 If a complex number z equals a plus bi, the complex conjugate, written as bar z, is a minus bi. The same terms just the imaginary part is subtracted instead of added. For example, let's find the conjugate of 8 plus 2i. The conjugate will have the same terms, but instead of adding, we'll subtract them. Now, in pre-calculus and lower-level courses, the conjugate is often introduced because the product of a complex number and its conjugate is real. This is true, but it's not that big a deal because it's also true for non-conjugates as well. For example, let's find a non-zero complex number a plus bi, where the product a plus bi times 8 plus 2i is real, but a plus bi is not the conjugate of 8 plus 2i. So, let's set that up. We want to consider the product, and let's go ahead and expand it, and collect our real and imaginary terms. Now, in order for the product to be real, it's sufficient that the imaginary part be equal to zero. And notice that there's an infinite number of possible solutions to this equation. For example, a equals 4, b equals negative 1, and that gives us a factor 4 minus i, where if we multiply it by 8 plus 2i, we get a real number. So, what's the big deal about the conjugate? Well, there's actually two key properties of the conjugate. The conjugate of a complex number z is a unique non-zero complex number that satisfies z plus its conjugate is a real number, and z times its conjugate is a real number. We can prove this, but you should do this. So, here's a hint. A general strategy to prove that something is unique is to assume that there's another one. So, if our complex number is a plus bi, the conjugate will be a minus bi, and suppose we have another complex number, a prime plus b prime i, that also satisfies the property that the product and sum are real numbers. What does that tell you about a prime and b prime? Now, in addition, the conjugate satisfies a number of other properties, let u and v be complex numbers, then the conjugate of a sum is the sum of the conjugates, the conjugate of a product is the product of the conjugates, the conjugate of the conjugate is the original number, and the conjugate of a power is the same as the power of a conjugate, at least for whole numbers n. That restriction is there so that this is something that we can prove at this point. Although I can't even begin to imagine how you'd prove something like that. Induction. Did you hear something? Well, please prove one of them. So, let's prove that the conjugate of a sum is the sum of the conjugates. So, remember, definitions are the whole of mathematics. All else is commentary. So, suppose we have two complex numbers. This statement is talking about their conjugates. So, let's start with the conjugate of their sum. And, since we know what the two complex numbers are, we can do the complex arithmetic to find an expression for the conjugate of the sum. Now, it's useful to think about a proof as building a bridge from one point to another, and sometimes it's convenient to work from the end. The important thing is that if you are starting at the end, you can't go past the end. You have to go backwards. So, we want to end with the sum of the conjugates of u and v. Definitions are the whole of mathematics. All else is commentary. We can always go backwards on a definition. So, we know what the conjugate of u and v are, and let's see if we can bridge the gap between our two lines. And we see that it is only a matter of rearranging our terms, and so that joins beginning to end, and we have our proof.