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Thought-Process to Discover Knowledge

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mathsComplex NumbersAlgebra of Complex Numbers

### Division of complex numbers

Complex numbers division is explained.

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Consider two complex numbers z_1=a_1+ib_1 and z_2 = a_2+ib_2. What is z_1 / z_2 in a+bi form?

• (a_1a_2+b_1b_2) + i (a_2b_1-a_1b_2)
• a_1/a_2+ib_1/b_2
• ((a_1a_2+b_1b_2)+i(a_2b_1-a_1b_2))/(a_2^2+b_2^2)
• ((a_1a_2+b_1b_2)+i(a_2b_1-a_1b_2))/(a_2^2+b_2^2)

The answer is '((a_1a_2+b_1xxb_2)+i(a_2b_1-a_1b_2))/(a_2^2+b_2^2)'

z_1=a_1+ib_1 and z_2 = a_2+ib_2. What is z_1 / z_2?
Solution :
z_3 = z_1 / z_2
quad quad = (a_1+ib_1)/(a_2+ib_2)
quad quad = (a_1+ib_1)/(a_2+ib_2) xx (a_2-ib_2)/(a_2-ib_2)
quad quad = ((a_1+ib_1)xx (a_2-ib_2))/((a_2+ib_2) xx (a_2-ib_2))
quad quad = ((a_1a_2+b_1b_2)+i(a_2b_1-a_1b_2))/(a_2^2+b_2^2)

Division of two complex numbers follows numerical expression laws and properties with i handled as (sqrt(-1))^2 = -1 to arrive at a+bi form.

Division of two complex numbers : For any complex number z_1=a_1+ib_1 in CC and z_2 = a_2+ib_2 in CC
z_1 / z_2 = ((a_1a_2+b_1b_2)+i(a_2b_1-a_1b_2))/(a_2^2+b_2^2)

Solved Exercise Problem:

Given z_1 = 10+5i and z_2=3-4i what is z_1/z_2?

• (6+8i)/5
• (3i+4)/25
• 10+11i
• (10+11i)/5
• (10+11i)/5

The answer is '(10+11i)/5'.

slide-show version coming soon