# Rational Multiplication Identity is One

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## Theorem

The identity of rational number multiplication is $1$:

- $\exists 1 \in \Q: \forall a \in \Q: a \times 1 = a = 1 \times a$

## Proof

From the definition, the field $\struct {\Q, +, \times}$ of rational numbers is the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.

From the properties of the quotient structure, elements of $\Q$ of the form $\dfrac p p$ where $p \ne 0$ act as the identity for multiplication.

From Equal Elements of Field of Quotients, we have that:

- $\dfrac p p = \dfrac {1 \times p} {1 \times p} = \dfrac 1 1$

Hence $\dfrac p p$ is the identity for $\struct {\Q, \times}$:

\(\ds \frac a b \times \frac p p\) | \(=\) | \(\ds \frac {a \times p} {b \times p}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac a b\) | Equal Elements of Field of Quotients |

Similarly for $\dfrac p p \times \dfrac a b$.

Hence we *define* the unity of $\struct {\Q, +, \times}$ as $1$ and identify it with the set of all elements of $\Q$ of the form $\dfrac p p$ where $ \in \Z^*$.

$\blacksquare$