The Mysterious Case of the Irrational Numbers

In class today we were talking about sets of numbers:

$$\mathbb{N}$$ : the natural numbers,

$$\mathbb{W}$$ : the whole numbers,

$$\mathbb{Z}$$ : the integers,

$$\mathbb{Q}$$ : the rationals,

and $$\mathbb{R}$$ : the real numbers.

We defined the real numbers, somewhat lamely, to be the rationals plus all the irrationals! We spoke about the fact that surds like $$\sqrt{2}$$ are irrational.  I’d like to say a bit more about that here.

If you would like a gentle introduction to the idea of irrational numbers, here’s a Discovery Channel video about them:

If you want something a bit meatier, here’s a proof of the fact that $$\sqrt{2}$$ is irrational:

Now here are proofs of two rather surprising facts: firstly, that there are exactly the same number (the same infinity) of rational numbers as natural numbers, and secondly that there are more real numbers than rational numbers.  Infinity is weird!

These different infinities were first explored by Georg Cantor.  Here is his tragic story.  You will discover than mathematicians can be very mean! The first three parts are about Cantor:

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