2557: 2553: 2552: 2549: 2546: 2542: 2541: 2538: 2535: 2531: 2530: 2527: 2524: 2520: 2519: 2516: 2513: 2509: 2508: 2505: 2502: 2498: 2497: 2494: 2491: 2480: 2477: 2476: 2475: 2465:David Eppstein 2441: 2438: 2437: 2436: 2435: 2434: 2433: 2432: 2431: 2430: 2429: 2428: 2413: 2409: 2377:David Eppstein 2359:SteveMcCluskey 2352: 2351: 2305: 2304:Quick Question 2302: 2301: 2300: 2299: 2298: 2281: 2280: 2270:SteveMcCluskey 2251: 2250: 2249: 2248: 2143: 2142: 2126: 2123: 2122: 2121: 2049: 2048: 2047: 1975: 1974: 1973: 1962: 1959: 1956: 1950: 1947: 1941: 1933: 1929: 1925: 1919: 1911: 1907: 1903: 1897: 1891: 1888: 1882: 1879: 1876: 1873: 1870: 1867: 1864: 1861: 1858: 1855: 1842: 1841: 1840: 1829: 1826: 1823: 1818: 1815: 1810: 1803: 1799: 1795: 1790: 1783: 1779: 1775: 1770: 1765: 1762: 1757: 1754: 1751: 1748: 1745: 1742: 1739: 1736: 1733: 1730: 1717: 1694: 1693: 1670: 1669: 1668: 1667: 1666: 1665: 1664: 1654:SteveMcCluskey 1640:David Eppstein 1622:SteveMcCluskey 1599: 1598: 1597: 1587:David Eppstein 1564:SteveMcCluskey 1559: 1543:David Eppstein 1534: 1533: 1532: 1531: 1521:SteveMcCluskey 1517: 1512: 1511: 1494:SteveMcCluskey 1490: 1489: 1475: 1474: 1473: 1385: 1384: 1366: 1353: 1342: 1339: 1336: 1331: 1328: 1323: 1316: 1312: 1308: 1303: 1296: 1292: 1288: 1283: 1278: 1275: 1270: 1267: 1264: 1261: 1258: 1255: 1252: 1249: 1246: 1243: 1233: 1212: 1209: 1198:SteveMcCluskey 1192: 1191:External links 1189: 1188: 1187: 1186: 1185: 1184: 1183: 1182: 1181: 1156:SteveMcCluskey 1152: 1133:David Eppstein 1103:David Eppstein 1097: 1094: 1093: 1092: 1091: 1090: 1089: 1088: 1073: 1069: 1060: 1059: 1058: 1057: 1047:David Eppstein 996: 993: 992: 991: 990: 989: 988: 987: 977:SteveMcCluskey 968: 967: 966: 965: 917: 914: 913: 912: 911: 910: 909: 908: 907: 906: 882:David Eppstein 860: 859: 858: 857: 847:David Eppstein 786: 783: 755: 754:External links 752: 730: 729: 728: 727: 703:David Eppstein 671: 668: 667: 666: 665: 664: 663: 662: 661: 660: 650:SteveMcCluskey 645: 635: 621:David Eppstein 606:SteveMcCluskey 588:SteveMcCluskey 575: 574: 564: 561: 545: 544: 543: 542: 541: 540: 539: 538: 537: 536: 535: 534: 519: 511: 507: 491:David Eppstein 434:David Eppstein 408: 405: 366: 363: 362: 361: 360: 359: 358: 357: 356: 355: 333: 332: 331: 330: 329: 328: 318:David Eppstein 308: 307: 306: 305: 272: 271: 261:David Eppstein 229: 226: 205: 183: 175: 170: 167: 164: 163: 160: 159: 156: 155: 144: 138: 137: 135: 118:the discussion 105: 104: 88: 76: 75: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 4032: 4021: 4018: 4016: 4013: 4011: 4008: 4006: 4003: 4001: 3998: 3996: 3993: 3991: 3988: 3986: 3983: 3982: 3980: 3967: 3964: 3961: 3956: 3953:If you go to 3952: 3948: 3947: 3946: 3942: 3938: 3933: 3932: 3931: 3927: 3923: 3919: 