Prove that `sqrt(2)` is irrational.

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Prove that `sqrt(2)` is irrational:

Assume that `sqrt(2)` is rational; then `sqrt(2)=p/q` for some `p,q in NN` . Without loss of generality, we can assume that gcd(p,q)=1 (They have no common divisor -- if there is a common divisor, divide both p and q by that divisor to reduce to simplest form)

Then:

`sqrt(2)=p/q ==>2=p^2/q^2`

`==>p^2=2q^2`

This means that `p` is a multiple of 2 (p is even). So let `p=2k` .

Now:

`p^2=2q^2==>(2k)^2=2q^2`

`==>4k^2=2q^2`

`==>2k^2=q^2`

Thus q is even. But this contradicts our asssumption that p and q had no common divisor.

Therefore `sqrt(2)` is irrational.

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