Irrational numbers cannot be represented in the real physical world, I’ve been told. So my question is: if I have a one meter by one meter square of wood, which is a perfect square precisely to the atom, is its diagonal length not sqrt2?
In: Mathematics
Yes and no. It depends on what you mean by “represent”, and correspondingly what numbers one does consider to be representable in the real physical world.
With regard to the square of wood, how do you know the side lengths are exactly one meter? It seems silly to say it’s possible to physically represent the number 1.00000… but not 1.41421356… You could also question the assumption the sides are known to be straight, parallel, and at right angles.
On the other hand, in the spirit of plane geometry, one can take the view like you do, that in the context of certain acceptable idealizations that are clearly evoked by physical things, numbers like sqrt(2) are perfectly representable. You just have to make sure you don’t directly equate such assertions with those regarding numerical precision of measurements.
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