3915: 3913: 3909: 3905: 3900: 3899: 3898: 3897: 3893: 3889: 3888:68.186.87.217 3880: 3878: 3877: 3873: 3869: 3861: 3859: 3858: 3853: 3848: 3847: 3836: 3832: 3829: 3825: 3824: 3823: 3816: 3810: 3806: 3802: 3798: 3792: 3787: 3782: 3778: 3774: 3772: 3768: 3764: 3762: 3758: 3754: 3753: 3752: 3750: 3746: 3742: 3737: 3731: 3721: 3717: 3713: 3709: 3704: 3703: 3701: 3697: 3696: 3695: 3691: 3687: 3683: 3679: 3675: 3671: 3670: 3668: 3664: 3660: 3656: 3649: 3648: 3647: 3643: 3639: 3635: 3634: 3633: 3632: 3628: 3624: 3616: 3612: 3608: 3604: 3603: 3602: 3600: 3596: 3592: 3588: 3578: 3570: 3566: 3562: 3558: 3554: 3553: 3552: 3551: 3550: 3549: 3540: 3536: 3532: 3528: 3527:minuta quarta 3524: 3520: 3519: 3518: 3517: 3516: 3515: 3514: 3513: 3506: 3502: 3498: 3494: 3490: 3489: 3484: 3483: 3482: 3481: 3480: 3479: 3474: 3470: 3466: 3462: 3461: 3460: 3459: 3456: 3452: 3448: 3444: 3440: 3439: 3438: 3437: 3433: 3429: 3420: 3419: 3418: 3412: 3406: 3402: 3398: 3394: 3393: 3392: 3388: 3384: 3379: 3375: 3369: 3364: 3363: 3362: 3361: 3357: 3353: 3349: 3345: 3333: 3329: 3325: 3321: 3317: 3316: 3315: 3311: 3307: 3302: 3298: 3294: 3293: 3292: 3288: 3284: 3280: 3279: 3278: 3277: 3273: 3269: 3257: 3254: 3251: 3250: 3246: 3243: 3240: 3239: 3232: 3229: 3226: 3225: 3218: 3215: 3212: 3211: 3204: 3201: 3198: 3197: 3190: 3187: 3184: 3183: 3176: 3173: 3170: 3169: 3165: 3162: 3159: 3158: 3151: 3148: 3145: 3144: 3137: 3134: 3131: 3130: 3123: 3120: 3117: 3116: 3112: 3109: 3106: 3105: 3098: 3095: 3092: 3091: 3087: 3084: 3081: 3080: 3073: 3070: 3067: 3066: 3059: 3056: 3053: 3052: 3048: 3045: 3042: 3041: 3034: 3031: 3028: 3027: 3020: 3017: 3014: 3013: 3006: 3003: 3000: 2999: 2992: 2989: 2986: 2985: 2981: 2978: 2975: 2974: 2967: 2964: 2961: 2960: 2953: 2950: 2947: 2946: 2939: 2936: 2933: 2932: 2928: 2925: 2922: 2921: 2914: 2911: 2908: 2907: 2900: 2897: 2894: 2893: 2886: 2883: 2880: 2879: 2875: 2872: 2869: 2868: 2861: 2858: 2855: 2854: 2850: 2847: 2844: 2843: 2836: 2833: 2830: 2829: 2822: 2819: 2816: 2815: 2811: 2808: 2805: 2804: 2797: 2794: 2791: 2790: 2786: 2783: 2780: 2779: 2775: 2772: 2769: 2768: 2761: 2758: 2755: 2754: 2747: 2744: 2741: 2740: 2733: 2730: 2727: 2726: 2722: 2719: 2716: 2715: 2708: 2705: 2702: 2701: 2697: 2694: 2691: 2690: 2683: 2680: 2677: 2676: 2672: 2669: 2666: 2665: 2661: 2658: 2655: 2654: 2647: 2644: 2641: 2640: 2633: 2630: 2627: 2626: 2622: 2619: 2616: 2615: 2608: 2605: 2602: 2601: 2597: 2594: 2591: 2590: 2586: 2583: 2580: 2579: 2575: 2572: 2569: 2568: 2561: 2558: 2555: 2554: 2550: 2547: 2544: 2543: 2539: 2536: 2533: 2532: 2528: 2525: 2522: 2521: 2517: 2514: 2511: 2510: 2506: 2503: 2500: 2499: 2495: 2493:Prime factors 2492: 2489: 2488: 2485: 2478: 2474: 2470: 2466: 2462: 2458: 2457: 2456: 2455: 2451: 2447: 2446:71.204.170.66 2439: 2427: 2423: 2419: 2414: 2412:and 48 h not. 2410: 2406: 2405: 2404: 2400: 2396: 2392: 2388: 2387: 2386: 2382: 2378: 2374: 2370: 2369: 2368: 2364: 2360: 2356: 2355: 2354: 2353: 2350: 2346: 2342: 2338: 2334: 2330: 2326: 2325: 2324: 2323: 2319: 2315: 2314:RandomDSdevel 2310: 2309:community? 