Precalculus question

Question

What makes a number algebraic? What did Charles Hermite discover? Is the number pi algebraic or transcendental, and why? Where did the term 'transcendental number' come from?

Answer 1

An algebraic number is a number that is a root of a polynomial equation with rational coefficients. Algebraic numbers may be rational or irrational and real or complex. For example, 1/2, √2, and 3 - i are all algebraic numbers.

Contrarily, transcendental numbers are non-algebraic, i.e. they are not the roots of a polynomial with rational coefficients. Transcendental numbers may be real or complex but by definition transcendental numbers are irrational.

The two most well-known transcendental numbers are π and e, Euler's number.

Although when most people think of examples of irrational numbers they think of algebraic numbers such as √2 or √3/2 etc., in fact, "most" irrational numbers are transcendental.

Charles Hermite first proved that e is transcendental, and the subsequent proof that π is also transcendental was modeled on Hermite's.