How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

The reason the 0.999 . . . problem is difficult is that it brings our intuitions into conflict. We would like the sum of an infinite series to play nicely with arithmetic manipulations like the ones we carried out on the previous pages, and this seems to demand that the sum equal 1. On the other hand, we would like each number to be represented by a unique string of decimal digits, which conflicts with the claim that the same number can be called either 1 or 0.999 . . . , as we like. We can’t hold on to both of these desires at once; one must be discarded. In Cauchy’s approach, which has amply proved its worth in the two centuries since he invented it, it’s the uniqueness of the decimal expansion that goes out the window. We’re untroubled by the fact that the English language sometimes uses two different strings of letters (i.e., two words) to refer synonymously to the same thing in the world; in the same way, it’s not so bad that two different strings of digits can refer to the same number.

Two words referring to the same thing: kettle and pot