# Algebraic Number

## Algebraic Number

An algebraic number is any number that is a root of a non-zero polynomial with rational coefficients.

**Put more simply**, when you have a polynomial like (for example):

### 2x^{2} - 4x + 2 = 0

Then **x**is algebraic.

Because:

- It is a non-zero polynomial
**x**is a root (ie**x**gives the result of**zero**for the function 2x^{2 }- 4x + 2)- the coefficients (the numbers 2, 4 and 2) are rational numbers

The polynomial could, of course, be simpler or more complicated than this example, just so long as the coefficents are rational.

## Not Algebraic? Then Transcendental!

If a number is not algebraic, it is called transcendental.

### Example: is √2 (the square root of 2) algebraic or transcendental?

√2 is the solution to x^{2} - 2 = 0, so is therefore algebraic

## Properties

All algebraic numbers are computable and therefore definable.

The set of algebraic numbers is countable.

The imaginary number **i** is algebraic (it is the solution to x^{2} + 1 = 0).

All rational numbers are algebraic, but an irrational number may or may not be algebraic.