# Definition:Addition/Rational Numbers

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## Definition

The addition operation in the domain of rational numbers $\Q$ is written $+$.

Let:

- $a = \dfrac p q, b = \dfrac r s$

where:

- $p, q \in \Z$
- $r, s \in \Z_{\ne 0}$

Then $a + b$ is defined as:

- $\dfrac p q + \dfrac r s = \dfrac {p s + r q} {q s}$

This definition follows from the definition of and proof of existence of the field of quotients of any integral domain, of which the set of integers is an example.

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 2$: Example $2.1$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 4$: Number systems $\text{I}$: The rational numbers - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**addition** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**addition**

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- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*: Introduction: $\S 4$