 So, while complex numbers are the solutions to quadratic equations, we have an arithmetic of complex numbers as well. It's convenient to view i as a variable so that all of our standard rules of algebra apply. So, for example, we can add 3x plus 5x to give us 8x, where we add the coefficients and keep the variables. Similarly, we can add 3i plus 5i to get 8i. Likewise, 12x minus 3x is 9x, same variable, subtract the coefficient, and likewise, 12i minus 3i equals 9i. Multiplication is similar, but we can simplify a resulting product by remembering that i squared is equal to negative 1. So, if I have x times 3x, I get 3x squared, and I can't do anything else with it. If I have i times 3i, I get 3i squared, but remember, i squared is equal to negative 1, and so this i squared can be replaced with negative 1, and so my final product is negative 3. So, for example, if I want to expand 8i times 5 plus i, I'll multiply those out using the distributive property. That gives me 40i plus 8i squared, but remember, i squared equals negative 1, so this 8i squared becomes 8 times negative 1, and we can do a little bit more arithmetic. And my final answer, 40i plus negative 8, or we typically write the real part of the complex number first, negative 8 plus 40i. We might have the product of two complex numbers, 3 plus i times 2 minus i. Multiplying this out, we get 6 minus 3i plus 2i minus i squared. We can simplify this minus 3i plus 2i, and i squared equals negative 1 allows us to simplify, and we get our final answer, 7 minus i. An interesting case occurs when we multiply 7 plus i times 7 minus i, and here's the thing to notice is that except for the plus and the minus, these two complex numbers have the same real part and have the same imaginary part. When we multiply them out, we get 50, and the thing to recognize here is that even though we were multiplying two complex numbers, our final answer was a real number. And this leads to the following definition. Let z equals a plus bi be some complex number. The conjugate of z, written z bar, with a little bar over it, is a minus bi. So for example, let's find the conjugates of the numbers 7 minus 3i, 4i, and 8. So we know that the conjugate of a plus bi is a minus bi, so we want to write our numbers in that form. 7 minus 3i is 7 plus negative 3i, and so our conjugate is going to be 7 minus negative 3i, otherwise known as 7 plus 3i. How about 4i? We want to write this in the form a plus bi, so 4i can be written as 0 plus 4i, so the conjugate will be 0 minus 4i, or just minus 4i. And I want to write 8 in the form a plus bi, so 8 is 8 plus 0i, so its conjugate will be 8 minus 0i, or just 8. We need the conjugate to be able to divide by a complex number. Let u and b be complex numbers. Then u divided by v is going to be u times the conjugate of v divided by v times the conjugate of v. The importance of this is that because we multiplied v by its conjugate, we are now dividing by a real number. For example, let's say we want to divide 3 minus i by 7 plus i. So our definition says we have to multiply numerator and denominator by the conjugate of the denominator. So our denominator is 7 plus i, the conjugate is going to be 7 minus i. So we'll multiply numerator and denominator by 7 minus i. Our numerator, 3 minus i times 7 minus i, is going to be, and our denominator will be 7 minus i times 7 plus i, and we should end up with a real number, in this case we do 50. And so the product 20 minus 10i over 50, but we need to express this as a complex number, so we'll split our fraction. And now this is in the form real number plus a real number times i. There's an optional reduction of fractions at this point, but that's not a precalculus problem. It's actually an arithmetic problem. And we can divide 8 by 5i, so we need to multiply by the conjugate of the denominator. Since 5i is 0 plus 5i, then the conjugate will be minus 5i. So we'll multiply numerator and denominator by minus 5i. In the numerator we get 8 times negative 5i, that's negative 40i. In the denominator we get 25, and we want to write our number in the form number times i, so we get our final form negative 40 over 25 times i, reduction of fractions optional.