Rational Addition is Closed

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 field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.

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

$\blacksquare$

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Introduction
• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$
• 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.1$ Real Numbers
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The Real Number System: $3$