Put negative 18 minus nine 𝑖 over three 𝑖 in the form 𝑎 plus 𝑏𝑖.
In order to write our fractional expression in the form 𝑎 plus 𝑏𝑖, we essentially need to find a way to get rid of this imaginary number on the denominator. And so there are two things that we could do. Let’s consider both methods. The first method involves rewriting our fraction somewhat. By reversing the process for adding fractions, we can write it as negative 18 over three 𝑖 minus nine 𝑖 over three 𝑖. And this is really useful because 𝑖 divided by itself will be one. So this second fraction becomes nine divided by three, which is of course equal to three. So this simplifies a little bit to negative 18 over three 𝑖 minus three.
Negative three then is the 𝑎 part of our expression. But what do we do with negative 18 over three 𝑖? Well, we can simplify it a little bit. 18 divided by three is six and three divided by three is one. So it simplifies further to negative six over 𝑖 minus three. We need to find a way to remove this 𝑖 from the denominator of our fraction. And so we go back to the definition of 𝑖. 𝑖 is the number such that 𝑖 squared is equal to negative one. So if we can square the denominator, we’ll end up with a simple integer that we can work with. Of course, we cannot just multiply the denominator of our expression. We need to do the same to the numerator. This is the equivalent then of multiplying by one. And so it doesn’t change the value of our fraction. We get then negative six 𝑖 over 𝑖 squared minus three.
But of course, we know that 𝑖 squared is equal to negative one. So we get negative six 𝑖 over negative one minus three. And since negative six divided by negative one is six, this becomes six 𝑖 minus three. We want to write the real part of our complex number first. So we just switch these numbers around. And we find that negative 18 minus nine 𝑖 over three 𝑖 is negative three plus six 𝑖.
Now there is an alternative way to answer this. And that is by multiplying both the numerator and the denominator by 𝑖 straightaway. When we do this, we get three 𝑖 squared as the denominator of our fraction. And the numerator is 𝑖 times negative 18 minus nine 𝑖. We distribute that 𝑖 across the parentheses on the numerator of our fraction to get negative 18𝑖 minus nine 𝑖 squared. Then we replace the 𝑖 squared on the denominator with negative one. And so the denominator of our fraction becomes three times negative one which is just negative three. We then replace the 𝑖 squared on our numerator with negative one.
And so we get negative 18𝑖 minus nine times negative one which is negative 18𝑖 plus nine. We now see that we can divide both parts of our numerator by that negative three. Negative 18𝑖 divided by negative three is positive six 𝑖, and nine divided by negative three is negative three. Once again, we’ve shown that writing our number in the form 𝑎 plus 𝑏𝑖 and we get negative three plus six 𝑖.