Question

prove square root 7.5 is a irrational no​

Answer 1

The problem I'm having with this proof is that I'm not sure if my proof actually proves the theorem correct or if I'm using circular reasoning.

Theorem:

Prove that the square root of any irrational number is irrational.

Proof:

=> Suppose not. The square root of any irrational number is rational.

=> Let m be some irrational number. It follows that m−−√ is rational.

=> By definition of a rational number, there are two positive integers p and q such that m−−√=qp

=> m=q2p2

=> q2 and p2 are integers, and by definition of a rational number, q2p2 is rational

=> m is irrational and is equal to the rational number q2p2. This is a contradiction.

=> Thus, the square root of any irrational number is irrational.