Old Babylonian mathematics and Plimpton 322: The remarkable OB sexagesimal system
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- Опубликовано: 2 июн 2024
- The Old Babylonian arithmetical system was powerful and logical, and gave this ancient civilization a huge computational capability. Many mysteries remain unanswered, for example how and why such an early culture was able to adopt such a sophisticated system.
In this lecture Daniel and Norman look at how the Babylonians adopted the Sumerian base 60 system, how the reciprocal table played a big role, their use of multiple tables, and how quadratic questions arose naturally in the context of Pythagoras' theorem, called the Diagonal Rule. We also look at the famous tablet YBC 7289 involving the OB approximation to a square root of 2.
Video Contents:
00:00 Introduction
01:13 Sexagesimal number system
08:33 The Standard Reciprocal Table
13:00 Factors used in multiple tables
16:17 A formula for Babylonians triples
21:57 YBC 7289 [ geometrical diagram]
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In 1961, when I was an undergraduate at Yale I took a History of Science course. As one of the assignments we each went to the library and were given a small tablet (enclosed in a plastic box) and had to translate it. They were all schoolboy exercises and so were not difficult to understand, but I still remember the thrill of having that in my hands.
Is Plympton 322 named after the skull and bones club?
Double and halve. Look at these two approximations for root2 : 99/70 and 140/99. If you add these together and halve the result you get 19601/13860 : root2 to 99,9999998698%. You can then continue, adding 27720/19601 to the previous decimal and dividing by 2. This can be done for all square roots from simple integers. For example, for root5, a first approximation is 5/2. Inverse the fraction and multiply the numerator by 5 (for root5). This gives 2x5/5, 10/5 which we add to 5/2 and divide the result by 2 = 45/20 = 9/4. Add to this result (5x4)/9. 9/4 + 20/9 = 161/36 = 4.4722... Divided by 2 (161/72) gives root 5 to 99.998%. To increase precision, continue by adding (72x5)/161 and dividing by two. Precision = 99.9999998%
I have been watching videos around mathematics everyday on RUclips.. Norman is by a long way the better i am enjoying to watch.. He did released me from the dogma of real numbers and axiomatic mathematics and i am now advancing in many subjects in a good part because of his teaching
I imagine that when he walks into a classroom, all the students say *NORM!*
15:42 The ambiguity between 1.15(=75) and 5/4 remembers me of the way we think of 125 as a percentage (5/4).
I love how nerdy they are, honestly. It's awesome
As I mentioned before I really like the courses of Norman Wildberger. Wild Linear Algebra is my favorite. This one is exciting in a magical Harry Potter way, but then it is real
I'm glad that you're uploading videos still!
Very enlightening. I love the 'scaleless fraction' idea that captures just the relation of a number to the 'whole' or the unit.
This is awesome.. Thank you for your time putting this together...
This is pure gold! Thank you. How doesnt this video have more views?
Interestingly, as sqrt(2)/2 is equal to 1/sqrt(2) this tablet contains both sqrt(2) and its reciprocal. Presumably this is why the sidelength of 1/2 was chosen for this problem.
That is a very nice observation!
sqrt(2)/2 is also equal to sqrt(1/2). (In general, sqrt(n)/n = 1/sqrt(n) = sqrt(1/n)).
I am wondering what the significance of the
It's difficult to say from those three numbers alone.
They did have tables of constants, which would contain an approximation for "the side of a square". So one possibility is that the scribe took the constant, wrote it down, and then rescaled it by
One bit of speculation I’ve seen was that this is a tablet written by a student, and that the value of √2 came from another tablet (not now available) where it had been computed by an expert. All the student had to do was multiply that constant by 1/2 = 30.
If the square on the tablet has sides of 1/2, the length of the diagonal is sqrt(1/2).
(1/2)^2 + (1/2)^2 = sqrt(1/2)^2.
Most likely, that's what is being depicted on the tablet.
You're right, multiplying by
Video Content
00:00 Introduction
01:13 Sexagesimal number system
08:33 The Standard Reciprocal Table
13:00 Factors used in multiple tables
16:17 A formula for Babylonians triples
21:57 YBC 7289 [ geometrical diagram]
Great video! I've been on a knowledge quest about all of this Sumerian and egyptian knowledge
Thank you! You are doing a great job by explaining these!
The Babylonian number system is quite attractive. The unsurprising something/nothing vertical stroke for number one is elaborated into a 3x3 matrix, leading smoothly into decimal, which is then extended to a multiple of ten which also happens to have a lot of divisors, which strongly benefits their amazing approach to division!
Phylogeny recapitulates ontology, as they say! ;)
Oh who am I kidding...sigh. I don't have the froggist idea what frick the friggin' video's on about, nor your comment, and last but not east (for all intensive purposes) whatever the hell I _"illedgidly"_ might or might not have written above (or not).
33 though: that's a rabbit hole right there. Flip, that's a flippin'....Intergalactic Space-Rabbit...Space-Porthole! That's the Super Bingo, Triple Whammy, Yahtzee!, Number of the Beast, Willy Wonka Golden Winning Ticket right there!
