 In the Fourier transform we have to do rather a lot of adding and multiplying of complex numbers. I covered those two operations in previous videos. In this video we're going to cover division. Now, division isn't really used in the Fourier transform, but I thought we should cover it here for the sake of completeness. After all, complex numbers are used in other places apart from just the Fourier transform. When we multiplied two complex numbers, we saw that we could use the FOIL method to find our answer. FOIL stands for first, outside, inside, last. We have to split the multiplication operation into four stages. We can use the same method for dividing. However, we're going to hit a problem, a problem that can only be solved by a special type of complex number. Now let's try to divide 3 plus 4i by 9 plus 2i. How about if we use the FOIL method to divide the numbers? Well, this time we really hit a problem. Look what happens. Dividing the two first terms in the brackets, 3 divided by 9, is no problem, as that simply gives us a third. No, it's the outside terms that are the difficult ones, as look what happens to the i. If we divide the two outside terms, 3 divided by 2i, the i ends up on the bottom as the denominator. This makes things very awkward. We've already had enough trouble with i being the square root of minus 1, let alone having to introduce a new term to cope with 1 over the square root of minus 1. If only there was some trick we could use to get the i out of the denominator. Well, as it happens there is. We use something called the complex conjugate. The complex conjugate is a nifty little number. Any complex number multiplied by its complex conjugate gives us a real number as the result, no i's to worry about. The complex conjugate of 9 plus 2i is 9 minus 2i. Now, if we multiply these two numbers together using FOIL, this is what we get. 9 times 9 equals 81. 9 times minus 2i equals minus 18i. 2i times 9 equals 18i. And 2i times minus 2i equals minus 4i squared. But i squared equals minus 1. So minus 4i squared equals minus 4 times minus 1, which simply equals 4. So grouping these terms together gives us 81 minus 18i plus 18i plus 4. The minus 18i and 18i cancel out, leaving us with no i's. So we're simply left with 81 plus 4 giving us 85 a totally real result. So how do we use this trick in our division calculation? What we can do is to multiply both the 3 plus 4i and the 9 plus 2i by 9 minus 2i. We can do this as 9 minus 2i divided by 9 minus 2i is equal to 1. So all we have done is multiplied our original calculation by 1, which doesn't affect the result. However, what it does do is allow us to use FOIL on both the numerator and the denominator. We already calculated the denominator as 9 minus 2i is the complex conjugate of 9 plus 2i, which we worked up before simply equaled 85. So without changing the outcome of the calculation at all, we have managed to get the i out of the denominator and can now treat the rest of the calculation as a multiplication. So using FOIL, 3 times 9 equals 27, 3 times minus 2i equals minus 6i, 4i times 9 equals 36i, and 4i times minus 2i equals minus 8i squared, or simply 8 as i squared equals minus 1. This makes 27 minus 6i plus 8 plus 36i. If we arrange this grouping the real and imaginary terms, we get 27 plus 8 minus 6i plus 36i, which gives us 35 plus 30i. So now we are left with the result 35 plus 30i over 85. But we're used to seeing complex numbers written out with the real and imaginary part. So let's rewrite this expression slightly. 35 over 85 plus 30 over 85i, or if we actually work at the division, 0.41 plus 0.35i. Although the FOIL method does allow us to get to an answer when dividing two complex numbers, we had to work very hard to get there. It is here the writing out two complex numbers in their exponential form really comes into its own. If we rewrite our calculation using the exponential form of the two complex numbers like we did when we multiplied them before, we can use a similar method to divide them. When multiplying, we first multiplied the 5 and the 9.2. Now that we are dividing, we simply divide them instead, giving us the answer 0.54. When multiplying, we added the 53.1 degrees and the 12.5 degrees. Now that we are dividing, we simply minus them instead, giving us 40.6 degrees. This gives us the overall result 0.54 times e to the 40.6 degrees times i. Using the polar form, we can convert this back into the Cartesian form, which gives us 0.54 times the cosine of 40.6 degrees plus 0.54 times i times the sine of 40.6 degrees, which gives us the result 0.41 plus 0.35i, the same result as we got before, but arrived at with much greater ease. So when dividing two complex numbers, we can use the FOIL method together with the complex conjugate. However, it might be easier to convert the two complex numbers to their exponential form and divide those instead. If you'd like to see the whole series on maths with complex numbers, you can find it here. Have you ever wondered why i is the square root of minus 1? You can find out in another of my videos. See how complex numbers are used in the Fourier transform by visiting my online course on how the Fourier transform works. Please consider supporting the making of these videos by becoming a patron.