Question

Prove that \[\sqrt{5}\] is irrational.........

Answer 1

I believe we would use a proof by contradiction.

Answer 2

similar to the one with \[\sqrt2 \]right? :)

Answer 3

yeah..

Answer 4

Could you do that for me?

Answer 5

we will do that together...:)

Answer 6

Of course... hehe

Answer 7

suppose that \(\sqrt{5}\) is a rational number so we can write \[\sqrt{5}=\frac{p}{q}\ \ \ \ \ \gcd(p,q)=1 \]square both sides\[p^2=5q^2\]

Answer 8

Yes.. I know it till now

Answer 9

now suppose \(q\) is an even number then \(p\) will be even and this is impossible.

Answer 10

then let \(q\) to be an odd number...\(p^2\) divides \(5\) so does \(p\) ... so \(p=5k\)

plug in the equation \(25k^2=5q^2\) and \(q^2=5k^2\)

so \(q^2\) divides \(5\) so does \(q\) and this is a contradiction

Answer 11

Oh!

Answer 12

Thank you so much!

Answer 13

yw :)