I'm sure it's got something to do with that bullshit, Beatles (AKA George Martin) filler track bullshit:
_"Number 9...Number 9...Number 9...Number 9"_
3x3 = 9? Oh yeah; guess that's it.
666 = 9? 6 + 6 +6 = 18; 1 + 8 = 9
23? Hrm...2/3 = .666 - oh snap. Number 9 again...trippy.
Okay, I'm off to Siruis, gotta see a Giant Pooka about a rabbit hole...something to do with Fnords too. Meh.
WANG CHUNG, BONY M FANS! ("By the Rivers of Babylon....blah blah de blah")
@@soundgardener4940 Read "The number of the beast" by Heinlein. Blow yer fricken' mind. And I don't think you are anywhere near Sirius... since it is ontogeny, not ontology...
The Babylonians certainly had fractions based around 6, some of which has unique signs, most interesting of which is "Parasrab" which means "big division" for 5/6 (Kingusila in Sumerian). Other common fractions are 1/3 (sushana/shalush), 2/3 (Sittan - a dual form of the word for two, ergo two twos > 4 i.e. 4/6), and a means of expression 1/x, which is usually written in Sumerian as Igi.x.gnal2, literally meaning "the face/surface has x".
A very interesting possibility arises regarding normalisation versus square rooting! Using proposition 14 of book 1 it is clear that a side of a square is constructed but not necessarily a whole number of sub units. . However by scaling the co struction so that the constructed square sude is set at 60 or some multiple , one can scale the other measures in a rectangle to normalise a diagonal. . Thus without having to measure irrational lengths they could scale accurately and then count
I think I would favour a base 8 counting system, with binary representations of the digits, for example a horizontal line with up to three lines sticking up out of it, eg. U shape would be 101 or 5, and with the three main fingers on one hand you could naturally count 0-7, and then 8-63 including the other hand, with the option of adding the fourth and fifth fingers to count up to 127 or 255. Or use all ten fingers if you want, have a decimal point between your hands maybe. Common things like doubling and halving would be so easy to do. Been thinking about this, it works out very nicely. Calculations often become very simple to do, and I like having the two spare fingers on the units hand, as when doing calculations individually using the "ones" and the "eights", things that carry over from the ones can be temporarily stored on the extra fingers while the eights are dealt with, and then added to the eights seperately.
This was a very insightful video, thank you!
What if the problem with mathematics is that we tend to stick with one base, be it base10, base60, base16, base2, whatever, instead of converting to the base needed to do the calculation exactly?
For instance, if you needed to divide by 7, convert to base420, but if you didn't need the 5, you could do the equation in base84. Working in base12 and you need to divide by 9? 12=3x2x2, 9=3x3, so your base needs another 3 -> 3x2x2x3 = 36, so convert to base36 and get an exact answer.
This could get interesting if you included fractal bases.
And that's why seven was a mystical magical even dangerous number. But anyway, they used both "decimal" (irrational floating point) and rational fractions. Often you can carry the seven divisor around and either cancel it out or leave the divisor at the end. For example, the famous Metonic cycle (discovered by the Babylonians but credited to Greeks) is left as 235 synodic months / 19 years.
11:00 the video says that they didn't or couldn't use fractions. Ba'al poop.
They used the sexagesimal system because of 3 6 and 9 and the golden spiral. You could use the golden spiral as a times table for the sexagesimal system. We use a square table because we work with groups of 5.
I love it. The magnitudeless math reminds me of slide rule multiplication.
Terrific series.....well done.
The alpha numerals are placed in a sequence position which determines the dimension size( metres, decimetres, centimetres , millimetres ) . Reciprocal are factors that multiply to a given value ie co factors. They reciprocate: as one increases the other decreases accordingly
Kind of gauge-theory-esque in some ways with the measurement fluidity. This is rad as hell.
*Thank you Norm & Danny!!*
Formulae are procedures! Of course we have to learn the procedural solutions to each form.. The Babylonians saw x as a guessing trial and error unit. . The unit size to start from for the square root of 2 is not 1 but 30 , ie not 60 but 30
Is it possible that there is a "Master Tablet or Tablets" for all Babylonian Mathematics?
It would seem there would have to be to facilitate teaching Math to a group of students instead of Individuals...
It's 1:51 am, I am German and watching this AMAZING video
I am pleased someone is looking at this seriously , but you start off at a disadvantage by using the number concept. . You need to start with katametresee, or counting by placing or putting down. .the patterns formed are mosaics eventually called Arithmoi. .
I am currently trying to do research in this subject, and I admit, I am not as well versed in theses areas as some other people are... but two things I wanted to point out that I noticed right off the bat.
1. I am pretty sure 1 over 2 does not equal point O 5... or, 0.05... I believe that is 0.5. I may be wrong, but it was probably just a slip of the lip.
2. I was not aware that the Mesopotamians developed/discovered multiplication yet, at least as we know it. I mean, as far as my research has uncovered, the Egyptians didn't have it either. They used a somewhat similar system, but it mostly involved a variation of what we would more closely associate with complicated addition in groups.
