 So we've learned how to add and subtract complex numbers. How does one multiply complex numbers? Well, it turns out that since complex numbers themselves are two dimensional numbers, it's a real part plus an imaginary part, multiplying them is gonna feel a lot like the foil method because you have a binomial times a binomial, essentially. The only thing that we have to keep track of is what happens when you multiply together powers of i. Like because the number i itself is the square root of negative one, when you take i to the first power, no big deal there, but when you get i squared, well, i squared is gonna be i times i. That's what it means to square something. And so you're taking the square root of negative one squared. If this truly is the square root of negative one, then the square should be negative one by definition. And so i squared is equal to negative one. If you know nothing else about i, that's basically the only thing you need to know is that i squared is equal to negative one. That definition right there takes care of everything else. Now, what if you had to take higher powers? So if you had to take like i cubed, for example, well, i cubed means you're gonna take i squared times i, which is actually equal to negative one. And if you take i to the fourth, that would look like i squared times i squared, which is gonna be negative one squared, which is just one right there. And so if I clean this thing up, this list up real quickly, let me kind of summarize what we just said. If you take i to the first, that's just gonna be an i, i squared is negative one. i cubed is negative i, and i to the fourth is one. The reason why I wanted to clean that thing up is look if we continue on with this list. We take i to the fifth. i to the fifth is i to the fourth times i, which since i to the fourth is one, this is just gonna give you an i. If you take i to the sixth, this would be i to the fourth times i squared. i to the first, i to the fourth is just a one, and then i squared is negative one. If you take i to the seventh, you're gonna end up with i to the fourth times i cubed, which i to the fourth is still one, and i cubed is negative i, and then i to the eighth is equal to, well, that's just equal to i to the fourth times i to the fourth, which would be one times one, which is one. And look here, if you look at these numbers, i negative one, negative i one, i negative one, negative i one, those are just the same numbers that have just repeated themselves. That's interesting. Well, that's actually not a coincidence. Let me kind of show you what happens if we look at the next couple of these. Let's do it again here, leaving off, picking up where we left off, I should say. i to the ninth, this would look like i to the eighth times i, which we already saw, i to the eighth was one, so this is an i. Let me write that again, that's an i. If you do i to the tenth, that would just be i to the eighth times i squared. You end up with one times i squared, which is negative one. You get i to the 11th, which would equal i to the eighth times i cubed. i to the eighth is one, might be observed. And then, so we get a negative i again. And then i to the 12th would equal i to the eighth times i to the fourth, which we already saw that i to the eighth and i to the fourth are one, so you get a one. And so look here, it's the exact same number sequence. Again, i negative one, negative i one, i negative one, negative i one. And this is what happens. As you take powers and powers and powers of i, always cycle through the same four numbers, always in this order, i negative one, negative i one, i negative one, negative i one. Arbitrally large powers of i, it really just comes down to, can we find the biggest multiple, whoops, the biggest multiple of four inside the power, take it away and then see what's left over. So for example, if you wanted to do i to the 27th, what we're gonna do is we're gonna look at what's the biggest multiple of four that goes into 27? We could write this as i to the 24 plus three. The idea here is that 24 is going to be four times six. And then you have this i cubed right here. Why do we care about multiples of four? But like we were seeing earlier, every time you take a multiple of four, every time you take a multiple of four, you're just gonna get i to the fourth, which is just a one. And if you have like one to the sixth, that's just gonna be one still, right? And so we can ignore multiples of four when it comes to exponents of i, because every four powers, you just get a one again and multiplication of one doesn't do anything. So the only thing we have to know is, once you ignore the multiple of four, what is i cubed? And that's gonna be a negative i, which gives us the solution right here. So i to the 27th power is negative one. What about i to the 101st power, right? That's a lot of dormations, but it's not a hard thing when it covers of i right here. Because what we can do is recognize the following. 100 is, sorry, 101 is just 101, right? 100 is a multiple of four, right? It's four times 25. And as such, we can ignore all multiples of four in the exponent of i. That just leaves i to the first, in which case then this product is just gonna equal an i. If we know that, we can then handle any power of i. It hence any multiplication that involves i. We truly need to work out. I mean, we're gonna in practice, we'll mostly just see i squared equals negative one. This all derives from that observation. So let's look at some multiplication in of arbitrary complex numbers. Like I said, it's basically the FOIL method, right? You're gonna take first, first, outside, inside, last. So what we're gonna do is we're gonna take five times two. So we take the product of the first two terms, five times two. We're gonna add that to the outside terms, five times seven i. Then we're gonna take the product of the inside terms. So we get three i times two. And then we get the product of the last terms right here. So we're gonna get three i times seven i. So performing these products, we see that five times two is 10. Five times seven i is 35 i. Three i times two is a six i. And then three i times seven i will be 21 i squared. For which then we try to, we're gonna try to combine like terms here. You'll notice that the 35 i and the six i, we can add to 41 i. You have a 10, but also notice that we have an i squared. Like we said above, i squared is the same thing as a negative one. And so having an i squared in there actually means we have a negative 21, which those are both real numbers, which we can add together, or I should say subtract here. And so we get the resultant negative 11 plus 41, 41 i, which would then be the product of these two imaginary numbers. So when you foil out the complex numbers, you just have to remember to replace i squared with negative i. What if we wanted to do like two plus i cubed? Well, two plus i cubed means two plus i times two plus i times two plus i. And so to compute this one, we're just gonna foil out the two plus i. If we do the first two, you'll end up with two times two, which is four, you're gonna get a two i, you're gonna get a two i, and then you're gonna get a negative one, the i squared, it took the liberty of replacing that one there. Combining like terms, you end up with four minus one, which is three, and then two i plus two i, which is a four i. And then we're gonna times that by two i again, foil this thing out, three times two is six, three times i is a three i, four i times two is eight i, and then four i times i is a negative four. Again, I took the liberty of writing i squared as a negative. Combining like terms, six take away four is a two, and then three i plus eight i should give us an 11 i. And that would be then two plus i cubed. Cubing just means we just have to foil it multiple times, so we had to do the first two, and then we did the third one.