Review question

# Why is at most one of these numbers rational? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8276

## Question

What does it mean to say that a number $x$ is irrational?

Prove by contradiction statements A and B below, where $p$ and $q$ are real numbers.

A: If $pq$ is irrational, then at least one of $p$ and $q$ is irrational.

B: If $p+q$ is irrational, then at least one of $p$ and $q$ is irrational.

Disprove by means of a counterexample statement C below, where $p$ and $q$ are real numbers.

C: If $p$ and $q$ are irrational, then $p+q$ is irrational.

If the numbers $e$, $\pi$, $\pi^2$, $e^2$ and $e\pi$ are irrational, prove that at most one of the numbers $\pi+e$, $\pi-e$, $\pi^2-e^2$, $\pi^2+e^2$ is rational.