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 function 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
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): algebraic number
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