Idempotent Element/Examples/One is Idempotent for Multiplication

Example of Idempotent Element

$1$ is idempotent under the operation of multiplication in the set of real numbers $\R$.

Proof

From Non-Zero Real Numbers under Multiplication form Abelian Group, $\struct {\R_{\ne 0}, \times}$ forms an abelian group whose identity element is $1$.

The result follows from Identity Element is Idempotent.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): idempotent
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): idempotent