Also, Division is not an operation. It violates the field axioms, as I am sure you are aware. It is, however, an inverse operation of multiplication, which I believe is what you are trying to explain... So, of course they multiplied the numbers in reverse... assuming that is what they were doing in the first place.
But, at least you got the symbols right. Excuse me if that sounds like I am being sarcastic, I am not. There is surprisingly very little material available on the subject (even though the British Museum has over 130,000 tablets in there collection alone.) but from what I have been able to gather, your symbols make sense. The other sources I have found literally use what I can only say are tally marks. Your's actually look like the stylus impressions I was expecting.
Now, I need to figure out how the numbers fit together. If it is anything like the language, they are kind of taken in groups, instead of distinct numbers like we know them. It is a fascinating, complex, yet beautiful system. I think it is a shame more people aren't interested in learning this stuff... they just take numbers for granted. I am looking into how they came to be... sort of like etymology but for maths.
Fascinating stuff all round. Who would ever had thought the Babylonians were so good at maths?
All I can say is 'keep taking the tablets'.
The Babylonian triples are based on a quadrant urge construction discussed in thevStoikeia, books 1,2,3 and 4. In constructing a quadrant urge for a rectangle the depicted triangle occurs.
I have tried this out, then thought, can this be applied for cube roots?
.5 ((a+b)/(1+a)) where a.a=b, for square root.
.5 ((a+b)/(1+a.a)) where a.a.a=b, for cube root.
Amazing. I wonder what happened. How did a community so specifically advanced just disappear? And then to devolve into the barbaric idiocy we are forced to contend with in this day and age, it is such a shame.
Wonderful presentation, gentlemen. It is phenomenal!
Thank you for sharing!!!
religion
18:20 4x20 is not 120, it's 80. What's not being made clear is that "4/3" is 4/3 of SIXTY, and "3/4" is 3/4 of SIXTY, referring back to the previous statement that "60 is one." so 4/3 should be 80, and 3/4 45.
Yeah that part messed me up they should correct it.
it's not 120, it's "1.20" where 1=60 so 1.20 =60+20=80
The place value system is is in general a systematic arrangement of Arithmoi in a sequence of scaled units. The magnitude of our numbers is not obtained in the Arabic numeral, but in the unit dimension.
The re,event quadrature is found in book 2 proposition 14 . Using any rectangle which factors 60, 30 etc we can construct its quadrature , and calculate roots by a
I'd love to see graphic or animation examples of these ideas. Can this system be used in computer generated images? Modern games use polygons a great deal - is it possible to apply this system or rational trigonometry to generating 3d geometry?
Yes, similar techniques (reciprocals and stuff) are very common practice in 3D rendering. One famous example is what the programmers did on Quake.
9:03 - Double and halving x 30 times equals x! :P Cool video! Wish you would do an update on it
How to compute approximate square roots? Generalize the algebraic formula for Babylonian triples.
((x-Q/x)/2)² + Q = ((x+Q/x)/2)²
The right-hand side is recognized as an iteration step for solving x² = Q approximately.
x' := (x+Q/x)/2
The formula tells us, that the error is itself a square. Can we take advantage of this fact?
Draw a rectangle containing both the current iterate (x+Q/x)/2 as its diagonal length and its exact error (x-Q/x)/2 as the length of its short side. Then the long side has quadrance Q. Can we iterate in this picture, and see geometrically that the long side will dominate?
The mathematician seems to indicate that the length of a side of a square is
In the slide about the square root of 2 the factor 30 ( 1/2 in your formula) appears on one of the sides because it is a square, your formula expresses the idea that Babylonian triples can be factored by 30
By a procedure of approximations using cofactors ( x and 1/x ) such as 80 by 75 for 360, or say for a product of 60^n .
It's interesting how, while the practical implications of their counting system is base 60 with a relationship between 6s and 10s, the way it's drawn is very clearly a 3x3 square that builds out line by line and then a clockwise simplistic spiraling pattern with 5 steps. Since the notation isn't base 10, it sometimes makes irrational numbers more elegant. The golden ratio in sexagesimal could be written as: 1; 37; 4; 57; 28; 24; 56; 0; 7; 59; (with 59; repeating from there on). Assuming you use the fractional place-value style. Hindu-Arabic have some geometry principles, but over time they've been altered to be convenient to write, whereas the geometric principles in relationships like 2s, 3s, squares, and ratios that form spirals or tiling polygons are apparent in the Sumerian representation. For instance, you can tesselate triangles, squares, and hexagons without gaps, and the hexagons are just sets of 6 triangles, if you drew a hexagon made of 6 triangles with the minimum number of segments possible, you would use 9 segments (6 perimeter and 3 interior). It makes sense that such an ancient writing system would try to represent physically relevant geometric relationships simplistically, since base 10 is a more abstract linguistic-based numbering system, but one presumes that most mathematical reasoning derived from drawings and measurements, initially, since physical relationships in space are relevant to primitive people even if they don't have formalized writing.
