Math 2.0

Functions of a complex variable Proposition

Equality of complex numbers

Builds on Real and imaginary parts

Statement

Let zz and ww be complex numbers. Then z=wz = w if and only if Re⁡z=Re⁡w\operatorname{Re} z = \operatorname{Re} w and Im⁡z=Im⁡w\operatorname{Im} z = \operatorname{Im} w.

Remarks

Here Re⁡\operatorname{Re} and Im⁡\operatorname{Im} are the real and imaginary parts. Thus one equation between complex numbers gives two equations between real numbers. For example, x+yi=3−2ix + y i = 3 - 2 i with x,yx, y real means x=3x = 3 and y=−2y = -2.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionEquality of complex numbers, statement, no evidence levelEquality of complex n…statement
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  1. Complex numbers object
  2. Real and imaginary parts definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Complex_number, Definition and basic operations
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Kindstatement, proposition
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MSC30A99
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UsesReal and imaginary parts

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