# How do you prove that the square root of 14 is irrational?

##### 2 Answers

A rational number is expressed by ratio of integers.

#### Explanation:

The only square roots that are rational numbers are those who are perfect squares.

Use proof by contradiction...

#### Explanation:

Suppose

Then

Without loss of generality, we can suppose that

#(p/q)^2 = 14#

So:

#p^2 = 14 q^2#

In particular,

If

So:

#14 q^2 = (2k)^2 = 4 k^2#

Dividing both sides by

#7 q^2 = 2 k^2#

So

#7 q^2 = 2 (7m)^2 = 7*14m^2#

Divide both sides by

#q^2 = 14 m^2#

So

So

Now

So our supposition is false and therefore our hypothesis that