Irrational Numbers

Cold mornings and bright-faced first year students have signified the start of the teaching year here at the ANU. This semester, I’m running a couple of tutorials for MATH1115 Mathematics and Applications 1 Honours; a first year unit which gives students their first glimpse at analysis and linear algebra, both of which lie at the core of modern maths.

I’ve just come back from a tutorial where part of the discussion was centred on rational and irrational numbers. For those who need a reminder, a rational number is any number which can be written as a fraction of integers. For example

\frac{2}{3}, \frac{4}{1}, \frac{-7}{5}, \ldots

are all rational numbers. An alternative and useful definition is as follows; a rational number is any number which has a decimal expansion which at some point becomes periodic onwards. For example,


is a rational number, as the 914 recurs forever. Have a go at writing the above number as a fraction of two integers – it’s a good little exercise!

With these two equivalent definitions in mind we turn to irrational numbers, which we define as “numbers which are not rational”.

Anyways, I asked the class to come up with some irrational numbers and here’s what I got:

\sqrt{2}, \sqrt{3}, \sqrt{5},..., \sqrt{\text{not square}} ...

\pi, 2\pi, 3\pi, ...

e, 2e, 3e, ...

Cool. I then asked if somebody could write down an irrational number without using square roots, \pi or e. The class was silent. I hinted at using the alternative definition, to write down a number which does not have a decimal expansion which at some point becomes periodic. We came up with this guy:


Sure, the decimal expansion here has a pattern; first there is one zero between the ones, then there are two, then three and so on. But this is not periodic! It’s not the same string occurring over and over again. And that is what makes the above number irrational. From then on, the students had no problem inventing all sorts of crazy irrational numbers. I think it was a useful exercise to give them a good feeling for irrationality.

Interested readers can click here for the slides for a talk on irrational numbers I gave once. I also made a video of the proof that e is irrational. You can find it here.

2 thoughts on “Irrational Numbers

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