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### Introduction

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Also ancient cultures used the letters of the alphabet in order to express digits Abjad numerals - Arabic alphabet; Hebrew numerals, Greek numerals, Armenian numerals.

See also the notation for sexatrigesimal base 36 http: In the sexagesimal notation capital or small letter matters A! Because is more intuitive for people and computers can do the conversion and sorting work very quickly. A similar notation is used with IBAN bank account validation: In this way we get some repeating "rules" and is easier to learn. See further "Rules" helping the learning curve 1. There are exactly 50 characters digits to learn expressing the numbers from 10 to These letters are grouped in two main groups: In this way we have a repeating rule.

We can group the 25 letters into 5 groups in this way we can learn at first 5 letters: The next 25 letters will be the same only as capital letters. We can group the 25 letters into 5 groups in this way we can get the same 5 letters only as capital letters: Compression of information Only notation, visual expression, the storage remains in bytes.

Nothing in this article speaks about storage. In the sexagesimal system we can express the same number large integers, float numbers in a more compact way. The greatest possible IPv6 number: Table ID-s identification numbersprimary keys, foreign keys could be expressed in sexagesimal. In such situation people don't have to compute the decimal number, it's important only to identify a row or an object in the database.

Which one is easier to process by a human brain? The system could be used in constructing ID numbers strings. Real numbers float numbers can be expressed with greater precision with fewer digits.

The Babylonians calculated as: The base 60 is divisible by the prime numbers: Results that a lot more fractions can be expressed by finite fractions. Date and time can be expressed in compact way: S in sexagesimal, 1 digit for hours, 1 digit for minutes, 1 digit for seconds Here we have a big advantage, because hours, minutes and seconds can be expressed by only one character.

Angles, longitudes, latitudes can be expressed in compact way when the context is clear and the code "0z" is missing: The code "0x" and "0o" can appear in a sexagesimal number. This can cause a wrong interpretation. Sometimes the letter "l" small L and "I" capital i is very similar to the digit "1" and the letter "O" is very similar to the digit "0".

These are old issues and should be solved by clear fonts. Other notation schemes for the sexagesimal system: There are also other proposals for the modern notation of sexagesimal numbers: Separating digits by the ": Separating digits by the "," comma and closing in square brackets. Separating digits by the "," comma and using ";" semicolon as the floating point.

S3 The Smith Sexagesimal System very difficult to learn and to adopt.

The actual square root of 2 is: Here the sexagesimal divisions continue past the "seconds" position, which is generally where modern base and Babylonian base notations diverge with the former inconsistently switching to tenths and hundredths of a second, etc.

A system with a base of 60 might appear cumbersome, but it has distinct advantages, as most modern users have come to recognize, especially with respect to the number of integer divisors and associated reciprocals, i.

Convenient divisors are not limited to integers alone, but although the equivalents of recurring decimal fractions such as 0. On a more practical note, the accurate but still rounded?

Babylonian mean synodic month of 29;31,50,8,20 days From a modern viewpoint the duration of one "tithi" would be obtained from the mindless division of 29;31,50,8,20 days by However, instead of starting with "29" and carrying out successive divisions in the modern manner, all that is necessary is to start at the other end, multiply by 2, and shift the sexagesimal place accordingly to obtain the required result of 0;59,03,40,16,40 tithi per day. This is a very trivial example; for a vastly more complex application see Friberg's "algorithm for the factorization of a regular sexagesimal number n or reduction of a semi-regular number " in his paper "On the Big 6-Place Table of Reciprocals and Squares from Seleucid Babylon and Uruk, and their Old Babylonian and Sumerian Predecessors," SUMER, Vol.

In the sexagesimal system, the extended use of reciprocals effectively reduces division and multiplication to a single operation, i. This concept was maximized by Babylonian multiplication tables which extended from 1 through 19 with further entries for 20, 30, 40, and 50 to cover the entire range between 0 and Babylonian mathematical tables were not, however, confined to simple values, as atypical multiplication table U91 Istanbul attests Aaboe, Journal of Cuneiform Studies, Vol.

Although unrecognized, U91 also appears to include the tabulated slope for the "trapezoid" mentioned in two Seleucid Era astronomical procedure texts for Jupiter ACT Section 5, and ACT Section 4 and quite possibly slopes for similar trapezoids associated with the four remaining planets.

Returning to the substitution of 30 tithi for the mean synodic month of 29;31,50,08,20 days, the use of the conveniently rounded year of 12;22,08 mean synodic months permitted Babylonian astronomers to replace the awkward unit of daily motion of 1;00,52,36,22,57,24. Applied to the motion of Jupiter this method nevertheless produced a mean synodic period of 13;30,27,46,40 mean synodic months from ACTSection 2 and 13;30,27,46 stated in Section 1 but oddly unrecognized by Neugebauer.

Once so obtained the period could finally be converted to days by multiplying by 29;31,50,08,20 the result is: The same method was employed to compute the times for the varying synodic velocities; it goes without saying that rounded parameters, convenient shortcuts, and simplified methodology suggest more than a passing acquaintance with the phenomena and problems under consideration.

More significantly and also a latter-day puzzlefew if any modern commentators appear to have recognized the most obvious feature of the above, i. In detail, the Babylonian mean synodic arc of Jupiter of 33;8,45 degrees is accomplished in Thus the mean daily motion will be 0;4,59,8,31,3,28,53.

Divided by the best estimate for the Babylonian year of It is also a slightly more accurate value 11;51,40,25,48 years than the period of As for the interval of In other words, the mean lap time i. For further details on the heliocentric nature of Babylonian planetary theory see the link below. Even years or so before the Seleucid Era Babylonian mathematicians were already utilizing tables of squares and cubes, and they were also capable of extracting cube roots, although not without peculiarities, as Sachs Journal of Cuneiform Studies, Vol.

