# Question Video: Dividing a Complex Number by an Imaginary Number Mathematics • 12th Grade

Simplify (2 + 4π)/π.

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### Video Transcript

Simplify two plus four π divided by π.

There are lots of ways of trying to simplify this expression. Whichever method we use, we need to try and get a real value on the denominator. From our knowledge of imaginary and complex numbers, we know that π squared is equal to negative one. This means that negative π squared is equal to positive one. One way of simplifying our expression is to therefore multiply the numerator and denominator by either π or negative π.

Multiplying the numerator by negative π gives us negative two π minus four π squared. Multiplying π by negative π on the denominator gives us negative π squared. As already established, we know that this is equal to one. The expression therefore simplifies to negative two π minus four π squared. As π squared is equal to negative one, this in turn can be rewritten as negative two π minus four multiplied by negative one, which gives us an answer of negative two π plus four.

We generally write complex numbers in the form π§ is equal to π₯ plus π¦π, where π₯ is the real part and π¦ the imaginary part of the complex number. The simplified version of two plus four π divided by π is therefore equal to four minus two π.