# Real Numbers form Ring

## Theorem

The set of real numbers $\R$ forms a ring under addition and multiplication: $\struct {\R, +, \times}$.

## Proof

From Real Numbers under Addition form Infinite Abelian Group, $\struct {\R, +}$ is an abelian group.

We also have that:

Real Multiplication is Closed:
$\forall x, y \in \R: x \times y \in \R$
Real Multiplication is Associative:
$\forall x, y, z \in \R: x \times \paren {y \times z} = \paren {x \times y} \times z$

Thus $\struct {\R, +}$ is a semigroup.

Finally we have that Real Multiplication Distributes over Addition:

 $\ds \forall x, y, z \in \R: \,$ $\ds x \times \paren {y + z}$ $=$ $\ds x \times y + x \times z$ $\ds \paren {y + z} \times x$ $=$ $\ds y \times x + z \times x$

Hence the result, by definition of ring.

$\blacksquare$