nLab quadratic irrational number

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.

Related entries

• quadratic formula

• rational number

• irrational number

• number field

References

• Wikipedia, Quadratic irrational number