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Sat, 04 Feb 2006
Approximations to pi
The entry concerns approximations to π, and in particular π ≅ (13/25)√146. Hardy says "If R is the earth's radius, the error in supposing AM to be its circumference is less than 11 yards." Hardy continues, mentioning the well-known approximations 22/7 and 355/113, about which I am sure I will have something to say in the future, in connection with continued fractions. He then says:
A large number of curious approximations will be found in Ramanujan's Collected papers, pp. 23-39. Among the simplest areAll of which, in my usual digressive style, is only an introduction to the main point of this note, which is that Hardy finishes the section by saying:
It is stated in the Bible (1 Kings vii. 23, 2 Chron. iv. 2) that π = 3.Let's look at what the Bible actually says:
1 Kings 7:23 And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.I think there are two arguments that must be made here in defense of the Bible. First, one can infer the supposed value of π only if one assumes that the molten sea was a geometrically exact circle. But the sea is not described as circular; it is described only as "round". It could have been an approximate circle; or it could have been a mathematically exact ellipse; or it could have had many other shapes. Veterans' Stadium in Philadelphia was often described as "round" also, and it was not a circle, but an octorad. Watermelons are round, but are not circular, or spherical. The other argument I would make is that it is not at all clear that any attempt was being made to state the sizes with mathematical exactitude. The use of round numbers throughout (no pun intended) supports this. If the Bible had said that the molten sea was thirteen cubits across and thirty-nine cubits around, I might agree that Hardy was right to complain. But if we suppose that the measurements are only being reported to one significant figure, we cannot conclude whether the value of π that was used was 3 or 3.1416—or 2.7 for that matter. If you say that your house is forty feet tall, you would be rightly annoyed to have G.H. Hardy to come and ridicule you for being unable to distinguish between the numbers 40 and 41.37.
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