10:35 "If you want to divide one by two, then that's point oh five." Haha!
Damn it, you got me.
only for extremely large values of 2
Good day! Please help me to answer my assignment. Thank you❤️
1.) Multiply the number 12,3;45,6 by 60. Describe the simple rule for multiplying any sexagesimal number by 60; by 60(squared)
2.)The Babylonians generally determined the area of a circle by taking it as equal to 1/2 the square of the circumference. Show that this is equivalent to letting π=3.
Have to say I have a hard time distinguishing between algebraic formulae and procedures and algorithms, to me they are all so similar, making it out they are different strikes me as a bit phony. Algebra is really just a compact way of expressing a procedure. If you spend any time writing code for symbolic algebra software like Maxima or Octave you understand this! I love that line... "The Procedure" instead of "The Answer". That's exactly how a mathematics student should think.
I need to know how to do multiplication and division using the Sumerian number system for a school project. What's the difference between Sumerain and Babylonian?
So which number base provides the maximum number of exact pythagorean triplets?
I don't understand why, in the drawing at the 23rd minute, the scribe wrote '30' down as three individual ten marks instead of one thirty mark? Same question goes for the root 2 number: why two individual 10 symbols to represent 20 instead of the 20 symbol?
13.37.10 is the ratio of the whole.... It isn't a quantity of units... That is defined inherently or in specifics in the "document".
:)
So they would construct a right triangle this way? And then go on to construct the rectangle ? The second example shows a table of constants was used? If so someone generated it by large scale geometrical construction.
A reciprocal of 6 is a co factor , not a multiplïcation by 10 . The cofactors records parts of a unit( any unit dimension) divided into 60 sub units.
Keeping track of the decimal place reminds me of a slide rule. You were always forced to maintain a sense of the expected magnitude. Interesting if decimal place was included in the dimension name such as 1 unit = 1 inch or 1 unit = 60 inches versus for english system 1 unit = 1 inch or 1 unit = 1 yard (36 inches). Imagine doing math in yards, feet, inches. It’s like having a different base for each place (first digit on right is in base 12, 2nd digit is in base 3. Babylonians would consider it primitively ineffective.
They used geometry and dimensions to calculate. Probably figured out our E8 theory of everything and had multidimensional thinking. Most of the “Temples” are actually depictions of math they found interesting. So in their design are formulas that were so mind blowingly interesting to them, they had to build the geometry of it in the 3d world they were living in. The pyramids are also NOT tombs and were build 12000years ago. They are actually power stations of some sort much like Nikola Tesla’s famous tower. We should show them more respect!
Where can I find a class that will teach this ways of multiplication, Division, adding, and subtraction? Anyone knows?
The formula is trivial to prove with algebra, but is there a straightforward geometric way to show it? Specifically where does "x" and "1/x" get constructed on the diagram and then how might the geometry proof go from there?
You need to encode things. Subtraction a-b is encoded as a+x=b, and multiplicative inverses are encoded by a multiplicative equation ax=b. In the next step you turn addition and multiplication into juxtaposition of parallel segments and areas. This is just a basic method.
Here's a picture:
drive.google.com/file/d/0BwVv1Z9Tmq_ycU1RTlJQZ05EWU0/view?usp=drivesdk
In the first figure you find a square on a side x and a designated unit area. Inside it are some extra labels that indicate the equation xy+1=xx and its solution y=x-1/x, which is needed to express the left-hand side of the babylonian identity. Another horizontal side has been drawn to complete some symmetry and thus prepare for the next step.
In the second figure two smaller squares have been drawn, namely XZNF and XYVW, inside the original square ACDF, by bisecting the sides called y. The babylonian identity corresponds, in my perspective, to the fact that the additional area between XYVW and XZNF is a unit. This fact is stated and motivated in the last row.
Also, you can note that the last fact above is independent of calling BCEF a "unit area". That term, "unit area", seems to be useful only when counting, and there is no counting involved in recognizing that that L-shaped area equals the area BCEF. But the Babylonians did a lot of counting in their arithmetic! That's, I would explain, why we augment the picture with the notion of a unit.
I find it a bit exciting to insert Babylonian hexagesimal measurements in my picture. The simplest way seems to be, to take a reciprocal pair from the table of reciprocals, such that their ratio relates approximately the perpendicular sides of the rectangle BCEF. Or conversely (and more exactly!) pick a reciprocal pair first and then construct BCEF and the rest of the picture from that ratio. The geometric argument then shows that said equality of numbers holds. Or pick an approximate pair of reciprocals, and the equality will hold approximately.
In summary, it is these that are in correspondence:
((x-1/x)/2)² + 1² = ((x+1/x)/2)²,
XYVW + BCEF = XZNF
- based on my vivid imagination, not just historical facts ;)
It is interesting to consider that a sexagesimal number system would accommodate counting on five or six fingered hands.
When it comes to plane trigonometry the ancient Babylonians probably have more efficient system of mathematics which is not angle-based. Their method is a combination of advance and fundamental aspects of mathematics.
