Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

A quadratic number is a number that is the root of some quadratic polynomial with rational coefficients. A quadratic irrational number is a quadratic number that is also irrational. The real quadratic numbers are quadratic numbers without a square root of $-1$ (imaginary unit), while complex quadratic numbers are quadratic numbers with a square root of $-1$. Real quadratic irrational numbers are real quadratic numbers that are irrational, while complex quadratic irrational numbers are complex quadratic numbers that are irrational.

## Properties

Every quadratic irrational number $x:K$ could be expressed as

$x = a + b\cdot\sqrt{c}$

where $a:\mathbb{Q}\subseteq K$, $b:\mathbb{Q}\backslash\{0\}\subseteq K$, and $c:\mathbb{Z}\backslash\{0\}\subseteq K$, and where the principal square root $\sqrt{c}$ is not a positive integer or $i$ times a positive integer. $x$ is a real quadratic irrational number if $0\lt c$ and $x$ is a complex quadratic irrational number if $c\lt 0$.

Every quadratic number field is a subfield of the complex quadratic numbers.