# Question Video: Simplifying Expressions That Include Complex Numbers Mathematics

Simplify (2 + 4𝑖)/𝑖.

01:56

### Video Transcript

Simplify two plus four 𝑖 over 𝑖.

To work out how to divide two plus four 𝑖 by 𝑖, we recall the definition of 𝑖. It’s a solution to the equation 𝑥 squared equals negative one. And we say that 𝑖 squared is equal to negative one. Or often 𝑖 is equal to the square root of negative one.

If we consider this fraction as two plus four 𝑖 divided by the square root of negative one, we can see that, to simplify, we’d need to perform the same process as rationalizing the denominator when we’re dealing with any other radical. We multiply both the numerator and the denominator of our fraction by the square root of negative one.

In fact, we know that the square root of negative one is 𝑖. So we’re going to be multiplying both the numerator and the denominator of this fraction by 𝑖. And we’re allowed to do that because multiplying by 𝑖 over 𝑖 is the same as multiplying by one. Essentially, we’re creating an equivalent fraction.

Let’s apply the distributive property for 𝑖 multiplied by two plus four 𝑖. 𝑖 multiplied by two is two 𝑖, and 𝑖 multiplied by four 𝑖 is four 𝑖 squared. Now of course 𝑖 squared is equal to negative one. So our expression becomes two 𝑖 plus four multiplied by negative one, which is negative four plus two 𝑖.

On the denominator, we have 𝑖 multiplied by 𝑖, which is of course 𝑖 squared, which is negative one. So we can rewrite two plus four 𝑖 over 𝑖 as negative four plus two 𝑖 over negative one. And now we’re simply dividing by a real number. And to divide a complex number by a real number, we divide the real part and then separately divide the imaginary part. Negative four divided by negative one is four, and two 𝑖 divided by negative one is negative two 𝑖. And we fully simplify two plus four 𝑖 over 𝑖. It’s four minus two 𝑖.