2303: 2297: 2293: 2289: 2285: 2284: 2283: 2282: 2279: 2275: 2271: 2266: 2265: 2264: 2263: 2259: 2255: 2180: 2179: 2178: 2177: 2176: 2173: 2172: 2168: 2164: 2158: 2157: 2153: 2149: 2148:Michael Hardy 2141: 2138: 2135: 2131: 2127: 2124: 2053: 2052: 2050: 1979: 1978: 1976: 1960: 1957: 1954: 1948: 1945: 1939: 1931: 1927: 1923: 1917: 1909: 1905: 1901: 1895: 1889: 1886: 1880: 1877: 1874: 1871: 1868: 1865: 1862: 1859: 1856: 1853: 1846: 1845: 1843: 1827: 1824: 1821: 1816: 1813: 1808: 1801: 1797: 1793: 1788: 1781: 1777: 1773: 1768: 1763: 1760: 1755: 1752: 1749: 1746: 1743: 1740: 1737: 1734: 1731: 1728: 1721: 1720: 1718: 1715: 1711: 1710: 1709: 1708: 1704: 1700: 1692: 1688: 1684: 1680: 1675: 1671: 1663: 1659: 1655: 1651: 1650: 1649: 1645: 1641: 1637: 1633: 1632: 1631: 1627: 1623: 1619: 1615: 1614: 1613: 1609: 1605: 1600: 1596: 1592: 1588: 1584: 1580: 1575: 1574: 1573: 1569: 1565: 1560: 1555: 1554: 1553: 1552: 1548: 1544: 1539: 1530: 1526: 1522: 1518: 1514: 1513: 1508: 1507: 1506: 1505: 1504: 1503: 1499: 1495: 1487: 1476: 1405: 1404: 1402: 1398: 1394: 1390: 1389: 1388: 1382: 1364: 1354: 1340: 1337: 1334: 1329: 1326: 1321: 1314: 1310: 1306: 1301: 1294: 1290: 1286: 1281: 1276: 1273: 1268: 1265: 1262: 1259: 1256: 1253: 1250: 1247: 1244: 1241: 1234: 1231: 1227: 1223: 1219: 1218: 1217: 1210: 1208: 1207: 1203: 1199: 1190: 1180: 1176: 1172: 1171:Wendy.krieger 1167: 1166: 1165: 1161: 1157: 1153: 1150: 1149: 1144: 1143: 1142: 1138: 1134: 1130: 1129: 1128: 1124: 1120: 1119:Wendy.krieger 1117:discussion. 1115: 1114: 1113: 1112: 1108: 1104: 1095: 1087: 1083: 1079: 1078:Wendy.krieger 1074: 1070: 1066: 1065: 1064: 1063: 1062: 1061: 1056: 1052: 1048: 1044: 1040: 1039: 1038: 1034: 1030: 1026: 1025: 1024: 1023: 1019: 1015: 1011: 1007: 1003: 994: 986: 982: 978: 974: 973: 972: 971: 970: 969: 964: 960: 956: 955:Wendy.krieger 951: 950: 949: 948: 947: 944: 940: 936: 932: 928: 915: 905: 901: 897: 896:Wendy.krieger 893: 892: 891: 887: 883: 879: 878: 877: 873: 869: 868:Wendy.krieger 864: 863: 862: 861: 856: 852: 848: 844: 843: 842: 838: 834: 829: 828: 827: 824: 820: 816: 812: 808: 800: 796: 790: 784: 782: 781: 777: 773: 772:90.176.211.48 769: 765: 761: 753: 751: 749: 745: 741: 737: 726: 722: 718: 714: 713: 712: 708: 704: 700: 699: 698: 697: 693: 689: 683: 679: 677: 669: 659: 655: 651: 646: 643: 642: 636: 632: 631: 630: 626: 622: 617: 616: 615: 611: 607: 603: 602:these changes 599: 598: 597: 593: 589: 