Old Babylonian Mathematics and Plimpton 322: How did the OB scribe construct P322?

[@PlatinumDragonProductions999]

Also, in case I'm sounding too negative in my comments. I genuinely appreciate you sharing your scholarship with us on KZbin. It allows for "arm chair" scholars like myself to interact with, react to, and contemplate the latest research and discoveries. Thank you very much for posting your findings! :-)

[@jryer1]

You guys are too cool. The language the Babylonians used to describe their trigonometry is refreshing. Base 60, I like it, very efficient!

[@njwildberger]

Yes base 60 is something we should think seriously about here in 2017.

[@heathensaplenty4184]

njwildberger While I tend to agree, may I humbly suggest base 120 instead? While the Babylonians wouldn't have gotten much mileage out of it (though they sometimes used it for dividing rations) our system of handling irregular reciprocals allows us to make use of it's relationship with 119 (7*17) and 121 (11^2). In other words, multiples of seven or seventeen will have reciprocals of one digit infinitely repeating, and multiples of eleven will have reciprocals with two digits infinitely repeating. When you add in the three digit reciprocals for multiples of 13 you get three repeating digits or fewer for those numbers whose prime factors are only 2, 3, or 5 to any power plus 7, 11, 11^2, 13, 17, or 1117 to the powers listed. The only caveat is that for those numbers which aren't properly divisors, the higher powers of a number are treated like a different factor. i.e. The reciprocal of 7^2 is 7 digits repeating. Edit: Corrected a number.

[@AdlerMow]

@@njwildberger I wrote a question on quora about using superscript simbols over the digits as to transform our base 10 into a base 60 one. Can you do a review of applying a modernized base 60 system into a metric system, and how it would simplify day to day math by allowing easy and fast mental calculation of big numbers. Here is a image explaining my system: qph.fs.quoracdn.net/main-qimg-69fd0ca0a54ee22ee8d37360bc0f5d64

[@apolloniuspergus9295]

@@heathensaplenty4184 Interestingly enough, 120 = 5!

[@mohannadmahmoud6674]

@@AdlerMow The concept that we just need to memorize 5 new symbols reminds me of the hexadecimal system.

[@hellenakinyi7942]

Video Content 00:00 Introduction 01:09 Babylonian triples 03:25 From reciprocals to right angles 06:00 Neugerbauer &Sachs(1945) -E.M.Bruins(1957)- theory -De Solla Price(1964) theory 09:26 38 values of S and r 10:20 Missing columns on Plimton 322 12:53 How row 1 was made 16:34 How the b&d values were obtained

[@CraniumDranium]

Thank you. These videos are very informative. Thumbs UP!

[@jehovajah]

I draw your attention to the Stoikeia book 2, propositions 11, 12,13 and 14. In each case the Igi is the whole line and the igib is the unequal cofactor of the line. The igib is chosen so as to either cut the line unequally (the Igi and the igib), or to extend the Igi unequally. The demonstration requires the use of the so-called Pythagorean theorem. And also the Igi igib are chosen as cofactors which multiply up to some multiple 60.

[@jehovajah]

In your worked example you multiply two bar numbers so effectively you square the bar or rather the denominator. The Normalised 1 is then a unit of the denominator squared ie 60^2. To complete the Pythagorean triple you square this denominator again! Huge numbers xx

[@jehovajah]

The use of the 1 or ratherv1 bar refers to the denominator or unit of the system of measurements . This unit has to be subdivided into its factors, whichever unit is chosen. The bars or reciprocals are chosen from these co factors,

[@jehovajah]

I have since looked at the proof of Pythagoras Theorem in book1 of the Stoikeia. It is a dynamic proof based on rotatations. , the gonos being a dynamic knee bend. . The parallelogram mic form is dynamic, based on this knee rotation . The gnomon is named after this knee form! . So the right parallelogram was well studied from Babylon to Egypt to Italy where the Pythagorean school based itself.

[@jehovajah]

Interesting how the algebra is tackled from a numerical point of view rather than a lineal point of view. The Rectilineal form is a right tri lineal form or trilateral. The quadrilateral form is the rectangular parallelogram, or the form contained by line segments a quarter turn to each other. . The Stoikeia book 2 delves into this Rectilineal form algebraically but is rhetorically presented.

[@DamianReloaded]

Amazing. I wonder how long it took the Babylonians to develop all these techniques.

[@njwildberger]

We have to remember that they inherited the remarkable sexagesimal system from the Sumerians, who used this possibly as far back as 1000 years earlier, but certainly 500 years earlier. The Sumerians ought to get more credit; they were a truly remarkable civilization. I hope that our concentration on Plimpton 322 and OB mathematics in this series does not obscure this point!

[@DamianReloaded]

Amazing! I completely forgot about the Sumerians.

[@Xardy5806]

@@njwildberger we should take our heads out of the sand and admit that the sexagesimal system sumerians also inherited. It was developed by sophisticated mind , not by somebody who came right from bloody stone age ... Base 60 is the best system . 60 is nowhere in nature and astrology,or astronomy .. But it is in geometry . Therefore this system has scientific background ,while our stupid decimal has not .

