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Thought-Process to Discover Knowledge

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mathsComplex NumbersAlgebra of Complex Numbers

Addition of two Complex numbers

Complex numbers addition 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 Addition:
`z_1+z_2`
`quad quad =a_1+ib_1 + a_2+ib_2`
`quad quad =a_1+a_2+ib_1+ib_2` (associative law of addition)
`quad quad = a_1+a_2+ i(b_1+b_2)` (distributive law of multiplication over addition)
`quad quad =(a_1+a_2) + i (b_1+b_2)` (real and imaginary parts of result)

Addition of two complex numbers is the addition of real and imaginary parts individually.

Addition 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 two complex numbers `3+2i` and `-1+i`, what is the sum?

  • `3+i`
  • `2+3`
  • `2+3i`
  • `2+3i`
  • `2-i`

The answer is '`2+3i`'

                            
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