Ramble, Part 62: The Real and the Rational

That the real numbers are not the same as the rational came to the attention of mathematicians as a result of a geometrical problem. In the middle of the nineteenth century, the question arose: can we talk about the real numbers without making any reference to geometry? Can we describe them purely in terms of our familiar rational numbers? The answer proved to be yes, and in two distinctly different ways. This post will discuss one of those ways, devised by Cauchy; the other, due to Richard Dedekind, will be the topic of the next.

( Bunching Up )

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