Video Transcript
Simplify four plus 𝑖 over four minus 𝑖.
Here we have the quotient of two complex numbers written in algebraic form. And so we recall that we can divide by a complex number by multiplying both the numerator and the denominator by the conjugate of that complex number. If we have a complex number 𝑧 in the form 𝑎 plus 𝑏𝑖, its conjugate, which is denoted by 𝑧 bar, is found by changing the sign of the imaginary part. So the conjugate of 𝑧 is 𝑎 minus 𝑏𝑖.
Now, the denominator, the divisor here, is four minus 𝑖. So the conjugate of four minus 𝑖 must be four plus 𝑖. And so to simplify our fraction, we need to multiply both the numerator and the denominator by four plus 𝑖. And that gives us four plus 𝑖 times four plus 𝑖 on the numerator and four minus 𝑖 times four plus 𝑖 as our denominator.
Let’s distribute the parentheses over on the right-hand side here. We begin by multiplying the first terms. That’s four times four, which is 16. Then, we multiply the outer terms. Four times 𝑖 is just four 𝑖. We multiply the inner terms, giving us another four 𝑖. And then we multiply the last terms. 𝑖 times 𝑖 is 𝑖 squared. But actually, we know that 𝑖 squared is equal to negative one. So this simplifies to 16 plus eight 𝑖 plus negative one, which is 15 plus eight 𝑖.
Let’s repeat this process for the denominator. This time, distributing the parentheses as before, and we get 16 plus four 𝑖 minus four 𝑖 minus 𝑖 squared. Notice that four 𝑖 minus four 𝑖 is zero. So this becomes 16 minus negative one, which is the same as 16 plus one, which is 17. We put this all back together. And we see that when we simplify four plus 𝑖 over four minus 𝑖, we’re left with 15 plus eight 𝑖 all over 17.
