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How do I divide complex numbers?


 

in Mathematics by Bronze Status (8,758 points) | 11 views

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Division is not like any of the previous operations in that it requires knowledge of the conjugate of a complex numbers. We can express the division of two complex numbers as:

$\dfrac{z_{1}}{z_{2}}=\dfrac{x+y i}{a+b i}$
process:
1. Find the conjugate of the denominator: In our case, the denominator is $z_{2}$, which is $a+b i . \bar{z}_{2}$ is $a-b i$. That is our conjugate

2. Times the top and bottom by the conjugate: $\dfrac{x+y i}{a+b i} \times \dfrac{a-b i}{a-b i}$

3. Multiply like normal: $\dfrac{x+y i}{a+b i} \times \dfrac{a-b i}{a-b i}=\dfrac{x(a-b i)+y i(a-b i)}{a(a-b i)+b i(a-b i)}$

4. Answer: $\dfrac{a x-b x i+a y i-b y i^{2}}{a^{2}-a b i+a b i-b^{2} i^{2}}=\dfrac{(a x-b y)+(a y-b x) i}{a^{2}+b^{2}}$
by Bronze Status (8,758 points)

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