Best dating site for sexagesimal notation algebra

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    Conversion to sexagesimal

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    How to Use Babylonian Numerals (Clay Tablets)


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    Babylonian Mathematics and Sexagesimal Notation

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    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.

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    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.

    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".

    However, the representation of these fractions as sexagesimal numbers does not depend on such an interpretation.
    The triples are too many and too large to have been obtained by brute force.

    Much has been written on the subject, including some speculation perhaps anachronistic as to whether the tablet could have served as an early trigonometrical table.

    Care must be exercised to see the tablet in terms of methods familiar or accessible to scribes at the time. Robson, "Neither Sherlock Holmes nor Babylon: Geometry[ edit ] Babylonians knew the common rules for measuring volumes and areas. The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases.

    Example cases for the Pythagorean theorem were also known to the Babylonians. There are no sources indicating that Babylonians were aware of the Pythagorean theoremwhich is a general statement. This measurement for distances eventually was converted to a "time-mile" used for measuring the travel of the Sun, therefore, representing time.

    Tablets found in the British Museum provide evidence that the Babylonians even went so far as to have a concept of objects in an abstract mathematical space. The tablets date from between and 50 B. The Babylonians used a method for estimating the area under a curve by drawing a trapezoid underneath, a technique previously believed to have originated in 14th century Europe.

    50 Centuries in 50 minutes (A Brief History of Mathematics)


    This method of estimation allowed them to, for example, find the distance Jupiter had traveled in a certain amount of time. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. October Learn how and when to remove this template message Since the rediscovery of the Babylonian civilization, it has become apparent that Greek and Hellenistic mathematicians and astronomersand in particular Hipparchusborrowed greatly from the Babylonians.

    Ptolemy had stated in his Almagest IV. However Kugler found that the periods that Ptolemy attributes to Hipparchus had already been used in Babylonian ephemeridesspecifically the collection of texts nowadays called "System B" sometimes attributed to Kidinnu.

    Apparently Hipparchus only confirmed the validity of the periods he learned from the Chaldeans by his newer observations. It is clear that Hipparchus and Ptolemy after him had an essentially complete list of eclipse observations covering many centuries. Most likely these had been compiled from the "diary" tablets: Preserved examples date from BC to ADbut probably the records went back as far as the reign of the Babylonian king Nabonassar: Ptolemy starts his chronology with the first day in the Egyptian calendar of the first year of Nabonassar, i.

    This raw material by itself must have been hard to use, and no doubt the Chaldeans themselves compiled extracts of e. This allowed them to recognise periodic recurrences of events. Among others they used in System B cf. This is now known as the saros period, which is useful for predicting eclipses. Various relations with yearly phenomena led to different values for the length of the year. Similarly various relations between the periods of the planets were known.

    All this knowledge was transferred to the Greeks probably shortly after the conquest by Alexander the Great BC. According to the late classical philosopher Simplicius early 6th century ADAlexander ordered the translation of the historical astronomical records under supervision of his chronicler Callisthenes of Olynthuswho sent it to his uncle Aristotle.

    Although Simplicius is a very late source, his account may be reliable. He spent some time in exile at the Sassanid Persian court, and may have accessed sources otherwise lost in the West. Anyway, Aristotle's pupil Callippus of Cyzicus introduced his year cycle, which improved on the year Metonic cycleabout that time. He had the first year of his first cycle start at the summer solstice of 28 June BC Proleptic Julian calendar datebut later he seems to have counted lunar months from the first month after Alexander's decisive battle at Gaugamela in fall BC.

    So Callippus may have obtained his data from Babylonian sources and his calendar may have been anticipated by Kidinnu. Also it is known that the Babylonian priest known as Berossus wrote around BC a book in Greek on the rather mythological history of Babylonia, the Babyloniacafor the new ruler Antiochus I ; it is said that later he founded a school of astrology on the Greek island of Kos.

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    In any case, the translation of the astronomical records required profound knowledge of the cuneiform scriptthe language, and the procedures, so it seems likely that it was done by some unidentified Chaldeans.

    Now, the Babylonians dated their observations in their lunisolar calendar, in which months and years have varying lengths 29 or 30 days; 12 or 13 months respectively. At the time they did not use a regular calendar such as based on the Metonic cycle like they did laterbut started a new month based on observations of the New Moon. This made it very tedious to compute the time interval between events.

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    What Hipparchus may have done is transform these records to the Egyptian calendarwhich uses a fixed year of always days consisting of 12 months of 30 days and 5 extra days: Ptolemy dated all observations in this calendar. Pliny states Naturalis Historia II.

    IX 53 on eclipse predictions: This seems to imply that Hipparchus predicted eclipses for a period of years, but considering the enormous amount of computation required, this is very unlikely. Rather, Hipparchus would have made a list of all eclipses from Nabonasser's time to his own.

    Other traces of Babylonian practice in Hipparchus' work are:
    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.

    Babylonian Mathematics and Sexagesimal Notation

    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. 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.

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    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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