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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. 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:

Origin[ edit ] It is possible for people to count on their fingers to 12 using one hand only, with the thumb pointing to each finger bone on the four fingers in turn.

A traditional counting system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. In this system, one hand usually right counts repeatedly to 12, displaying the number of iterations on the other usually leftuntil five dozens, i.

Throughout their many centuries of use, which continues today for specialized topics such as time, angles, and astronomical coordinate systems, sexagesimal notations have always contained a strong undercurrent of decimal notation, such as in how sexagesimal digits are written.

Their use has also always included and continues to include inconsistencies in where and how various bases are to represent numbers even within a single text. In ancient texts this shows up in the fact that sexagesimal is used most uniformly and consistently in mathematical tables of data. The early shekel in particular was one-sixtieth of a mana, [3] though the Greeks later coerced this relationship into the more base compatible ratio of a shekel being one-fiftieth of a mina.

Apart from mathematical tables, the inconsistencies in how numbers were represented within most texts extended all the way down to the most basic Cuneiform symbols used to represent numeric quantities. But within the same texts in which these symbols were used, the number 10 was represented as a circle made by applying the round end of the style perpendicular to the clay, and a larger circle or "big 10" was used to represent Such multi-base numeric quantity symbols could be mixed with each other and with abbreviations, even within a single number.

The details and even the magnitudes implied since zero was not used consistently were idiomatic to the particular time periods, cultures, and quantities or concepts being represented. While such context-dependent representations of numeric quantities are easy to critique in retrospect, in modern time we still have "dozens" of regularly used examples some quite "gross" of topic-dependent base mixing, including the particularly ironic recent innovation of adding decimal fractions to sexagesimal astronomical coordinates.

Instead, the cuneiform digits used ten as a sub-base in the fashion of a sign-value notation: The value of the digit was the sum of the values of its component parts: Numbers larger than 59 were indicated by multiple symbol blocks of this form in place value notation.

Because there was no symbol for zero in Sumerian or early Babylonian numbering systems, it is not always immediately obvious how a number should be interpreted, and its true value must sometimes have been determined by its context. Without context, this system was fairly ambiguous. For example, the symbols for 1 and 60 are identical. The same stem and branch repeat every 60 steps through this cycle. This number has the particularly simple sexagesimal representation 1,0,0,0,0.

Later scholars have invoked both Babylonian mathematics and music theory in an attempt to explain this passage. The sexagesimal number system continued to be frequently used by European astronomers for performing calculations as late as The Greeks limited their use of sexagesimal numbers to the fractional part of a number and employed a variety of markers to indicate a zero.

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By the seventeenth century it became common to denote the integer part of sexagesimal numbers by a superscripted zero, and the various fractional parts by one or more accent marks. The same minute and second nomenclature is also used for units of time, and the modern notation for time with hours, minutes, and seconds written in decimal and separated from each other by colons may be interpreted as a form of sexagesimal notation.

In modern studies of ancient mathematics and astronomy it is customary to write sexagesimal numbers with each sexagesimal digit represented in standard decimal notation as a number from 0 to 59, and with each digit separated by a comma.

When appropriate, the fractional part of the sexagesimal number is separated from the whole number part by a semicolon rather than a comma, although in many cases this distinction may not appear in the original historical document and must be taken as an interpretation of the text. Sexagesimal degree Unlike most other numeral systems, sexagesimal is not used so much in modern times as a means for general computations, or in logic, but rather, it is used in measuring anglesgeographic coordinates, electronic navigation, and time.

One hour of time is divided into 60 minutesand one minute is divided into 60 seconds. Thus, a measurement of time such as 3: Similarly, the practical unit of angular measure is the degreeof which there are six sixties in a circle. There are 60 minutes of arc in a degree, and 60 arcseconds in a minute. In some usage systems, each position past the sexagesimal point was numbered, using Latin or French roots: To this day we call the second-order part of an hour or of a degree a "second".

### sex·a·ges·i·mal

However, the representation of these fractions as sexagesimal numbers does not depend on such an interpretation.

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### sexagesimal

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