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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].

    How to Add Angles and Time Measures (Ice Cream Mathematics)


    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:
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    But in some cases, they may cause the program or the entire system to " hang ", becoming unresponsive to input such as mouse clicks or keystrokes, to completely fail, or to crash. He gave a successful demonstration of its use in sexagrsimal tables in Since ancient times, simple manual devices like the abacus aided people in doing calculations.

    In most cases, computer instructions are simple: While the subject of exactly which device was the first microprocessor is contentious, partly due to lack of agreement on the exact definition of the term "microprocessor", it is largely undisputed that the first single-chip microprocessor was the Intel[52] designed and realized by Ted HoffSedagesimal Fagginand Stanley Mazor at Intel.

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    The input of programs and data was to be provided to the machine via punched cardsa method being used at the time to direct mechanical looms such as the Jacquard loom. The input of programs and data was to be provided to the machine via punched cardsa method being used at the time to direct mechanical looms such as the Vating loom.

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