Transcendental Numbers

By Sally Itani and Issam Mneimneh

Two types of numbers:

In mathematics, there are two types of numbers: Algebraic and transcendental.


An algebraic number is any number that can be written as a root of a polynomial with rational coefficients.


Where did transcendental Numbers come from?

Transcendence theory is concerned with the converse question of algebraic numbers:

if we are given a complex number "a", can we find a polynomial P with rational coefficients such that P(a) = 0?


If not, then the number is called transcendental.

Properties of Transcendental Numbers

1) A transcendental number cannot be written as a root of a polynomial equation with rational coefficients.

2) Every transcendental number is irrational, but the opposite is not true.

3) A transcendental number cannot be written as a sum of algebraic terms.

4) For any two transcendental numbers "a" and "b", at least one of "ab" or "a+b" is transcendental.

Examples of Transcendental Numbers