Video Transcript
Simplify 13 over 13 plus two 𝑖.
In this question, we’re looking to divide a real number by a complex number. And so we recall that to divide by a complex number, we write our quotient of the fraction and then we multiply both the numerator and the denominator of that fraction by the conjugate of the denominator. Then we find the conjugate of a complex number by changing the sign of the imaginary part. So if we have a complex number 𝑧, which is 𝑎 plus 𝑏𝑖, the conjugate, which we denote as 𝑧 bar or sometimes 𝑧 star, is 𝑎 minus 𝑏𝑖.
Now, the complex number that we’re dividing by is 13 plus two 𝑖. So we’re going to multiply by the conjugate. And we find that by changing the sign of the two 𝑖. This means the conjugate is 13 minus two 𝑖. And we need to multiply both the numerator and the denominator by this number. Let’s begin with doing that on the denominator. We’re going to multiply 13 by 13 first. That’s 169. Then we multiply 13 by negative two 𝑖. And we get negative 26𝑖. Two 𝑖 minus 13 is positive 26𝑖. And finally, we multiply the last term in each expression. Two 𝑖 times negative two 𝑖 is negative four 𝑖 squared.
Next, we spot that negative 26𝑖 plus 26𝑖 is zero. But of course, we also know that 𝑖 squared is equal to negative one. So this simplifies to 169 minus four times negative one. And that’s 169 plus four, which is 173. And so we have the denominator of our fraction.
The numerator is a little more straightforward. We simply take the 13 here and we multiply it by both parts of our complex number. 13 times 13 is 169, and 13 times negative two 𝑖 is negative 26𝑖. We’re going to represent this using the standard form of a complex number, 𝑎 plus 𝑏𝑖. And when we do, we get 169 over 173 minus 26 over 173𝑖.