[@QuestforaMeaningfulLife]

Wonderful to see that literal translation so you can imagine the way the ancients would have looked at doing arithmetic. Add = heap, subtract = tear off, divide = break, square = "make butt itself", square root = "equal-sided"

[@jehovajah]

The "Euclid " formula for pythagorean triples avoids line segments purposely. It is based on squares and rectangular parallelograms . Squaring these forms requires the Arithmoi, and is therefore an inductive step above the lineal or lateral forms. Using factors is more hands on

[@jehovajah]

Ifvr,s is 2,1 then the denominator is 2 rs that is 4 , the cofactors/ reciprocals are factors of 4 . Thus ß is 2bar( 2x1bar- 1x2bar) or 2(2x4-1x2)

[@jehovajah]

So a x ā has to be set to 1bar, that is one count of the denominator ,2 bar is a cofactor of the denominator , that is the number required to produce the denominator.by multiplication.

[@jehovajah]

Making the correct square equal to a rectangular parallelogram was sought rather than square rooting. For some rectangles this was not possible to do with a whole count of sub units, no matter how subdivided and multiplied the co factors were. For example the rectangle 7x8

[@jehovajah]

I apologise , the Stoikeia book2 propositions 9,10,11 deal with the igi and igib directly. Proposition4 deals with the Baff. . The previous posts references deal with the Cosine ratio rule for obtaining the diameter/ diagonal.

[@jehovajah]

224 seems out of place. I would expect 225 to be the cofactor as 16 x 225 gives a multiple of 60, (60^2)

[@jehovajah]

The use of the bar indicates that the bar is squared. Thus the denominator is squared to 16 by the reciprocal/ co factor process.. Do now the § or delta value is 2x10 , giving 12,16,20 as the Pythagorean triple . Clearly the original denominator 4 can be factored out , giving the 3,4,5 trip,e, and the procedure. For 5,12 the denominator is 120! Squaring that gives a huge 14400denominator .2 bar is actuallyb60 in this scale and 12 bar is 10 while 5 bar is 24, giving 14280,14400,20280

[@jehovajah]

The bar notation created an algebraic problem, fir me at least. While the form may represent a Babylonian calculation it is in fact a hybrid, and leads to a breakdown in the equality. Reciprocal/ cofactor numbers have to be set in a context called the denominator.

[@jehovajah]

224 becomes interesting in an astrological sense, as it is a factor of 7x360. Thus roughly every 7 years a jubilee was declared in ancient Aramaic cultures. Other prime number years also have significance. Pragmatically they may simply have been calendar adjustment years( leap years)"

[@Velzen5]

I wonder why they did not divide the 11th row by the obvious common factor 15 to get an even more simple b en d value.

[@njwildberger]

The reason has to do with their preference to have the long side of a right triangle normalized to 1 (aka 60). So 45, 60, 75 is actually more in line with their way of thinking, and there are other known examples where this is their preferred take on the 3,4,5 triangle. In modern times, we like to normalize this to be the 3/5, 4/5,1 triangle so it fits into our cultural framework of right triangles inside a unit circle. The Babylonians did not involve circles in their study of triangles: but they did like to normalize to have a long side/height of 1.

[@kevinolson1102]

Does normalizing the long side tabulated value to 1 simply facilitate computation? In other words, with the long side normalized to 1, it becomes obvious what is the correct order of magnitude of the tabulated values. The hypotenuse must be greater than 1 (it's even longer), and by the convention, the short side is less than 1. Sounds like something I'd prefer, too.

[@CyFr]

Would it be possible that using base 60 is the theoretical answer to properly quantify pi?

[@heathensaplenty4184]

Cyber-Freak Sadly, no. Pi is an irrational number (it cannot be expressed as a ratio or fraction) and therefore could be extended out to an infinite number of digits in any base. In base 60 it starts out 3.8:29:44:0:47:25:53:7:24:57:36:17... and keeps going infinitely.

[@Xardy5806]

@@heathensaplenty4184 you are expresing our Pi in base 60 , sumerians had different Pi.

[@PlatinumDragonProductions999]

5:03 also, it's not clear what the place values are. I can only guess at this because of my self-research into this system online, "butt" 1.01 = 61, "make butt itself" is "square that," which would make that total 3,721 (01.02.01) . "Tearing off" the one would mean that the the higher order "01" of 01.02.01 must be the 60^2 place and therefore 3,600. So subtracting 3,600 from 3,721 would yield 121, which gets to the 11^2 part that is the result. An explanation of the place values in Babylonian numeric arrangement would be most helpful in following this exciting discovery. :-D

[@mujtabaalmodhafar1027]

So thy also had an idea of inverse functions if you can call it tha way, off courese thy did not have an understanding of functions. But with reciprocals they created the beginning of inverse operations.

[@PlatinumDragonProductions999]

5:05 Ok, I think further explanation should accompany this statement: "2.01 is 11 squared." This is true if and only if "2" represents the "sixties" place giving 120, and the "1" is in the ones place yielding 120 + 1 = 121, which is 11 squared.

[@brendawilliams8062]

The numbers could be many places with 121. As in 1111186412. Many places

[@jehovajah]

The triple119,120,169 is the lowest result.

[@Lybra60]

Bonsoir et Merci ;Dommage que cela ne soit pas en Français .Mais je pense que nos Responsables Politique ne souhaitent pas que le citoyen soit plus instruit que les SUMERIENS ! Auriez vous trouvez une tablette qui expliquerait à c'est Responsables comment ne pas faire de Déficit Budgétaire chaque année ??!!Merci Cordialement.(ph) (Paix Amour Santé ici maintenant et dans tout l'Univers)02.2021