Rational Number is Algebraic

Theorem

Let $r \in \Q$ be a rational number.

Then $r$ is also an algebraic number.

Proof

Let $r$ be expressed in the form:

$r = \dfrac p q$

Consider the linear polynomial in $x$:

$q x - p = 0$

which has the solution:

$x = \dfrac p q$

Hence the result, by definition of algebraic number.

$\blacksquare$

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): algebraic number