Title: Course Presentation

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Division of Complex Numbers

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The following rule applies to dividing two complex numbers:

expand a fraction by the conjugate number of the complex number in its denominator.

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\(\displaystyle\frac{a+bi}{c+di}=\frac{(a+bi)\cdot\color{green}{(c-di)}}{(c+di)\cdot\color{green}{(c-di)}}=\frac{ac-adi+cbi-bdi^2}{c^2+d^2}=\frac{ac+bd}{c^2+d^2}+\frac{bc-ad}{c^2+d^2}i\)

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Calculate \(\;\displaystyle\frac{2-i}{1+i}\)

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Example 1

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Practice

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\(\displaystyle\frac{2-i}{1+i} = \frac{(2-i)(1-i)}{(1+i)(1-i)} = \frac{2-2i-i+i^2}{1-i^2} = \frac{1-3i}{2} = \frac{1}{2}-\frac{3}{2}i\)

Multiplication Fill in the Blanks

Calculate

Q: 1) \(\frac{25}{3-4i}\) = [...]

2) \(\frac{25i}{3-4i}\) = [...]
A: 3+4i; -4+3i
Feedback (0 to 100%): Solution: \(1)\;\frac{25}{3-4i} = \frac{25(3+4i)}{(3-4i)(3+4i)} = \frac{25(3+4i)}{3^3+4^2} =\frac{25(3+4i)}{25}=3+4i;\hskip8cm\) \(2)\;\frac{25i}{3-4i} = \frac{25i(3+4i)}{(3-4i)(3+4i)} = \frac{25i(3+4i)}{3^3+4^2}=\frac{25i(3+4i)}{25}=(3+4i)i=-4+3i\).

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Calculate \(\;\displaystyle\frac{5}{1-2i}=\frac{5\cdot(1+2i)}{(1-2i)\cdot(1+2i)}=\frac{5+10i}{5}=1+2i\)

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Example 2