Simplify two over three plus 𝑖.
And in the context of this question, 𝑖 isn’t just any normal variable. It’s representing a complex number or more specifically, the imaginary part of a complex number. Now at first sight, this looks like a pretty simple fraction in the first place, two over three plus 𝑖. But when we talk about simplifying complex fractions, we mean that we don’t want complex numbers in the denominator. So we have to look at ways of in- eliminating that 𝑖 from the denominator.
And before we go on and answer the question, let’s just remember that 𝑖 squared is defined as being equal to negative one. And that means that the value of 𝑖 is the square root of negative one. Overall, what we’re trying to do is find an equivalent fraction to two over three plus 𝑖 but without a complex part in the denominator.
And the way that we approach this is to think, is there something that we can multiply the top by and the bottom by in order to get an equivalent fraction which is going to eliminate the 𝑖 from this denominator here. Well, what we’re gonna do is pick the complex conjugate of the denominator. And the complex conjugate is the same expression. It’s got the same magnitude of real part and same magnitude of imaginary part except that the imaginary part is going to be negative. So for three plus 𝑖 or three plus one 𝑖, the complex conjugate is three minus 𝑖 or three minus one 𝑖. So I’m gonna multiply the numerator by three minus 𝑖 and the denominator by three minus 𝑖.
Now for the moment, I’m not actually gonna multiply out the numerator because we may have a factor there with which we’ll cancel with something on the denominator later. But the denominator is gonna consist of three plus 𝑖 times three minus 𝑖. And to work that out, I’m gonna do three times three then three times negative 𝑖 then 𝑖 times three then 𝑖 times negative 𝑖. Well, three times three is nine, three times negative 𝑖 is negative three 𝑖, 𝑖 times three is positive three 𝑖, and 𝑖 times negative 𝑖 — well positive times negative is negative and 𝑖 times 𝑖 is 𝑖 squared.
Now in simplifying that expression on the denominator, I’ve got negative three 𝑖 plus three 𝑖. They’re gonna cancel out. And that was the point of multiplying by the complex conjugate there. We knew that that was gonna cancel out those 𝑖s. Now that does leave us with this 𝑖 imaginary term here. But remember we said that 𝑖 squared is defined as being negative one, so I’m gonna replace 𝑖 squared with negative one. So that gives us nine minus negative one on the denominator. So nine take away negative one is the same as nine plus one, which is 10. So we’ve now got two times three minus 𝑖 on the numerator and 10 on the denominator. And two is a factor of the numerator and the denominator. If I divide two by two, I get one. If I divide 10 by two, I get five. So we’ve simplified our fraction here to three minus 𝑖 over five.
So there is our answer, which is essentially a simplified fraction that has no imaginary part in the denominator.