Question

prove that : 1-2√3 is irrational​

Answer 1

1+23–√ is rational by contradiction. So we have

1+23–√=pq

for p,q∈Z. Notice we can move 1 to the other side to obtain

23–√=pq−1=p−qq

Now, divide by 2 to obtain

3–√=p−q2q

But this is a rational number. Hence 3–√∈Q. And we have reached a contradiction since 3–√ is not rational.