Why is base 60 more precise for trigonometry, can you give an example?

Question

I came across this article 3,700-year-old Babylonian tablet rewrites the history of maths - and shows the Greeks did not develop trigonometry.

However unlike today’s trigonometry, Babylonian mathematics used a base 60, or sexagesimal system, rather than the 10 which is used today. Because 60 is far easier to divide by three, experts studying the tablet, found that the calculations are far more accurate.

Giving aside pearls like because 60 is far easier to divide by three.

Why are base 60 calculation more accurate, and why 60 being divisible by 3 helps?

Answer 1

One of the important claims the paper makes is that this is the only known trigonometric table that does not use any approximations. When using the table, approximations are only introduced when calculating the final result.

As I understand it there are two main reasons why the table has no approximations:

• The first one is that they think of trigonometry in terms of ratios of lengths (eg. length of the short side of a right angle triangle over the length of the diagonal) - while nowadays it is much more common to think of trigonometry in terms of angles ;
• The second reason is because they use the sexagecimal system.

Because 60 is a multiple of 2, 3 and 5 you can write 1/2, 1/3 and 1/5 as exact numbers.

For example, when using base 10 we can write 1/2 = 0.5 - but we can't write 1/3, because 10 is not divisible by 3. When using base 60, you can write 1/2=0."30" but also 1/3=0."20" (Where "30" and "20" are the symbols for the corresponding decimal values)

So to recapitulate:

• They considered trigonometry as ratios ;
• The sexagesimal system allowed them to write more exact ratios than the decimal system does.

The paper gives examples showing how the table is more precise than other known tables (though not as precise as what we get using computers). Whether this is only because of the two reasons above is not in itself demonstrated.

Answer 2

The article is nonsense; the choice of base 60 has nothing to do with geometry. There are a number of excellent books on Babylonian mathematics, e.g. by Neugebauer, Hoyrup, Friberg and Robson.

Answer 3

The sentences you quoted from The Telegraph are probably a garbled and misunderstood version of the following (correct) bits from the article (emphasis in last sentence added by me):

4.1. Reciprocal tables

Regular numbers [defined earlier as numbers of the form $2^a \times 3^b \times 5^c$] had a special place in OB [Old Babylonian] arithmetic and were used for division. Specifically, division by the regular number $n$ (the igi) was performed by looking up its reciprocal $\bar{n}$ (the igibi) on a table of reciprocals, followed by another look up on the multiplication table for n. The ‘standard’ table (see Table 2) lists 30 reciprocal pairs (Neugebauer and Sachs, 1945, 11), but there are variations which omit the reciprocal pairs $(1.12, 50)$, $(1.15, 48)$, $(1.20, 45)$ or add $(\frac23, 40)$ at the beginning (Friberg, 2007, 68).

Reciprocals of irregular numbers, such as $\dfrac{1}{7}$, do not exist in this system. Instead, division by an irregular number would be performed by multiplication by an approximate reciprocal, such as those found in YBC 10529 (Neugebauer and Sachs, 1945, 16). Nevertheless, this sexagesimal system allows for more division calculations to be performed exactly compared with our decimal system.

Note the last sentence. All this says is the fact that more reciprocals can be represented exactly in the sexagesimal system than can be represented exactly as decimal digits.

For example, in the decimal system, the reciprocal of $3$ (namely $\frac13$) cannot be represented exactly except as a fraction: which is fine. In the system that the Babylonians used, the reciprocal of $3$ would be represented by what we could write as as $0.\langle 20 \rangle$, or simply just “$20$”. This was a kind of floating-point representation where we don't bother to represent powers of $60$. A nice exposition of this system is the following paper of Knuth:

Donald E. Knuth, Ancient Babylonian Algorithms, Communications of the ACM, Volume 15 Issue 7, July 1972 (Pages 671-677) (PDF)

Answer 4

The main justification for saying it's more accurate I think is that with base 60 there are many more regular numbers whose reciprocals are expressible in finite form, so this sexagesimal table is richer (more entries) than one, say, using base 10. If you use the table to do interpolation, the interpolated values will be more accurate with more entries. A crucial restriction is that in Plimpton 322, the second-longest side of the triangle has to be a regular number as it's used as a divisor to get the first column, which was hypothetically used as an index into the table.

Plimpton 322 has 15 rows covering slopes equivalent to an angle of 45.24°-54.11°, average spacing well under 1°. If under its same constraints it was expanded to cover angles up to 90°, there'd be 44 rows, averaging 1° spacing. But a similar table requiring the second longest side to be regular using base 10 would only allow 6 rows, at 53.13°, 54.22°, 64.01°, 65.24° 76.00° and 77.32°. I.e., fail.

Answer 5

Before attempting to answer the question about the advantages of the sexagesimal base for trigonometry, one should ask whether Plimpton 322 is dealing with trigonometry at all, as claimed in that paper. Many respected experts believe this is total nonsense and the media coverage an unjustified hype. I only read the original paper in one hour, which is too little to judge, but my first impression was that it adds very little real insights over previous scholarship (Britton, Proust, Høyrup, Friberg, Robson, ...), while only adding their own highly-speculative thin layer of interpretation on top of that solid foundation. The part at the end about possible numerical uses of the table in Plimpton 322 to solve practical problems that we associate with trigonometry might be worth consideration, but it cannot be safely backed by the historical record from that period. Stricto sensu it is not trigonometry, which by definition is about measuring angles in triangles. No mathematician or astronomer between the Seleucid period and the 18th century would have considered this to be trigonometry, by the way.