# Why is the square root of 5 an irrational number?

##### 1 Answer

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#### Explanation:

Here's a sketch of a proof by contradiction:

Suppose

Without loss of generality, we may suppose that

Then by definition:

#5 = (p/q)^2 = p^2/q^2#

Multiply both ends by

#5 q^2 = p^2#

So

Then since

So

So we have:

#5 q^2 = p^2 = (5m)^2 = 5*5*m^2#

Divide both ends by

#q^2 = 5 m^2#

Divide both ends by

#5 = q^2/m^2 = (q/m)^2#

So

Now

So our hypothesis that