ThmDex – An index of mathematical definitions, results, and conjectures.

Result R3379 on D370: Set of irrational numbers

Product of irrational numbers is not necessarily irrational

Formulation 1

Let $x \in \mathbb{R}$ be a D993: Real number such that

(i)	\begin{equation} x = \sqrt{2} \end{equation}

Then

(1)	$x$ is an D1243: Irrational number
(2)	\begin{equation} \sqrt{2} \sqrt{2} = 2 \end{equation}

Also known as

Class of irrational numbers is not closed under multiplication, Counterexample: two irrational numbers whose product is not irrational

Proofs

Proof 0

Let $x \in \mathbb{R}$ be a D993: Real number such that

(i)	\begin{equation} x = \sqrt{2} \end{equation}

Given result R590: Square root of two is irrational, this is clear. $\square$