Definition:Addition/Rational Numbers

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

• Rational Addition is Well-Defined

• Rational Addition is Commutative
• Rational Addition is Associative

• Rational Addition Identity is Zero
• Inverse for Rational Addition

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

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