They were using a fundamentally nonzero type of advance sexagesimal (base 60) number system. Sub base 10 was employed to facilitate the counting numbers.
In conventional type number systems, the convenience of using zeros as place holders or numbers are commonly known.
Going back to further fundamental type number system, there is a method where zero is not use as place holder or number.
For illustration of a decimal number system that do not use zero as place holder or number,
click on
sites.google.com/site/wanderinginmath/
and open the menu of "Sample Decimal Counting Numbers without Zero Digits".
So the procedure involves multiplying by a cofactor to place the magnitudes in the same unit size( denominator) . The unit1 is a power of 60 in this case.
I got completely confused at the end for a moment. At the beginning when you showed the numbers, you depicted them as it's own symbol, rather than a grouping of symbols. So when, at the end, you showed what would be 30 as essentially
Those are both good questions. 1. How did they come up with it? and 2. why did we get away from it?
The short answer to 1. is we don't know. The 10-symbol "probably" indicates an older base-10 system (which somehow seems more "natural", but that may be cognitive bias on our part) that somehow got combined or maybe just sort of grew into the base-60 system. Maybe the system even grew up as job security for the administrative/priest class.
We know quite a bit more about 2. Everybody had their own number system (and there's quite a bit of literature about the borrowings because they, along with astronomy, can be used to date cultural exchanges). In particular, there has always been at least a little conflict between the scientific uses of numeration and commercial uses. In general, commercial uses avoided fractions. (One doesn't set a dime equal to a tenth of a dollar, for example, but rather 10 dimes to the dollar. Or 3 obols to the drachma, as the case may be.) For commercial transactions, we don't need to calculate with fractions. For computing when the next eclipse will be (or what the position of the heavens was on the date of your birth [remember, there were no star charts, the astrologer had to calculate the position of the heavenly bodies by hand]), there was no way to avoid fractions. Tycho Brahe calculated with tables of sines and tangents (in Greek, not Babylonian) in sexagesimal notation in the 1590s. (That's probably worth checking. I could be misremembering.) Then Napier changed everything with his logarithms, and the big seller was the table of logarithms of the trig functions. (To either 14 or 20 decimal places, I forget which.)
didnt they just use a circle? Exactly like a clock. And their map. And geometry. It's very simple to perform these calculations on a clock
لقد فعلها اجدادي البابليون
You must study Dtoikeia books 5 to 7 to understand how the concept of multiplication and division are designed. . The notion of multiplication is built on duplication . And the notion of multiplication gives rise to factors Ann the tables of factors and their relationship are what you call regular numbers.
it's also probably convenient for solar/lunar based calendars. numbers like 365 366 and 30, 12
Hi. Why is there no 7, 11 or 13 on the table of multiples?
it looks like a time system i think that we still using this system in watches
Your time machine going back to the Babylonians would use Babylonian notation to keep track of how far to go back in time.
Not just the clock is 24(12) by 60 minutes by 60 seconds but GPS and astronomical calculations are done with the exact same method. Babylonian math rules!
The Babylonians had precise clocks (maybe water clocks, we don't know). They measured each USh from sunset (or sunrise when anticipating a solar eclipse) and from the rear (east most) named star clusters (three star ANU zodiac). An USh is exactly 1° and 4 modern minutes of Earth's rotation. Each zodiac is exactly 30° and exactly 2 modern hours of rotation.
@@vegahimsa3057 This just blew my mind; it makes so much sense.
@@JT-iw2cw except for precession and leap second, it would have made for a more logical clock than we use today. Hours and time zone (and a unit of distance a bit like nautical miles) would have made more intuitive sense. And of course sexagesimal had several advantages over our decimal system.
It would be interesting to know the intellectual process that led to the very clever and useful base 60 base and a system which effectively had place value fractions way before its time. Almost all other systems came up independently with a base of 10 (or in the case of the Mayan Indians 20 (or more precisely of 4 x 5) but still apparently arising from the number of fingers on ones hands including the thumb. ( it is a bit surprising that none used base 8 (the fingers without the thumb). Among the Babylonian tablets relating to computation were fragments of what appeared to be a table of squares with the whole number root on the left and the square on the right. It was proposed that it was used to compute square root, but I read that someone proposed that it was used to multiply two numbers by the difference of squares identity. Given two odd number or two even numbers you calculate half their difference and add it to the smaller number and go to the square of that number and then subtract from it the square of half the difference computed earlier. For example to multiply 21 x 29 you'd get a difference of 8 and half that is 4 when added to 21 give you 25 the square of which is 625. If you now subtract 16 (the square of 4) from 625 you get 609 which is the product of 21 and 29. So the table of squares could be used to yield the product of two number without a multiplication process based on a table of multiples. Only subtraction and addition and a division by 2 would be required to find the product of small numbers within the range of your table of squares . There is no evidence that the Summarians actually used the table of squares for this purpose, but it has been proposed as a possibility. And they were clever people and no doubt knew of the this possibility.
