# Rational Addition is Closed

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## Theorem

The operation of addition on the set of rational numbers $\Q$ is well-defined and closed:

- $\forall x, y \in \Q: x + y \in \Q$

## Proof

Follows directly from the definition of rational numbers as the quotient field of the integral domain $\struct {\Z, +, \times}$ of integers.

So $\struct {\Q, +, \times}$ is a field, and therefore a priori $+$ is well-defined and closed on $\Q$.

$\blacksquare$

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$ - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The Real Number System: $3$