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Thought-Process to Discover Knowledge

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mathsComplex NumbersProperties of Complex Number Arithmetic

### Distributive Property

Complex multiplication is distributive over complex addition.

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What does 'distribute' mean?

• to restrict; to arrest

Consider three complex numbers z_1, z_2, z_3 in CC, and the following

add first z_2+z_3 and multiply that by z_1.
multiply z_1 xx z_2 and z_1 xx z_3 and added the results.
Will they give the same result?

• Yes, the results will be the same
• Yes, the results will be the same
• No, the results will differ

The answer is "Yes, the results will be the same".

The complex numbers addition and multiplication is considered to be operations on numerical expressions of real numbers.

Given z_1 = a_1+ib_1, z_2 = a_2+ib_2, and z_3 = a_3+ib_3. What is z_1 xx (z_2 + z_3)?

• z_1 xx z_2 + z_1 xx z_3
• z_1 xx z_2 + z_1 xx z_3
• z_1 xx z_2 + z_3

The answer is 'z_1 xx z_2 + z_1 xx z_3'

z_1 xx (z_2 + z_3)
quad quad = z_1 xx z_2 + z_1 xx z_3

proof:
z_1 xx (z_2 + z_3)
quad quad = (a_1+ib_1) xx (a_2+a_3 + i (b_2+b_3))
quad quad = a_1a_2-b_1b_2 + i(a_1b_2+a_2b_1)
quad quad quad quad +a_1a_3-b_1b_3 + i(a_1b_3+a_3b_1)
quad quad = z_1 xx z_2 + z_1 xx z_3

•  Product with sum of complex numbers is sum of, products with the complex numbers - distributive.

Distributive Property of Complex Numbers: For any given complex numbers z_1, z_2, z_3 in CC
z_1 xx (z_2 + z_3)
quad quad = z_1 xx z_2 + z_1 xx z_3

Solved Exercise Problem:

Given z_1z_2= 3+4i and z_2z_3=1-3i what is (z_1+z_3)z_2?

• 5i
• 4+i
• 4+i
• 4+7i
• -5i

The answer is '4+i'

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