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The students from St Stephens School, Australia, found the following examples:
Sina Sanaizadeh from Hinde House Secondary School, Sheffield, sent in the following explanations:
A Brief Introduction to Complex Numbers
Age 14 to 18
Challenge Level 
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.
(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.
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.
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