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Thought-Process to Discover Knowledge

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mathsComplex NumbersAlgebra of Complex Numbers

Subtraction of Complex Numbers

Complex numbers subtraction 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-z_2`?

  • `(a_1-a_2) + i (b_1-b_2)`
  • `(a_1-a_2) + i (b_1-b_2)`
  • `a_1-ib_1`
  • `a_2-ib_2`

The answer is '`(a_1-a_2) + i (b_1-b_2)`'. This is from the associative and distributive laws of real numbers extended to numbers with `sqrt(-1)`.

Complex number Subtraction:
`z_1-z_2`
`quad quad =a_1+ib_1 - (a_2+ib2)`
`quad quad =a_1-a_2+ib_1-i_b2` (associative law)
`quad quad = a_1-a_2+ i(b_1-b2)` (distributive law)
`quad quad =`(a_1-a_2) + i (b_1-b_2) (real and imaginary parts of result)`

Subtraction of two complex numbers is the subtraction of real and imaginary parts individually.

Subtraction of 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-z_2 = (a_1-a_2)+i(b_1-b_2)`

Solved Exercise Problem:

Given `z_1 = 2.1+i` and `z_2 = -2.1+i` What is `z_1-z_2`?

  • `4.2+2i`
  • `4.2`
  • `4.2`
  • `4.2-2i`
  • `-4.2+2i`

The answer is '`4.2`'

                            
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