The complex numbers are thefield of numbersof the form, where and arereal numbers andi is theimaginary unit equal to thesquare root of,. When a single letter is used to denote a complex number, it is sometimes called an "affix." In component notation, can be written. Thefield of complex numbers includes thefield ofreal numbers as asubfield.

The set of complex numbers is implemented in theWolfram Language asComplexes. A number can then be tested to see if it is complex using the commandElement[x,Complexes], and expressions that are complex numbers have theHead ofComplex.

Complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions. However, recognition of this fact is one that took a long time for mathematicians to accept. For example, John Wallis wrote, "These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen) are reputed to imply that the Case proposed is Impossible" (Wells 1986, p. 22).

Through theEuler formula, a complex number

(1)

may be written in "phasor" form

(2)

Here, is known as thecomplex modulus (or sometimes the complex norm) and is known as thecomplex argument orphase. The plot above shows what is known as anArgand diagram of the point, where the dashed circle represents thecomplex modulus of and the angle represents itscomplex argument. Historically, the geometric representation of a complex number as simply a point in the plane was important because it made the whole idea of a complex number more acceptable. In particular, "imaginary" numbers became accepted partly through their visualization.

Unlike real numbers, complex numbers do not have a natural ordering, so there is no analog of complex-valued inequalities. This property is not so surprising however when they are viewed as being elements in thecomplex plane, since points in a plane also lack a natural ordering.

Theabsolute square of is defined by, with thecomplex conjugate, and the argument may be computed from

(3)

Thereal andimaginary parts are given by

			(4)
			(5)
			(6)
			(7)

de Moivre's identity relatespowers of complex numbers for real by

(8)

Apower of complex number to a positive integer exponent can be written in closed form as

(9)

The first few are explicitly

			(10)
			(11)
			(12)
			(13)

(Abramowitz and Stegun 1972).

Complex addition

(14)

complex subtraction

(15)

complex multiplication

(16)

andcomplex division

(17)

can also be defined for complex numbers. Complex numbers may also be taken to complex powers. For example,complex exponentiation obeys

(18)

where is thecomplex argument.

See also

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References

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Complex Number." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/ComplexNumber.html

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