Give an example of multiplication of 2 imaginary numbers. The result of multiplication has to be a real number.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

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