The Real Number System

Multiplication is being distributed over addition. It is the one axiom that links the two operations.

$\sqrt{25} = 5$, which is an integer and therefore rational. A root sign alone does not make a number irrational.

One counterexample is enough to disprove closure. Any pair where the second does not divide the first will do.

The multiplicative inverse returns the number to 1: $\dfrac35 \times \dfrac53 = 1$. Turn the fraction upside down.

Let $x = 0.2727\ldots$, so $100x = 27.2727\ldots$ and $99x = 27$. Then $x = \dfrac{27}{99} = \dfrac{3}{11}$.

The irrationals are not closed under multiplication. A single counterexample settles it; $\sqrt2 \times \sqrt3 = \sqrt6$ shows the product can also be irrational.