Why we cannot divide 3+5i by 1+i?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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The answer is easy!

The divisor 1 + i cannot divide 3 + 5i because it is not a real number. The division of complex numbers follows the next procedure. First, we'll have to multiply both numerator and denominator by the conjugate of divisor.

`(3+5i)/(1+i) = ((3+5i)(1-i))/((1+i)(1-i))`

We notice that the special product from denominator returns a difference of two squares:

`((3+5i)(1-i))/(1- i^2) = ((3+5i)(1-i))/(1+1) = ((3+5i)(1-i))/2`

Since the divisor is a real number we can perform now the division.

We'll remove the brackets from numerator:

`((3+5i)(1-i))/2 = (3 - 3i + 5i - 5i^2)/2`

`((3+5i)(1-i))/2 = (3 + 2i + 5)/2`

`` `((3+5i)(1-i))/2 = (8 + 2i)/2`

`((3+5i)(1-i))/2 = (2(4 + i))/2`

`` `((3+5i)(1-i))/2 = 4 + i`

The result of division of the complex numbers 3 + 5i and 1 + i is the complex number 4 + i.

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