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'A Brief Introduction to Complex Numbers' printed from http://nrich.maths.org/

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The students from St Stephens School, Australia, found the following examples:
  • Pairs of complex numbers whose sum is a real number:
2+i, 2-i
4+5i, 3-5i
  • Pairs of complex numbers whose sum is an imaginary number: 
2i+1, -i-1
i+2, i-2
  • Pairs of complex numbers whose product is a real number: 
1+2i, 1-2i
2+3i, 4-6i
  • Pairs of complex numbers whose product is an imaginary number:
3+3i, 3+3i
1, i

Sina Sanaizadeh from Hinde House Secondary School, Sheffield, sent in the following explanations:
  • In general, what would you need to add to a+bi to get a real number? 
We want to remove the imaginary part of a+bi to get a real number, so we want to add a  complex number of the form c-bi.
  • Or an imaginary number?
We want to remove the real part of a+bi to get an imaginary number, so we want to add a complex number of the form -a+ci.
  • In general, what would you need to multiply by a+bi to get a real number?
Consider the product of the complex numbers a+bi and c+di:
(a+bi)(c+di) = (ac-bd) + (ad+bc)i
For this product to be real, the imaginary part must be 0, so ad+bc = 0
As a and b are fixed, must have $\frac{c}{d}$ = $\frac{-a}{b}$.
So, in general, for the product of two complex numbers to be real, the ratio of the real to imaginary parts of each complex number must be equal up to a minus sign.
  • Or an imaginary number?
Again, consider (a+bi)(c+di) = (ac-bd) + (ad+bc)i
For this product to be imaginary, the real part must be 0, so ac-bd = 0
As a and b are fixed, must have $\frac{d}{c}$ = $\frac{a}{b}$.
So, in general, for the product of two complex numbers to be imaginary, the ratios of the real to imaginary parts of each complex number must be the reciprocal of the other.