585: 581: 577: 576: 571: 567: 566: 562: 560: 559: 555: 551: 550:Wendy.krieger 533: 529: 525: 524:Wendy.krieger 520: 512: 508: 504: 503: 502: 501: 500: 496: 492: 487: 483: 482: 481: 477: 473: 472:Wendy.krieger 468: 467: 466: 462: 458: 454: 450: 445: 444: 443: 439: 435: 431: 426: 425: 424: 423: 419: 415: 406: 404: 403: 399: 395: 394:Wendy.krieger 390: 387: 383: 379: 375: 371: 364: 354: 350: 346: 345:Wendy.krieger 341: 340: 339: 338: 337: 336: 335: 334: 327: 323: 319: 314: 313: 312: 311: 310: 309: 304: 300: 296: 292: 287: 282: 281: 276: 275: 274: 273: 270: 266: 262: 258: 254: 253: 252: 250: 246: 242: 238: 227: 225: 223: 219: 215: 214:72.89.188.197 211: 203: 202: 199: 194: 192: 188: 182: 180: 174: 168: 153: 149: 143: 140: 139: 136: 119: 115: 111: 110: 102: 96: 91: 89: 86: 82: 81: 77: 71: 68: 65: 61: 56: 52: 46: 38: 37: 27: 23: 18: 17: 3884: 3865: 3843: 3840: 3815:source check 3794: 3788: 3785: 3738: 3735: 3699: 3659:68.175.11.48 3653:— Preceding 3638:Double sharp 3623:68.175.11.48 3619: 3607:Double sharp 3585:— Preceding 3582: 3526: 3522: 3492: 3487: 3486: 3447:Steve Summit 3428:Steve Summit 3424: 3416: 3397:Double sharp 3383:Double sharp 3340: 3324:Double sharp 3306:Double sharp 3300:calculation? 3268:Double sharp 3266: 2482: 2461:oxford comma 2443: 2418:Double sharp 2311: 2307: 2288:85.193.197.7 2254:83.28.151.17 2252: 2246:≈ 1.414212… 2174: 2159: 2144: 2134:Arthur Rubin 2129: 2119:≈ 1.414212… 2045:≈ 1.414212… 1713: 1695: 1678: 1604:Matthiaspaul 1535: 1491: 1471:≈ 1.414212… 1386: 1214: 1194: 1146: 1099: 998: 925:— Preceding 919: 916:Modern Usage 805:— Preceding 801: 797: 791: 788: 763: 757: 731: 685: 681: 673: 639: 546: 452: 448: 410: 391: 388: 384: 380: 376: 372: 368: 290: 278: 241:67.180.39.64 231: 204: 195: 189: 186: 176: 172: 169:Symbols used 148:Mid-priority 147: 107: 73:Mid‑priority 51:WikiProjects 34: 3741:Sexagesimal 3166:0;01,06,40 3140:09,13,50,46 2970:32,18,27,41 2876:0;01,52,30 2812:0;02,13,20 2800:18,27,41,32 2636:04,36,55,23 2395:Incnis Mrsi 2391:leap second 2373:mixed radix 2341:Incnis Mrsi 2163:Mark viking 1683:Incnis Mrsi 1401:unit square 1230:unit square 734:—Preceding 522:integers.-- 235:—Preceding 208:—Preceding 123:Mathematics 114:mathematics 70:Mathematics 3979:Categories 3951:Northgrove 3922:Northgrove 3852:Report bug 3488:definitely 3374:duodecimal 3348:WP:BALANCE 3297:duodecimal 1145:Ptolemy's 584:Neugebauer 3960:jacobolus 3835:this tool 3828:this tool 1583:WP:RETAIN 1002:primorial 931:TimWooley 573:notation? 295:Joe Kress 39:is rated 3841:Cheers.