### Quick description

If and are two distinct rational numbers written in their lowest terms, then , which implies that , which implies that . Therefore, if is a real number and we can find a sequence of rational numbers (in their lowest terms, with denominators tending to infinity) such that (which is equivalent to saying that ), then cannot be rational.^{♦} Loosely speaking, if you can approximate well by rationals, then is irrational. This turns out to be a very useful starting point for proofs of irrationality.

### Example 1

Let us construct inductively a sequence of rationals that approximate . (This is not necessarily the best proof of the irrationality of but it gives an easy illustration of the technique.) We begin with , and observe that . If were , then we would have , so the "" at the end of this is our error term in the first approximation. Now suppose we have defined and in such a way that . Then set and . (The justification for this choice is that if were then would be too, as can easily be checked.) Then

Thus, we have constructed a sequence of rationals , with denominators tending to infinity, such that for every . But from this we deduce that , and therefore that . Since

### Example 2

To prove the irrationality of , we start with the power-series expansion:

We then set to be . This is a fraction with denominator that divides . It differs from by

Also, this difference is strictly positive and not zero. Therefore, , so is irrational.

### General discussion

The two proofs given so far can easily be, and usually are, presented in other ways that do not mention the basic principle explained in the quick description. However, sometimes that basic principle plays a much more important organizational role: one is given a number to prove irrational, and one attempts to do so by finding a sequence of good rational approximations to .

Another point is that if is irrational then such a sequence always exists: one can take the convergents from the continued-fraction expansion of . But this observation is less helpful than it seems, since for many important irrational numbers (such as ) there does not seem to be a nice formula for the continued-fraction expansion. The point of the method explained here is that it is much more flexible: *any* sequence of good approximations will do (and sequences are indeed known that prove the irrationality of ).

## Comments

## Inline comments

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## One probably needs to insist

Tue, 05/05/2009 - 19:30 — emertonOne probably needs to insist that to avoid the triviality of approximating a rational number by the constant sequence . I'm not sure exactly how this is usually phrased in this context, so I will let someone else make the changes in the text.

## I was careful about that in

Tue, 05/05/2009 - 21:57 — gowersI was careful about that in the examples, but as you imply it is annoyingly difficult to find a good formulation for the quick description. I've gone for denominators tending to infinity when the fractions are written in their lowest terms. But in the case of it is slightly easier to use the not-equal-to- formulation. If anyone has strong views about this, feel free to change it.

## Perhaps this could be

Thu, 07/05/2009 - 13:43 — emertonPerhaps this could be explicitly discussed in the article; not in the quick description, but perhaps further down, in the general discussion. One could explain (just as in your comment) that there are various slightly different approaches to avoiding : either by having a positive lower bound on the difference, or by having tend to , staying coprime to .

Presumably at some point there will also be other articles on Diophantine approximation, including Liouville numbers and Roths' theorem, which will be linked to this article. Then one will be able to point out that even if is already known to be irrational (so that in particular is guaranteed to be positive), the question of bounding this quantity from above or below is still of great interest.

Another example that would be nice to include on this page eventually would be Apery's proof of irrationality of . (I suspect that this has been suitably massaged by this stage that one could write down a pretty simple sequence of fractions that does the job.) I don't have the resources to hand at the moment to do this, but if I get a chance at some later point, I might add it. If someone else wants to beat me to it, please do!

## Inline comments

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## e+e^(-1) has sequence of

Thu, 18/02/2010 - 22:15 — winstonsmithe+e^(-1) has sequence of approximations 2(1 + 1/2! + 1/4! + ... 1/(2n)!) and no immediately obvious (to me) formula for the continued-fraction expansion.

## Apery Constant

Sun, 25/04/2010 - 12:29 — ogerardThe use of this method for Zeta(3) is more involved but goes deeper into the principles of this method. I suggest that someone try to write up a description of this as Example 3.