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# complex number

The ring of complex numbers $\mathbb{C}$ is defined to be the quotient ring of the polynomial ring $\mathbb{R}[X]$ in one variable over the reals by the principal ideal $(X^{2}+1)$. For $a,b\in\mathbb{R}$, the equivalence class of $a+bX$ in $\mathbb{C}$ is usually denoted $a+bi$, and one has $i^{2}=-1$.

The complex numbers form an algebraically closed field. There is a standard metric on the complex numbers, defined by

$d(a_{1}+b_{1}i,a_{2}+b_{2}i):=\sqrt{(a_{2}-a_{1})^{2}+(b_{2}-b_{1})^{2}}.$ |

Related:

Complex

Synonym:

$\mathbb{C}$

Type of Math Object:

Definition

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Reference

Groups audience:

## Mathematics Subject Classification

12D99*no label found*30-00

*no label found*32-00

*no label found*46L05

*no label found*18B40

*no label found*46M20

*no label found*17B37

*no label found*22A22

*no label found*81R50

*no label found*22D25

*no label found*

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