I understand the common people used base 10 and that base 60 was only used by the scientists. In fact, that's implied by the sexagesimal notation itself. Similarly, the Sumerian language continued to be written (and chanted in hokus pokus rituals) into the Roman era despite (presumably) no one speaking it daily for 2000 years, since Akkad.
@@vegahimsa3057 And what is the source of your information with regards to the common people using base ten? I have studied the history of Math and number systems for years and I never heard that the common people of Summaria or Babylonia used base 10. It is something I would be inclined to believe, but I never saw any evidence of it?
@@LawrencRJUTube i appreciate your scholarly question. I'll have to dig after this day of rest. :)
We know that the Mesopotamians used many different systems, units of measure (much like English twelve inches, 16 ounces, etc) and different names of numbers and quantities depending on the thing counted (live animals were counted with a different number system (or names) than for dead animals -- heads not carcass weight). Early there were separate notation for fractions, such as half, quarter, third, etc (and the Egyptians had a bizarre notation based on sections of divine symbols) whose names again often differed depending on the thing (not unit) measured. These fractional symbols were replaced with sexagesimal (not to say that sexagesimal didn't predate other notation) floating point.
In the Babylonian record it seems (to me) that Sumerian was consistently sexagesimal despite a long and confusing mix in spoken and written Semetic.
I'll have to find a paper on decimal specifically. But perhaps you are aware of a thesis (which I consider silly) that there were two internet counting systems: base 6 and base 10, later merged into base 60. I find that incredible and unlikely. More likely was a decimal system with sixths, thirds, and halves (floating point base 6).
Anyway, yes, I'll have to back up the assertion that decimal was used by the common people.
I don't have access to the book, but in "Numerical Notation: A Comparative History", Stephen Chrisomalis claims that the Sumerian system was indeed sexagesimal with the familiar combinations of up to nine 1s and several < . (it seems to me, however that the Sumerian text, math, and records were preserved for science and ritual for thousands of years, while very few civil and common Sumerian records exist after Akkad). Chrisomalis refers to a later standardisation he calls Assyro- Babylonian (aka Akkadian or Old Babylonians) with separate symbols for 1, 10, 100, and 1000. Numbers were written with a string of ones followed by the "place". For example 2021 would be 11 (1000) 11 (10) 1 and written as we would express similar to English: two thousand two ten one. Sumerian "scientific" (my term) would be 33.41, I believe a stacks of:
Hildegard Lewy (1949) implies that Sumerian entered Semetic language later in Babylonian history, which is a perspective that I'm fairly certain no scholar maintains today. However, it seems plausible that the names of large Sumerian numbers (like 3600) were re-named by Babylonians.
doi.org/10.2307/595393
... both the qu and
the simdu eventually became the basis of a decimal
system of measures, the primary set being the
qu-sutu-imeru system comprising measures of 1, 10,
and 100 qu, and the secondary being the
simdu-kurru-10 kurru system reckoning with units of 1,
10, and 100 simdu ...
Gemdet Nasr tablet (measuring area of land):
buru : iku : 1/1O iku : musarum =
100 : 10 : 1 : 1/10 : 1/100
... (1 simdu = 10 kurru) in exactly
the same way as the Assyrian measures of surface
qu, (1 epinnu = 10 qu), (1 imeru = 100 qu) are de-
rived from the primary set of measures of capacity
(qu, sutu, imeru).
... average man's daily grain ration and the
secondary based on a man's monthly grain ration
were blended into one single system which, accord-
ingly, comprised the following units:
1 qu, 10 qu, 30 qu, 100 qu, 300 qu, 3000 qu...
... subsequent
Babylonian sexagesimal system of numeration,
namely the introduction of a secondary unit (here
30, in the later system 60) which, while not be-
longing in the decimal system of powers of 10,
was treated numerically in exactly the same man-
ner as the unit 1 ...
... Thureau-Dangin called ...
system of numerals used under the kings of Akkad
contained the following symbols:
1, 10, 60, 600, 6000
All the [Akkadian, Babylonian] numerals up to 3600 are formed from the
names of the numbers 1, 2, 3, 4, 5, 10, and 20.
Six, for instance, is named 5+1, seven 5+2, nine
5+4.
Thirty is called 3 times 10, and forty 20
times 2. Fifty, in turn, has a name meaning
40+10.
... The names by which the numbers are
designated in the Sumerian language show no trace
of the development outlined in the preceding pages.
For the number sixty, Sumerian texts
use two words, namely ges or gesta and sussu. The
former, being identical with the Sumerian name of
the numeral one, characterizes the number sixty as
' the large one '; the latter, on the other hand, is a
Semitic loan-word. Now the choice of identical
designations for the numbers 1 and 60 was possible
only at the time when the development had reached
the stage represented by the series. In other
words, the creation of the primary series 1, 10, 100,
... preceded the choice of the name ges for the
number 60. But we can go still a step further
and assert that even the creation of the series
was a matter of the past when the Sumerian lan-
guage made its appearance in Babylonia. In agree-
ment with the absence of an individual name of
the number hundred, the language proceeds strictly
according to the sexagesimal system when it names
hundred-and-twenty (120): 60 times 2,
hundred-and-eighty 180): 60 times 3,
and so on until 600 which is
called 60 times 10.