— 3672:We need 3655:unsigned 3587:unsigned 3493:Almagest 3193:17,08,34 3191:0;01,04, 3113:0;01,12 3088:0;01,15 3049:0;01,20 3009:25,42,51 2982:0;01,30 2929:0;01,40 2917:42,51,25 2825:08,34,17 2787:0;02,24 2776:0;02,30 2736:51,25,42 2698:0;03,20 2673:0;03,45 2650:17,08,34 2587:0;06,40 2576:0;07,30 2564:08,34,17 2490:Fraction 2440:An error 1958:1.414212 1825:1.414212 1477:Because 1397:diagonal 1355:Because 1338:1.414212 1226:diagonal 1148:Almagest 1043:original 953:range.-- 939:contribs 927:unsigned 819:contribs 807:unsigned 736:unsigned 237:unsigned 210:unsigned 179:ISO 8601 3862:Base 60 3745:my edit 2333:decimal 2329:encoded 1714:writing 1652:Fine. 1538:MathJax 1096:Ptolemy 1010:Base 30 760:Base 60 717:meshach 688:meshach 407:Symbols 257:sources 150:on the 41:C-class 3485:It is 3378:senary 2408:issue. 2137:(talk) 2130:almost 1636:WT:WPM 1579:WT:WPM 1484:is an 1379:is an 1029:Robo37 1014:Robo37 811:Komaba 457:Robo37 414:Robo37 198:ais523 47:scale. 3260:00,59 3247:0;01 3244:2 3 5 3219:0;01, 3205:0;01, 3177:0;01, 3138:0;01, 3124:0;01, 3060:0;01, 3035:0;01, 3007:0;01, 3004:2 3 7 2968:0;01, 2954:0;01, 2915:0;01, 2901:0;01, 2887:0;01, 2851:0;02 2848:2 3 5 2823:0;02, 2798:0;02, 2748:0;02, 2734:0;02, 2723:0;03 2662:0;04 2648:0;04, 2623:0;05 2598:0;06 2551:0;10 2540:0;12 2529:0;15 2518:0;20 2507:0;30 2241:21600 2235:30547 2114:21600 2108:30547 2051:(2A) 2040:21600 2034:30547 1949:21600 1946:30547 1844:(1A) 1817:21600 1814:30547 1466:21600 1460:30547 1399:of a 1330:21600 1327:30547 1228:of a 792:: --> 580:Aaboe 570:added 430:WP:OR 28:This 3941:talk 3926:talk 3908:talk 3892:talk 3872:talk 3716:talk 3690:talk 3663:talk 3642:talk 3627:talk 3611:talk 3595:talk 3565:talk 3535:talk 3501:talk 3469:talk 3451:talk 3432:talk 3401:talk 3387:talk 3376:and 3356:talk 3328:talk 3310:talk 3287:talk 3272:talk 3252:1/61 3241:1/60 3227:1/59 3216:2 29 3213:1/58 3202:3 19 3199:1/57 3185:1/56 3174:5 11 3171:1/55 3160:1/54 3146:1/53 3135:2 13 3132:1/52 3121:3 17 3118:1/51 3107:1/50 3093:1/49 3082:1/48 3068:1/47 3057:2 23 3054:1/46 3043:1/45 3032:2 11 3029:1/44 3015:1/43 3001:1/42 2987:1/41 2976:1/40 2962:1/39 2951:2 19 2948:1/38 2934:1/37 2923:1/36 2909:1/35 2898:2 17 2895:1/34 2884:3 11 2881:1/33 2870:1/32 2856:1/31 2845:1/30 2831:1/29 2817:1/28 2806:1/27 2795:2 13 2792:1/26 2781:1/25 2770:1/24 2756:1/23 2745:2 11 2742:1/22 2728:1/21 2717:1/20 2703:1/19 2692:1/18 2678:1/17 2667:1/16 2656:1/15 2642:1/14 2628:1/13 2617:1/12 2603:1/11 2592:1/10 2469:talk 2450:talk 2422:talk 2399:talk 2381:talk 2363:talk 2345:talk 2318:talk 2292:talk 2274:talk 2258:talk 2167:talk 2152:talk 1977:(2) 1719:(1) 1703:talk 1687:talk 1679:both 1658:talk 1644:talk 1626:talk 1608:talk 1591:talk 1568:talk 1547:talk 1525:talk 1498:talk 1391:The 1220:The 1202:talk 1175:talk 1160:talk 1137:talk 1123:talk 1107:talk 1082:talk 1051:talk 1033:talk 1018:talk 981:talk 959:talk 935:talk 900:talk 886:talk 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