PlanetMath: complex number

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complex number

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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 . For $a,b \in \mathbb{R}$ , the equivalence class of in $\mathbb{C}$ is usually denoted , and one has .

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

$\displaystyle d(a_1+b_1 i, a_2+b_2 i) := \sqrt{(a_2-a_1)^2 + (b_2-b_1)^2}.$

"complex number" is owned by djao.

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See Also: complex

Other names:

$\mathbb{C}$

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Cross-references: standard metric, field, algebraically closed, equivalence class, principal ideal, reals, variable, polynomial ring, quotient ring, ring
There are 87 references to this entry.

This is version 4 of complex number, born on 2001-10-23, modified 2002-08-26.
Object id is 471, canonical name is ComplexNumber.
Accessed 20138 times total.

Classification:

AMS MSC:	12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
	30-00 (Functions of a complex variable :: General reference works )
	32-00 (Several complex variables and analytic spaces :: General reference works )

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