Video: Dividing a Complex Number by an Imaginary Number

Simplify (2 + 4𝑖)/𝑖.

01:45

Video Transcript

Simplify two plus four 𝑖 divided by 𝑖.

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 𝑖 squared is equal to negative one. This means that negative 𝑖 squared is equal to positive one. One way of simplifying our expression is to therefore multiply the numerator and denominator by either 𝑖 or negative 𝑖.

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

We generally write complex numbers in the form 𝑧 is equal to 𝑥 plus 𝑦𝑖, where 𝑥 is the real part and 𝑦 the imaginary part of the complex number. The simplified version of two plus four 𝑖 divided by 𝑖 is therefore equal to four minus two 𝑖.

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