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Sat, 04 Feb 2006

Approximations to pi
In an earlier post I mentioned G.H. Hardy's astonishment when he first encountered Ramanujan's approximation to π:

[ Addendum 20060402: I inexplicably put in the wrong formula here. The one I meant to put in is in this followup article. ]

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Pure Mathematics
Pure Mathematics
with kickback
no kickback
I'm planning to write a blog article about Gaussian integers, and in the course of my research I picked up my old, battered copy of G.H. Hardy's Pure Mathematics. I haven't spent as much time reading this book as I should have; it's full of good stuff. There didn't seem to be anything in there about the Gaussian integers (digression: What's next in the sequence 1, 2, 4, 6, 10, 14, 16, 24, 26?) but while scanning the index I noticed there was an entry for Ramanujan, so I checked it out.

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 are

;

these are correct to 3, 3, 8, and 9 places respectively.

All 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.

2 Chronicles 4:2 Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; 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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A Mathematician's Apology
A Mathematician's Apology
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Hardy was an atheist, and was very strongly anti-religious. (C.P. Snow says, in the preface to A Mathematician's Apology, that "On a quiet and lovely May evening at Fenner's, round about the same period, the chimes of six o'clock fell across the ground. 'It's rather unfortunate', said Hardy simply, 'that some of the happiest hours of my life should have been spent within sound of a Roman Catholic church.'") He was only too glad to take little potshots at the Bible at any opportunity, even in his pure mathematics textbook—or especially so, since he could get in an additional dig through the implied comparison with Ramanujan. It's certainly true that the ancient Hebrews were not mathematically sophisticated. But this particular potshot, which Hardy is far from the only person to take, seems to me to be unearned.


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