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Thought-Process to Discover Knowledge

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Complex multiplication is distributive over complex addition.



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What does 'distribute' mean?distributive property illustration

  • to share; to spread
  • 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
  • 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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