An example of two imaginary numbers which when multiplied give a real number is of the form a+ bi and a - bi. They are called complex conjugates.
When two complex conjugates are multiplied, the imaginary portions cancel out and only the real numbers are left behind.
(a+ bi)*(a - bi)
=> a^2 + a*b*i - a*b*i - b^2*i^2
we see that a*b*i and - a*b*i can be eliminated. As i^2 = -1
=> a^2 + b^2
Complex conjugates when multiplied by each other give a real number
For instance, if we'll multiply an imaginary number by it's conjugate, we'll get a difference of squares and the result will be a real value.
z = a + b*i and the conjugate is z' = a - b*i
We'll apply the formula:
(a-b*i)(a+b*i) = a^2 - b^2*i^2, but i^2 = -1
(a-b*i)(a+b*i) = a^2 + b^2
We'll put a = 4 and b = 2i
(4+2i)(4-2i) = 4^2 - (2i)^2
(4+2i)(4-2i) = 16 - 4i^2
But i^2 = -1
(4+2i)(4-2i) = 16-(-4)
(4+2i)(4-2i) = 16+4
(4+2i)(4-2i) = 20
The result of multiplication of 2 imaginary numbers, (4+2i)(4-2i), is the real number 20.