Tuesday, December 26, 2006

Who's Got Your Number?

More from Richard Preston's fascinating 1992 piece on pi:

Pi is a transcendental number. A transcendental number is a number that exists but can't be expressed in any finite series of either arithmetical or algebraic operations ... Pi is a transcendental number because it transcends the power of algebra to display it in its totality ... [Pi] can't be written on a piece of paper, not even on a piece of paper as a big as the universe. In a manner of speaking, pi is indescribable and can't be found.
Preston goes on:
In 1873, Georg Cantor, a Russian-born mathematician who was one of the towering intellectual figures of the nineteenth century, proved that the set of transcendental numbers is infinitely more extensive than the set of algebraic numbers. That is, finite algebra can't find or describe most numbers. To put it another way, most numbers are infinitely long and non-repeating in any rational form of representation. In this respect, most numbers are like pi.

Cantor's proof was a disturbing piece of news, for at that time very few transcendental numbers were actually known ... [His] proof of the existence of uncountable multitudes of transcendental numbers resembled a proof that the world is packed with microscopic angels -- a proof, however, that does not tell us what the angels look like or where they can be found; it merely proves that they exist in uncountable multitudes ...

Cantor's proof disturbed some mathematicians because, in the first place, it suggested that they had not yet discovered most numbers, which were transcendentals, and in the second place that they lacked any tools or methods that would determine whether a given number was transcendental or not.
Wish I'd made more use of transcendental numbers when I was struggling with algebra in junior high school. After all, my math teachers sure felt my answers were irrational ...

--"The Mountains of Pi," by Richard Preston, The New Yorker, March 2, 1992, pp. 39 & 60.

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