Math Interactive
13. Complex Numbers
13.1. Complex number basics
13.1.1. Complex Arithmetic I
Important

Real and imaginary parts

Given a complex number z=x+yiz = x+y\mathrm{i}, we call xx the real part of z,z, denoted Re(z)=x\textrm{Re}(z)=x and yy the imaginary part of zz, denoted Im(z)=y\textrm{Im}(z)=y.

Addition and subtraction

Addition and subtraction are relatively straightforward: we add/subtract the corresponding real and imaginary parts.

For example,

(1+2i)+(3−5i)=4−3i(1+2i)−(3−5i)=−2+7i\begin{align*} (1+2\mathrm{i})+(3-5\mathrm{i}) &= 4-3 \mathrm{i} \\ (1+2\mathrm{i})-(3-5\mathrm{i}) &= -2+7\mathrm{i} \end{align*}

Powers of i\mathrm{i}

i2=−1\mathrm{i}^2 = -1

Because i\mathrm{i} can be thought of as ‘‘−1"``\sqrt{-1}", we have

i2=−1,i3=i2⋅i=−i,i4=(i2)2=1,⋯\begin{align*} &\boxed{\mathrm{i}^2 = -1}, \\ &\mathrm{i}^3 = \mathrm{i}^2 \cdot \mathrm{i} = -\mathrm{i}, \\ &\mathrm{i}^4 = (\mathrm{i}^2)^2 = 1, \\ &\cdots \end{align*}

Multiplication

Multiplication of complex numbers can be treated as an exercise in algebraic expansion, with the additional fact that i2=−1.\mathrm{i}^2 = -1.

For example,

(1+2i)(3−5i)=3−5i+6i−10i2=3+i+10=13+i\begin{align*} &(1+2\mathrm{i})(3-5\mathrm{i}) \\ &= 3-5\mathrm{i}+6\mathrm{i}-10\mathrm{i}^2 \\ &= 3 + \mathrm{i} + 10 \\ &= 13 + \mathrm{i} \end{align*}

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