No new number-name is intro-
duced until 60 square, the numbers 1200, 1800,
etc., being referred to as 600 times 2, 600 times 3,
etc. As 3600, finally, is called by a new name,
namely sar, we realize that the Sumerian language
composed its number-names according to series...
1:34 digits
8:37 division
16:25
Wow! I can read Babylonian tablets now! Or at least understand the Babylonian computational side of it.
I get the feeling Norman & Daniel are slowly recruiting us as amateur sleuths to join them in their mathematical & archaeological venture. They make it look so easy but with practice I am sure we can learn to decipher the symbols. cdli.ucla.edu/dl/photo/P254790.jpg
Chris Pyves We were taught Babylonian and Egyptian numeric systems at school, in the sixties, but they probably are too clever for that, now days.
The Incas used a 1,2,3,5,8 computing system. Next row is 20 (1,2,3,5,8) &c.
It's a bit like music. Scale by 60, move an octave.
perhaps we can invent a "sexigesmal tone row system" and create a new musical paradigm? Let us make music and let the magic begin gentleman!
Joel Sjögren I notice that the first four irregular numbers that are omitted from the reciprocal table are 7, 11, 13 and 14, which are the out-of-tune harmonics.
Harmonics 2, 3 and 5 make up a major chord. (1st, 5th, major 3rd)
By the way, while talking about frequencies, the formulas cos(at)+cos(bt)=2cos(t(a+b)/2)cos(t(a-b)/2) and cos(at)-cos(bt)=-2sin(t(a+b)/2)sin(t(a-b)/2) can also be expressed with msets as the identities
>>> [a b -b -a] = [(a+b)/2 -(a+b)/2][(a-b)/2 -(a-b)/2]
>>> [a _b _-b -a] = [(a+b)/2 _-(a+b)/2][(a-b)/2 _-(a-b)/2].
In the formula for sine, it is important to follow the computation rules
>>> _a + b = _(a+b)
>>> a + _b = _(a+b)
where _ means anti. In particular, addition of (anti-)msets [] and _[] is like the xor operation on boolean symbols F and T, or the == operation on T and F.
It should be noted that this is only exactly correct if one considers a Just Intonation chord, not the approximations available in the standard Western 12-TET scale, where the frequency values for tonic, fifth and third are actually 1, 2^(7/12), 2^(1/3).
15:45 very easy to understand if you think in Hours! 1.15 is of course 1 hour and 15 minutes = 75 minutes, 1.30 is one and a half hour =90 minutes (without the added time 😊)
Keep that line of thought. Then someone will suggest two clocks are going on
I think if I watch this a few more times I might actually begin to understand those %$&#@¥ reciprocal tables!
especially if you say this place was a school that these were found grading system makes sense. now if your theory is right now what would those numbers be used for? to my knowledge to the power of numbers are usually really high as to the reason why its simplified to the power of'. so these totals would have to have been used for something big but if it were grades then not sooo much understand?
Hey, 4x20 shouldn't be 1.33.., instead of 1.20? Since it would get 80 in decimal, so it's 60+20. Since 20 is a third of 60, shouldn't it be calculated as actual 0.33..?
Secondo le mie pluridecennali ricerche, questi tipi di tavolette cuneiformi a contenuto matematico hanno in comune un unico e formidabile strumento algebrico-geometrico scoperto dai sumeri grazie all'invenzione del mattone da costruzione che ho coniato come: il Diagramma di argilla.
il Digramma di argilla era fatto di mattoni rettangolari da costruzione movimentabili e sovrapponibili avente alla base un modulo detto: a modulo quadrato. Un gioco algebrico inventato dai sumeri e conosciuto da tutte le Civiltà potamiche e poi importato dai Greci.
il Plinton 322, nel caso in specie, era molto probabilmente il loro "teorema di Carnot dell'antichità " e che ritroviamo poi nelle proposizioni 12 e 13 nel Libro II degli Elementi di Euclide.
Chi volesse vedere le mie ricerche veda qui :
www.atuttascuola.it/collaborazione/bonet/
Scriba con il diagramma di argilla.
3.bp.blogspot.com/-O553c5rl8bI/U4Gmr-afMYI/AAAAAAAAqnU/Z7gKYEKedJE/s1600/copertina_2.jpg
Dear sir, can we actually reintroduce this lost Babylonian number system by replacing a Hindu Arabic number system in mordern day arthematic???
The numbers are representation of solar system movements in mathematics
Thankyou. If it works, it works.
Did they delve into primes or simply consider them evil?
I don't think we have any evidence that they considered primes. They were very practically minded, and abstract number theory probably was not considered very much ---at least as far as we know!
That system for Sumerian 6000 BC.
It is named 60 system. This system we using it now for times.
What happens if you set the equality to -1/12 ?
