I understand that different techniques can make calculations easier. I’m more dubious at the claim that base 60 permits more accurate fractions than base 10. The original paper in full can be found here.
My glance at the original shows they praise the Babylonians for using exact ratios for calculation, but that method also had problems dealing with irrationals.
Could one of mathperts explain the significance of approaching trig this way?
The big advantage of base 10 over base 60 is that base-10 numbers can be written using a set of ten simple symbols, distinct from alphabetic letters. To come up with 60 such distinct symbols would probably be unwieldy; instead the Babylonians built up 59 numeral symbols from just two symbols: one for 1 and one for 10. (Eventually they had a zero symbol also.)
Another area where base-60 would seem very inconvenient is multiplication. We memorize the 10x10 multiplication table and go from there. A 60x60 table would be hard to memorize and would require a huge amount of clay bricks to carry around!
However the Babylonians did have a very elegant method of multiplication. As shown at Wikipedia, as early as 2000 BC Babylonians apparently multiplied numerals using two table lookups, two subtractions and one addition. In my own experiments I have found that programming their method allowed multiplication significantly faster than the multiply instruction on many processors, including some of the late 1980’s (), and that today, 4000 years later, very few programmers are familiar with the technique. ( - Most processors have fast multiply hardware by now.)
Utter nonsense. It’s possible to produce a perfectly precise trig table in any base, provided that you limit yourself to certain particular angles. And if you don’t limit yourself to those angles, then it’s not possible in any base.
Like, the part of the paper where it says that this is “the only trigonometric table that is precise”? Or where it says that “Irrational numbers… are not actually necessary for trigonometry”?
How about basing our opinions on well established mathematical truths, like** Chronos** did? He’s not saying anything that’s remotely in doubt. You might as well claim that the square root of 2 is a rational number.
So I did read the paper and I am an honest to goodness real mathematician and everything.
The table contains a list of pythagorean triples (a, b, c) where a, b, c are integers, a < b < c, and a^2 + b^2 = c^2. (I’m not in the mood for sup tags)
The interesting part is that b is always a number that can be written 2^x 3^y 5^z. This means that 1/b has a finite representation in base 60. This also means that the tangent (a/b) and the secant (c/b) have finite representations in base 60.
That’s the whole story. It is pretty impressive that ancient people were able to generate these, but the authors of the paper are seriously blowing the importance of this out of proportion.
And, as Chronos alluded to earlier, it really is the authors of the paper doing this puffery and not just the journalists sensationalizing it.
“… with potential modern application because the base 60 used in calculations by the Babylonians permitted many more accurate fractions than the contemporary base 10 …”
I grabbed my book of trig tables published in the 1950’s and the angle measures are all in base-60 … minutes- and seconds-of-arc … were we really that backwards in the 20th Century? …
I suppose it IS interesting that 1000 years before Hipparchus, there existed a table of pythagorean triples such that useful trigonometric ratios could be precisely calculated. Presumably useful in building things utilizing right triangles if you weren’t all focused on a particular angle. Clearly, using base-60 helps in accomplishing such a table, since 60 has so many more integer factors than 10 does.
Of course, as long as this particular cuneiform tablet’s information isn’t ever connected to anything else we know about the Babylonians (for example: do their structures show that they used these triples in building things?), then it’s just a quaintly curious bit of calculation by someone with way too much time on their hands.
One more thing. The bit about the ‘only’ exact trig table is straight from the authors and pure bullshit.
When I taught calculus, I would tell the students early in the semester that I expected them to know have at least some trigonometry immediately recallable to their brain. This included some a small number of values for trig functions for common angles. Then I would put this trig table on the board as an easy way to remember those values.
t sin t cos t
0 sqrt(0/4) sqrt(4/4)
pi/6 sqrt(1/4) sqrt(3/4)
pi/4 sqrt(2/4) sqrt(2/4)
pi/3 sqrt(3/4) sqrt(1/4)
pi/2 sqrt(4/4) sqrt(0/4)
That is, without a doubt, an exact trig table and it’s been around a long time. I learned it in high school and I’m sure it wasn’t new then.
Base 60 gives access to more exact fractions, since 60 is evenly divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. And I would say that all the angles we care about most fit in a 60 degree circle (hell, they fit in a 12 degree circle). You know, degrees of 30, 45, 60, 90: angles used in the simplest right triangles.
So, using integers, it makes sense that they may have been able to calculate more triples exactly. It would just basically be trying them out. It would be inherent to measure things in groups of 60, due to their system, just like we measure things in groups of 10.
Maybe this was the most exact table up until that point in history. That would make sense. And, yes, in a way, they are doing some very basic trigonometry without the use of irrational numbers. The first might be a huge deal, but the second not so much.
Speaking of going back to the original, I am still curious about the tablet itself. Various descriptions and transcriptions of it present a table with four columns: a ratio, two integers, and finally a row number. (There are also a few errors in the table!!) But, looking at color and black-and-white photographs, I see what looks like five columns, and the table is headed by a couple of rows of text at the top. Can anyone describe/transcribe/translate the additional bits?
One more thing. At least one of the paper’s authors, N. J. Wildberger, is a real math professor. However, rational trigonometry is a fringe idea that is a bit of a hobby horse for him. Twelve years ago he wrote Divine Proportions: Rational Trigonometry to Universal Geometry.
I didn’t find anything about the other author, but I also didn’t look very hard.
It’s not “exact” because, while sqrt 2 is an exact number, it’s not a number you can use to create an “exact” answer with (since it’s irrational). I’m presuming they mean that the table is “exact” in the sense that all the values in the table are rational.
But if the sines and cosines are rational, then the angles won’t be, in any system of measuring angles you care to name. The whole point of trigonometry is that you have both the ratios and the angles.
