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mathsComplex NumbersAlgebra of Complex Numbers

### Multiplication of two Complex numbers

Complex numbers multiplication 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 xx z_2?

• (a_1a_2-b_1b_2) + i (a_2b_1+a_1b_2)
• (a_1a_2-b_1b_2) + i (a_2b_1+a_1b_2)
• a_1a_2+ib_1b_2
• (a_1a_2+b_1b_2)+i(a_1b_2+a_2b_1)

The answer is '(a_1a_2-b_1b_2) + i (a_2b_1+a_1b_2)'. This is from the associative and distributive laws of real numbers extended to numbers with sqrt(-1).

z_1=a_1+ib_1 and z_2 = a_2+ib_2. What is z_1 xx z_2?
Solution :
z_3 = z_1 xx z_2
quad quad = (a_1+ib_1)xx(a_2+ib_2)
quad quad = a_1 xx(a_2+ib_2)
quad quad quad quad +i b_1xx (a_2+ib_2)
quad quad = a_1a_2+ia_1b_2+ib_1a_2
quad quad quad quad +i^2b_1b_2
quad quad = (a_1a_2-b_1b_2)
quad quad quad quad + i(b_1a_2+a_1b_2)

Multiplication of two complex numbers follows numerical expression laws and properties with sqrt(-1) handled as per the property (sqrt(-1))^2 = -1

Multiplication 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 xx z_2 = (a_1a_2-b_1b_2)
quad quad quad quad quad +i(a_2b_1+a_1b_2)

Solved Exercise Problem:

Given z_1=1+2i and z_2=3-i what is z_1 xx z_2?

• 5+5i
• 5+5i
• -5+5i
• 5-5i
• -5-5i

The answer is '5+5i'

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