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Thought-Process to Discover Knowledge

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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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