Math 2.0

definition

Real and imaginary parts

Definition

Let z=a+biz = a + b i be a complex number, with aa and bb real. The real part of zz is Re⁡z=a\operatorname{Re} z = a. The imaginary part of zz is Im⁡z=b\operatorname{Im} z = b.

Remarks

The imaginary part is a real number: Im⁡(2+3i)=3\operatorname{Im}(2 + 3 i) = 3, not 3i3 i. The maps Re⁡\operatorname{Re} and Im⁡\operatorname{Im} are additive and R\mathbb{R}-linear, but they do not respect products. With the conjugate, Re⁡z=(z+zˉ)/2\operatorname{Re} z = (z + \bar{z})/2 and Im⁡z=(z−zˉ)/(2i)\operatorname{Im} z = (z - \bar{z})/(2 i). A complex number zz is real when Im⁡z=0\operatorname{Im} z = 0, and purely imaginary when Re⁡z=0\operatorname{Re} z = 0.

Explanations

L1 human

Each complex number has two real coordinates. For z=4−7iz = 4 - 7 i, the real part is 44 and the imaginary part is −7-7.

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  1. Web https://en.wikipedia.org/wiki/Complex_number, Definition and basic operations

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