In sexigesimal
great video, btw
Take 5÷7= .714285. Using the base 60 convert, 5(x)=60. this results in x=12..and as such..take..7 is =5+2,. Using base 60, the 7 is equal to5(x)+2(x),....thus 7=12+30....So now converted to base 60, 5÷7 becomes 12÷(12+30)=.285714,......now base 60 approaches 1. Thus 1-(.285714)= .714286......note the 6th place decimal is off by one digit. Using the base 60 system proves that all numbers approach 1. And it also illustrates that our arabic numerals are less precise when calculating decimals. It could be that the 1 in the plimpton tablet annotates the use of decimals by approaching a solution as one example in 5÷7 converted to 12÷42 approching 1 gives you the same result to the 6th decimal iff by one digit!
I guess.. with 20 system taken by Maya and 60 with Babylonians so how would a 40 system look like ? or the double of 16 (hexadecimal). ( and then there is 12 system)
Nice. And surprise, you did it without using the original triangle.
10:42 thru 10:46 "point 0 5", "point 0 2" is not how I've heard anyone say it in North America, we mean "0 point 5" and "0 point 2" i.e. 0.5 and 0.2 -- was this a tiny error or a difference in pronunciation of numbers across the English-Speaking world?
I'd be curious to see an example of actually writing in practice. I've tried with sourdough crackers. It's not as easy as it sounds.
thanks
everyone always says they had no zero and point at our numbers yet where do we use zero anywhere? if i write the number 2020 there is no zero in it we have just decided to use the same 0 symbol to denote a multiple of 10 or two 0 for a multiple of hundred and so on - it is not beng used as zero - even decimal fractions 1.0008 means that the otherside of the . one zero is a divisable by 10 two zeros a hundred and so on - none of them mean zero - we hardly ever use that symbol to actually zero anywhere
The Sumerians are one of the Pre-Turkic peoples who lived in Anatolia. They laid the foundation of mathematics 2000 years before the Greeks. The works they wrote formed the basis of the Torah and other holy books.
I don't know if there exist equation to compare with the results.
But I can read it although in an another way.
a = 300 = 10•3
b = 21 = 3•7
c = 10 = 1•10
----------
a•b•c = 63,000
======
Cut the be so?
NG
Sven
Didn't Neugebauer already discover and publish this translation back in 1945? How are you guys saying you just figured this out?
Read the preface of the book here: books.google.com/books?id=i-juAAAAMAAJ&printsec=frontcover
Neugebauer and Sachs published Plimpton 322 in their 1945 book. They suggested that the tablet encoded information of a list of right triangles. A trigonometric interpretation has been suggested by a number of authors (pretty well any mathematician studying the table would be inclined to this view). But historians have dismissed this interpretation as anachronistic since angle measurement had not been invented at this time, and would take more than one thousand years to be developed. Our new point of view says that Plimpton 322 is indeed a trigonometric table, but it is a different, pre-astronomical type of trigonometry, in which angles and circular functions play no role, and which is strongly rooted in the OB sexagesimal arithmetic. And we explain just how the form of the table supports this.
The point is that when this was being promoted in the press, it was made to seem like nobody knew Plimpton 322 dealt with right triangles or that it was known at least by 1945 that the Babylonians understood the concept. It would have been more appropriate to mention the earlier work on this by the likes of Neugebauer.
If you look at the Wikipedia entry for Plimpton 322 you will see that it is dominated by the interpretation championed by Robson and others: that the tablet is just a scribal tool to create quadratic exercises for students. Our point is that the trigonometric interpretation only becomes clearer once you let go of our current understanding of angle-based trigonometry.
Am I to understand they had an intuitive understanding of the relationship of numbers boiled down to a "paint by numbers" system? That's pretty cool. I've never been any good at math. I'm only here because I was pondering the imperial system and wondering why feet are divided by 12 inches and inches by 16 and remembered hearing something about the Babylonians having some crazy math system and wondered if it was related. Super interesting stuff but a bit over my head. Maybe I shouldn't feel so bad though. pretty sure if I asked a college student what the sexagesimal system is they would probably think it had something to do with the new wave feminist rape culture rhetoric. lol
or maybe theres a text that describes the numbers as stat shape like when selecting pokemon on pokemon go maybe those are angles and the shape of the triangle will show a persons strengths maybe ones basic skills ones i.q and one is physical perhaps? just because it includes math doesnt mean its for math. usually the simplest answers are the right answers and soon as i see that your trying to add to the power it makes me think of the stat shape instantly dont know what it is but we know things we were never taught for a reason i know why but do you?
Top of the financial statement: amounts in dollars (thousands of dollars, millions of dollars, possibly billions of dollars). So maybe 12,407 doesn't always mean 12,407.
This will be handy when they return to reestablish the earth.
It looks to me like sexagesimal is being expressed in decimal as a polynomial. Then there is another underlying pattern there.
I like to help and say that I fell that Babylonian 10 is equal to 5+1 and not to 9+1 and i will say they got this system from dividing a circle bay its radius ....The Babylonian did not have fractions because they did not need it like 1/3 =0.333 for infinity.....
100/20=5 ok 100*3=300/60=5 ok. very interesting