Examples

Complex multiplication

Compute the products:

(a) $(1-i)(4-7i)$ $$ (b) $(2+5i)(2-5i)$

Solution

(a) $(1-i)(4-7i)$

Expand:

$$(1-i)(4-7i)\gg \gg 4-7i-4i+7i^2$$

Simplify $i^2$:

$$\gg \gg 4-7i-4i+7(-1)\gg \gg -3-11i$$

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(b) $(2+5i)(2-5i)$

Expand:

$$(2+5i)(2-5i)\gg \gg 4-10i+10i-25i^2$$

Simplify $i^2$:

$$\gg \gg 4-10i+10i-25(-1)\gg \gg 29$$

Complex division

Compute the following divisions of complex numbers:

(a) ${\displaystyle \frac{1}{-3+i}}$ $$ (b) ${\displaystyle \frac{1}{i}}$ $$ (c) ${\displaystyle \frac{1}{7i}}$ $$ (d) ${\displaystyle \frac{2+5i}{2-5i}}$

Solution

(a) ${\displaystyle \frac{1}{-3+i}}$

Conjugate is $-3-i$:

$$\frac{1}{-3+i}\gg \gg \frac{1}{-3+i}\cdot \frac{-3-i}{-3-i}$$

Simplify:

$$\gg \gg \frac{-3-i}{9+1}\gg \gg \frac{-3}{10}+\frac{-1}{10}i$$

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(b) ${\displaystyle \frac{1}{i}}$

Conjugate is $-i$:

$$\frac{1}{i}\gg \gg \frac{1}{i}\cdot \frac{-i}{-i}\gg \gg -i$$

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(c) ${\displaystyle \frac{1}{7i}}$

Factor out the $1/7$:

$$\frac{1}{7i}\gg \gg \frac{1}{7}\cdot \frac{1}{i}$$

Use $\frac{1}{i}=-i$:

$$\gg \gg \frac{1}{7}\cdot (-i)\gg \gg \frac{-1}{7}i$$

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(d) ${\displaystyle \frac{2+5i}{2-5i}}$

Denominator conjugate is $2+5i$:

$$\frac{2+5i}{2-5i}\gg \gg \frac{2+5i}{2-5i}\cdot \frac{2+5i}{2+5i}$$

Simplify:

$$\gg \gg \frac{4+20i+25i^2}{4+25}\gg \gg \frac{-21}{29}+\frac{20}{29}i$$