Question

Show that 3-underroot2 is irrational​

Answer 1

Answer:

Step-by-step explanation:

Hello,

Basically, we need to prove that $\sqrt{2}$ is irrational

Let s assume that $\sqrt{2}$ is rational

it means that we can find p and q (q different from 0) two integers with no common factors other than 1

so that

   $\sqrt{2}=\dfrac{p}{q}$

And then we can write that

   $2=\dfrac{p^2}{q^2}\\ p^2=2q^2$

So $p^2$ is even so it means that p is even

so $p^2$ is divisible by 2*2=4

as  $p^2=2q^2$ it means that $q^2$ is even, meaning q is even

wait, p and q are then even !? but by definition they have no common factors. This is not possible.

so our assumption that $\sqrt{2}$ is rational is false

So it means that this is irrational

and then $3-\sqrt{2}$ is irrational too

Hope this helps