Sexagesimal number system

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The size difference between the equations and the surrounding text in Version 1 is a little odd, but the equation is readable. Really it should have the decimal representations too for comparison, but I don't have time to add those just yet.

For example, one hour can be divided evenly into sections of 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, 6 minutes, 5 minutes, 4 minutes, 3 minutes, 2 minutes, and 1 minute. Similarly, the practical unit of angular measure is the , of which there are six sixties in a circle.

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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 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 '. What are the rules concerning examples in articles? I think there needs to be a paragraph on that. 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 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. India acquired the zero by way of the arabs, who got it from the greeks. 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. 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. Neugebauer also gives the number of the sumerians for the multiplication scale. 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. If we included letters we will be able to show repeating decimals more easily. We should be following standard conventions here, not trying to make up new and more logical conventions: see. Yes, we do express time under this format, but this article isn't about time; it's about numbers. Many of the things that i see written of this system is exactly what one would expect of an alternating-base division system. 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. 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. 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. 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. 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. 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. 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 , , and others. Does anyone know of a rationale for this change? 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. 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. 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, , p. 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. They are almost verbatim of the first item from. 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? Because this example seems almost exactly like the ones in that section to me. But I guess I see how it is possible. Cheers, 18:44, 12 November 2010 UTC Just say base 10 decimal numbers and it will not be a self reference. Is there any reason to keep the other link to , 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? 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. In short, this is nothing to do with sexagesimal and I suggest that this paragraph be removed. 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. Even though there is a base 20 substrate in french, the change at sixty should be noted. A list of numbers in OE does show the change at sixty. 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. 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 it means it's one of the best at representing fractions, especially as it's also a , so I'm all for a table like the one on to show this, though, again I think it would look overwhelming with deciamal digits, though that might just be me. Still, my point on having a propper table still stands. I'd prefer not to make up new or not-well-used notations when we have a perfectly good standard notation to use. The babylon use of two digits to represent numbers base 60, is probably no different to the romans using two digits I, V. 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. I've seen in a number of references a calculation of sixty-number, laid out with the digits evenly spaced, like 3. 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. 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. The value 6;17 is not given anywhere in the discussion. That would certainly explain why it's 360 degrees in a circle. Thanks for the reference. As to the 360 degrees in a circle, that goes back to Babylonian usage and AFAIK, the origins of that are not clear. If anyone else feels it's superfluous, go ahead and delete the section and I won't be the least offended. 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. 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. 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. 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. 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 math 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. 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. In the meantime, as for how to resolve an impasse with too few editors to declare a consensus: the default Wikipedia convention e. 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. Hope this helps bring things to resolution. I've added a note there. Note that for unregistered users both are unavailable. 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. 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. If that's what they use on math. Is it still not the default here? Some of the meatier math articles were taking 15-20 seconds to render. Looks like we still need to hunt an optimum format. In a fashion virtually identical to. 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. In half of the positions, only digits 0-5 are used, rather than the full system of digits 0-9. 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. Really it should have the decimal representations too for comparison, but I don't have time to add those just yet. Are these useful for something nowadays? Do you have a source for the 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? But the reciprocals of all numbers as you list it above is, I think, going to be hard to source so, likely , 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 : the size of the table is far out of proportion to its significance as a part of the story of sexagesimal numbers. The reason is that and 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. That ought to be enough to get the idea, without bloating the page excessively. Did I make this up, or did I read it somewhere? I didn't make this up, and it's mentioned at. There is a citation needed flag in the Degree article. I am viewing a version translated by R. 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. 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. He explains the Roman Numeral superscript notation explicitly. 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. 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. But since 6 comes for free with 3 4 5, it doesn't need to be a reason. According to some sources I've found, the argument that 60 was chosen because of its divisibility goes back to , 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 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. 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. Please take a moment to review. 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Semicolons are used to indicate columns, are not in the solo. If the current number displayed is not an integer, the result is the factorial of the integer value of the current number. The early in particular was one-sixtieth of a mana, though the Greeks later coerced this relationship into the more base-10 compatible ratio of a gusto being one-fiftieth of a. This process rounds to the nearest integer value corresponding to our input assuming the input in positive. I've done this myself and seen it in the wild more than once. Throughout their many centuries of use, which continues today for specialized topics such as met, angles, and astronomical coordinate systems, sexagesimal notations have always contained a strong undercurrent of decimal notation, such as in how sexagesimal digits are written. The Parisian version of the ca. Indeed the important ones are just 3, 4, and 5, but the others do also come as a civil bonus for free. Example: Obtain a random number between 0 and 100. Do you want to allow both 0H and 24H?.