Real Numbers form Field

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Theorem

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


Proof

From Real Numbers under Addition form Abelian Group, we have that $\struct {\R, +}$ forms an abelian group.

From Non-Zero Real Numbers under Multiplication form Abelian Group, we have that $\struct {\R_{\ne 0}, \times}$ forms an abelian group.

Next we have that Real Multiplication Distributes over Addition.

Thus all the criteria are fulfilled, and $\struct {\R, +, \times}$ is a field.

$\blacksquare$


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