nLab quadratic irrational number




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 1-1 (imaginary unit), while complex quadratic numbers are quadratic numbers with a square root of 1-1. Real quadratic irrational numbers are real quadratic numbers that are irrational, while complex quadratic irrational numbers are complex quadratic numbers that are irrational.


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

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

where a:Ka:\mathbb{Q}\subseteq K, b:\{0}Kb:\mathbb{Q}\backslash\{0\}\subseteq K, and c:\{0}Kc:\mathbb{Z}\backslash\{0\}\subseteq K, and where the principal square root c\sqrt{c} is not a positive integer or ii times a positive integer. xx is a real quadratic irrational number if 0<c0\lt c and xx is a complex quadratic irrational number if c<0c\lt 0.

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


Last revised on May 9, 2021 at 05:24:00. See the history of this page for a list of all contributions to it.