Sant'Anna and Adonai S. It is uncertain how the Babylonians obtained their approximation for the square root of two, but it has been suggested that a Babylonian predecessor of Newton's iterative method may have been employed, albeit predating the latter by some years. Nevertheless, it seems possible that the likely combination of standard Babylonian procedures could have produced most profound results, e. Such problems in themselves are simple enough, e. The attested Babylonian solution is similar to the modern equivalent of solving a quadratic equation, albeit by algorithmic means.

Thus firstly in sexagesimal notation to the 8th sexagesimal place: Take one half of the difference 1, the result is 0;30 [ Hold the result in your hand ] 2. Take the half-difference and square it, the result is 0;15 3. Take the 0;15 and add it to the area 1, the result is 1;15 4.

Take the square root of 1;15, the result is 1;7,4,55,20,29,39,6,54 5. Add the half 0;30 from step1 to the square root, the result is 1;37,4,55,20,29,39,6,54 6. What value when multiplied by 1;37,4,55,20,29,39,6,54 gives 1 the area? Take one half of the difference 1, the result is 0. Take the half-difference and square it, the square is 0. Take the square root of 1.

What value when multiplied by 1. Thus the length and the width are found to be 1. Moreover, with unity in between, one also obtains three consecutive values from the Phi-Series itself. Not that this is to be taken as historical fact by any means, but the precise determination undoubtedly remains feasible using attested methods from the Old Babylonian period [ BCE].

On a technical note, the above treatment is linguistically imprecise and more verbose than the instructions given line-by-line in mathematical texts of the Old Babylonian Era; the use of the semi-colon to denote the equivalent of the decimal point is also a modern addition for clarity. Rounding at the sixth and fifth places would also provide useful if less accurate sexagesimal approximations for Phi of: As for the Babylonian approach itself, from a modern viewpoint one could suggest that it results from an awareness that for problems of this nature the quadratic formula: Thus the Babylonian algorithm is essentially the simplified form: Which although still unattested in the present context suggests in turn that the simplest way to obtain Phi is from the addition of one half to the square root of five-fourths.

This is not to say that this was how the procedure was arrived at by Babylonian mathematicians.

### RE: Decimal Vs Sexagesimal Notation

But what can be suggested here is that it takes a fair degree of competency whichever way one looks at it, and this proves to be true in other aspects of Babylonian methodology, including the techniques laid out in the Babylonian astronomical cuneiform texts of the much later Seleucid Era [ BCE - 75 CE ]. The Babylonians utilized "pythagorean" sets well before the time of Pythagoras, applied algorithms to solve a variety of linear equations including cubics and quadraticspossessed a knowledge of logarithms and also carried out mathematical operations of still unknown significance during the earliest period see: In some respects certain Babylonian mathematical tables from the later Seleucid Era are quite mystifying, especially those given to more than twelve sexagesimal places see: To put the latter in meaningful perspective, working to even the ninth sexagesimal place corresponds to an accuracy of better than to Babylonian computational methodology may be considered merely "arithmetical" by some, but this is surely a vast over-simplification and there are in addition enormous time-scales involved in its possible refinement and development.

Moreover, it is not known what percentage of Babylonian methodology and understanding is represented by the extant material. Consequently, what is known with respect to the astronomical cuneiform texts of the Seleucid Era should, perhaps, be judged mainly on its practical merit, namely the simple yet successful description of complex celestial phenomena associated with the mean, varying, and apparent motions of the five known planets, Sun and Moon.

Lastly, for more on the sophisticated nature of Babylonian methodology see:

So we need to be able to convert from There are a fair number of subtleties that lead to gotchas in naive code that performs these kinds of conversions.

First lets consider transformation from sexagesimal to decimal. Seems like it should be straightforward: But what about 30 We need to make the minutes and seconds add to the magnitude of the degrees. What about 30 30? Typically will be read the same as 0 so as soon as we naively converted the first string to integer we made an error. The code will misrepresent positions just south of the equator. I've done this myself and seen it in the wild more than once. To handle these values we need to explicitly look for the sign and if we find a - sign then invert the sign of our result.

This is a really easy bug to miss since it only affects a small fraction of the sky. There are a few. There's the obvious counterpart to the the one we just handled. The variable d is going to be 0 and we've lost the sign.

### sexagesimal

Bad things will happen. What we want to do is print out the sign first and then work on the absolute value of the input angle. And there is a more subtle issue that has to do with rounding Suppose we are looking at transforming Ok we have 10 degrees.

So we could represent this as 10d 29' I could try 10d 29' 59" but that's silly we should be rounding to the nearest arcsecond. So we need to round up to 10d 29' 60" Ooops I need to go back and increment the arcminutes to get 10d 30' 00" If the arcminutes had been 59 prior to the increment then I'd need to go back and increment the degrees This is a bit of a pain but it needs to be done to format the data properly. If you are doing the longitude or right ascension, you may also want to deal with the special case of what happens when you round up to degrees or 24 hours.

Do you want to allow both 0H and 24H? Rather than having code to respond to the increments when they happen we can convert the input value to a scaled integer early on and make sure the increments don't happen. We take the input angle we've already dealt with the sign so we can assume it to be positive and convert to units of the precision we want.

This process rounds to the nearest integer value corresponding to our input assuming the input in positive. Now we can use integer division and modulus operators to get the degrees and minutes and seconds. We'll need to insert the decimal point appropriately if we're displaying at fractions of seconds. One final issue is what about objects just south of the equator? We can keep them distinct or handle this special case, but we should probably do it consciously!

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