Algebraic number: Difference between revisions
imported>Catherine Woodgold (inserting "and", and a link) |
imported>Catherine Woodgold (Adding part of Greg Woodhouse's explanation, with slight modification, as a footnote) |
||
| Line 1: | Line 1: | ||
An '''algebraic number''' is any [[complex number]] that is a root of a [[polynomial]] with rational coefficients. Any polynomial with rational coefficients can be converted to one with integer coefficients by multiplying through by the least common multiple of the denominators, and every complex root of a polynomial with integer coefficients is an algebraic number. | An '''algebraic number''' is any [[complex number]] that is a root of a [[polynomial]] with rational coefficients. Any polynomial with rational coefficients can be converted to one with integer coefficients by multiplying through by the least common multiple of the denominators, and every complex root of a polynomial with integer coefficients is an algebraic number. | ||
The algebraic numbers include all rational numbers, and both sets of numbers, rational and algebraic, are [[countable set|countable]]. The algebraic numbers form a [[field (mathematics)|field]]; in fact, they are the smallest [[algebraically closed field]] with characteristic 0. | The algebraic numbers include all rational numbers, and both sets of numbers, rational and algebraic, are [[countable set|countable]]. The algebraic numbers form a [[field (mathematics)|field]]; in fact, they are the smallest [[algebraically closed field]] with characteristic 0. <ref>If 1 + 1 = 0 the characteistic is said to be 2; if 1 + 1 + 1 = 0 the characteristic is said to be 3, and forth. If there is no <math>n</math> such that adding 1 <math>n</math> times gives 0, we say the characteristic is 0. A field of positive characteristic need not be finite. </ref> | ||
Real or complex numbers that are not algebraic are called [[transcendental number]]s. | Real or complex numbers that are not algebraic are called [[transcendental number]]s. | ||
| Line 8: | Line 8: | ||
<math> \sqrt{2}</math> is an algebraic number, as it is a root of the polynomial <math>x^2-2</math>. Similarly, the imaginary unit <math>i</math> is algebraic, being a root of the polynomial <math>x^2+1</math>. | <math> \sqrt{2}</math> is an algebraic number, as it is a root of the polynomial <math>x^2-2</math>. Similarly, the imaginary unit <math>i</math> is algebraic, being a root of the polynomial <math>x^2+1</math>. | ||
==Notes== | |||
<references/> | |||
Revision as of 11:05, 29 April 2007
An algebraic number is any complex number that is a root of a polynomial with rational coefficients. Any polynomial with rational coefficients can be converted to one with integer coefficients by multiplying through by the least common multiple of the denominators, and every complex root of a polynomial with integer coefficients is an algebraic number.
The algebraic numbers include all rational numbers, and both sets of numbers, rational and algebraic, are countable. The algebraic numbers form a field; in fact, they are the smallest algebraically closed field with characteristic 0. [1]
Real or complex numbers that are not algebraic are called transcendental numbers.
Examples
is an algebraic number, as it is a root of the polynomial . Similarly, the imaginary unit is algebraic, being a root of the polynomial .
Notes
- ↑ If 1 + 1 = 0 the characteistic is said to be 2; if 1 + 1 + 1 = 0 the characteristic is said to be 3, and forth. If there is no such that adding 1 times gives 0, we say the characteristic is 0. A field of positive characteristic need not be finite.
