Question

When asked to prove that the difference of any irrational number and any rational number is irrational, a student began, "Suppose not. That is, suppose the difference of any irrational number and any rational number is rational." What is wrong with beginning the proof in this way?

Answer 1

First you have to define what a rational and irrational number is and they would be:

a / b, where b can never be 0.

Irrational are the numbers with non-periodic decimals, how the roots are not exact, the number pi, etc.

The student says "the difference of any irrational number and any rational number is rational"

The student's mistake is to use "any" since not any rational number would be the case that fulfills this.

That is, the correct thing is that there is an irrational number and a rational number in such a way that their difference is rational.