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Properties of Complex Number Arithmetic

Properties of Complex Number Arithmetic

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 »  Complex Multiplication is distributive over complex addition.

    →  `z_1 xx (z_2 + z_3) = z_1 xx z_2 + z_1 xx z_3`

Distributive Property

plain and simple summary

nub

plain and simple summary

nub

dummy

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

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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Complex multiplication is distributive over complex addition.


Keep tapping on the content to continue learning.
Starting on learning "Distributive Property". ;; In this page, you will learn that the complex multiplicative is distributive over complex addition.

What does 'distribute' mean?distributive property illustration

  • to share; to spread
  • to restrict; to arrest

The answer is 'to share; to spread'

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
  • 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_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`

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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

  • `5i`
  • `4+i`
  • `4+7i`
  • `-5i`

The answer is '`4+i`'

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What does 'distribute' mean?
share;spread
to share; to spread
restrict;arrest
to restrict; to arrest
The answer is 'to share; to spread'
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;s;same
Yes, the results will be the same
no;differ
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 z1 = a1+i b1, z2 = a2+i b2, and z3 = a3+i b3. What is z1 multiplied, z2 + z3?
1
z1 into z2 + z1 into z3
2
z1 into z2 + z3
The answer is 'z1 into z2 + z1 into z3'
z1 into, z2 + z3 ;; equals z1 into z2 + z1 into z3 ;; proof: z1 into, z2 + z3 ;; equals a1+i b1 xx, a2+a3 + i b2+b3 ;; equals a1 a2-b1 b2 + i a1 b2+a2 b1 +a1 a3-b1 b3 + i a1 b3+a3 b1 ;; equals z1 multiplied z2 + z1 multiplied z3
Product with sum of complex numbers is the sum of, products with the complex numbers- distributive.
Distributive Property of Complex Numbers: For any given complex numbers z1, z2, z3 in complex numbers ;; z1 multiplied, z2 + z3 ;; equals z1 multiplied z2, + z1 multiplied z3
Given z1 z2= 3+4i, and z2 z3=1-3i what is z1+z3, multiplied by z2?
1
2
3
4
minus 5i
The answer is ' 4+i '

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