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stoutfellowThat 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.
Take an irrational number - say, the square root of two. We can talk about it using its defining property - that it is the positive number whose square is two; but if we actually want to calculate with it, we'll generally resort to its decimal expansion, 1.4142.... For most purposes, the latter is sufficient, and indeed we can verify (more or less) the defining property on that basis as well. But what is it?
I've pointed out before that this infinite decimal expansion is actually shorthand for a sequence of finite - and hence rational - expansions: 1, 14/10, 141/100, 1414/1000, etc.; and we say that Sqrt[2] is the limit of this sequence. Augustin Cauchy took the bold step of declaring that Sqrt[2] is, not the limit of the sequence, but the sequence itself. And why not? Virtually everything that we need to do with the number, we can do with the sequence!
Can we capture every real number in this fashion? Certainly; every real number has a decimal expansion, and therefore can be regarded as being a sequence of rational numbers. (We can also regard rational numbers that way. There's an apparent danger of circularity, but we can distinguish between a rational number in itself and the same number regarded as a real number. It works.)
There are a couple of problems here, though. First, if we're thinking of real numbers as sequences of rational numbers, how do we tell which sequences to use? After all, there are sequences of rational numbers (e.g., 1, 2, 3, 4, 5, ...) which don't represent any real number. We could say something about "sequences which have a limit" - but, until we've defined the real numbers, sequences like 1, 1.4, 1.41, 1.414, ... don't have limits! Second, we gave a sequence for Sqrt[2] using its decimal expansion; but why decimal? Its binary expansion, 1.0110..., also gives an appropriate sequence: 1, 1, 5/4, 11/8, 11/8, .... For that matter, we could use the sequence 1, 3/2, 17/12, ... that arises from Heron's method of approximating square roots; and there are a host of others that work just as well.
Cauchy solved both problems at once. First, to say that a sequence converges to a limit is to say (loosely) that its members "bunch up" around that limit. But if they bunch up around a limit, they bunch together, and the rigorous notion of limit Cauchy devised can be brought to bear. Such a sequence has the following property: for any desired degree of closeness, there is a point in the sequence past which all of the numbers are that close to each other. A sequence with that property is called, today, a Cauchy sequence; this definition uses only the rational numbers, and provides us with the sequences we need. For the second problem, if two sequences converge to the same limit, then their terms must converge on one another; for any desired degree of closeness, there is a point past which all terms of both sequences are that close to each other.
This, then, is the proposal. Consider the collection of all Cauchy sequences of rational numbers. Regard two sequences as "the same" if they bunch together, as described above. Then the set of real numbers is simply the set of those (classes of) Cauchy sequences. We can define the usual algebraic operations in a perfectly natural way, and all the beloved properties of the real numbers emerge without too much difficulty.
And there's not a drop of geometry in it.
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Take an irrational number - say, the square root of two. We can talk about it using its defining property - that it is the positive number whose square is two; but if we actually want to calculate with it, we'll generally resort to its decimal expansion, 1.4142.... For most purposes, the latter is sufficient, and indeed we can verify (more or less) the defining property on that basis as well. But what is it?
I've pointed out before that this infinite decimal expansion is actually shorthand for a sequence of finite - and hence rational - expansions: 1, 14/10, 141/100, 1414/1000, etc.; and we say that Sqrt[2] is the limit of this sequence. Augustin Cauchy took the bold step of declaring that Sqrt[2] is, not the limit of the sequence, but the sequence itself. And why not? Virtually everything that we need to do with the number, we can do with the sequence!
Can we capture every real number in this fashion? Certainly; every real number has a decimal expansion, and therefore can be regarded as being a sequence of rational numbers. (We can also regard rational numbers that way. There's an apparent danger of circularity, but we can distinguish between a rational number in itself and the same number regarded as a real number. It works.)
There are a couple of problems here, though. First, if we're thinking of real numbers as sequences of rational numbers, how do we tell which sequences to use? After all, there are sequences of rational numbers (e.g., 1, 2, 3, 4, 5, ...) which don't represent any real number. We could say something about "sequences which have a limit" - but, until we've defined the real numbers, sequences like 1, 1.4, 1.41, 1.414, ... don't have limits! Second, we gave a sequence for Sqrt[2] using its decimal expansion; but why decimal? Its binary expansion, 1.0110..., also gives an appropriate sequence: 1, 1, 5/4, 11/8, 11/8, .... For that matter, we could use the sequence 1, 3/2, 17/12, ... that arises from Heron's method of approximating square roots; and there are a host of others that work just as well.
Cauchy solved both problems at once. First, to say that a sequence converges to a limit is to say (loosely) that its members "bunch up" around that limit. But if they bunch up around a limit, they bunch together, and the rigorous notion of limit Cauchy devised can be brought to bear. Such a sequence has the following property: for any desired degree of closeness, there is a point in the sequence past which all of the numbers are that close to each other. A sequence with that property is called, today, a Cauchy sequence; this definition uses only the rational numbers, and provides us with the sequences we need. For the second problem, if two sequences converge to the same limit, then their terms must converge on one another; for any desired degree of closeness, there is a point past which all terms of both sequences are that close to each other.
This, then, is the proposal. Consider the collection of all Cauchy sequences of rational numbers. Regard two sequences as "the same" if they bunch together, as described above. Then the set of real numbers is simply the set of those (classes of) Cauchy sequences. We can define the usual algebraic operations in a perfectly natural way, and all the beloved properties of the real numbers emerge without too much difficulty.
And there's not a drop of geometry in it.
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Ramble Contents
