Complex Numbers and Arithmetic
October 20, 2026

[00:00:05] Okay, so this, all this review we just did.
[00:00:12] Using the square root principle, we ended up with a negative under the square root.
[00:00:15] There are no real solutions. There are no real numbers that when you square it, is negative one.
[00:00:20] We make up a number
[00:00:24] that is a solution. And we call that number i. Call the
[00:00:32] number i. Right. Technically, it's just a numeral that stands for the number. And what property does i have?
[00:00:43] i squared equals negative one. This is the definition of the number i. It is the number that when you square it is negative one.
[00:00:55] It is called an imaginary number. Okay. Because it's not a real number in the sense of, on our real number line. Does i exist?
[00:01:08] Yeah. Yes. Just because we call it imaginary, doesn't mean it doesn't exist. Okay. When we say real numbers,
[00:01:15] we're talking about numbers on the real number line.
[00:01:20] This is the definition of i.
[00:01:22] Sometimes we say i equals the square root of negative one, and that is okay. Like that.
[00:01:31] Now, this equation does have two solutions. Okay. By the square root principle, x is plus or minus the square root of negative one.
[00:01:45] That is i, so the solutions are plus and minus i. The solution set is negative i and positive i.
[00:01:58] So, it has two solutions.
[00:02:01] It has two solutions over the set of complex numbers. I'll explain that in a minute.
[00:02:11] Okay.
[00:02:14] So, this is the real number line. Right? Zero, one, two, negative one, negative two. This is called the real number line.
[00:02:24] So, let me show you what it really is.
[00:02:29] The complex numbers are represented on
[00:02:34] a plane. This is called the complex
[00:02:42] number plane.
[00:02:45] Okay.
[00:02:48] This looks like our x-axis. This is the real number line.
[00:02:56] Okay. This is zero.
[00:03:00] One, two, three, four, and so on.
[00:03:11] Okay. Okay. Those are the real numbers. Everything on that line. And then we have a whole axis of imaginary numbers. This right here is i.
[00:03:23] What's this right here? Two i. Two i.
[00:03:28] Three i.
[00:03:31] Four i, and so on. This would be negative i.
[00:03:36] Negative two i. Negative three i. And so on.
[00:03:46] This point right here would be one and a half i, or 1.5i. Pi i would be right here.
[00:03:58] So,
[00:04:01] what is this point right here?
[00:04:06] Two i's right here. Two i. Close.
[00:04:13] Okay. This is two plus i.
[00:04:17] Okay. It's a number. It's not an ordered pair.
[00:04:25] That point represents the number two plus i. Just like this point represents the number three.
[00:04:33] What is this point right here? One plus three i.
[00:04:43] What is this point right here? Negative three plus two i. Negative three plus two i.
[00:04:54] Okay.
[00:04:58] This point would be negative two minus two i.
[00:05:06] Okay. There are so many complex numbers.
[00:05:11] It fills up an entire plane.
[00:05:19] And again, every point just represents
[00:05:27] a complex number. So, let's talk about complex numbers.
[00:05:45] Every complex number can be written in this form: a+bi, where
[00:05:53] a and b are real numbers.
[00:06:01] Okay.
[00:06:06] So,
[00:06:08] the set of complex numbers
[00:06:13] is a bold face c to represent it.
[00:06:21] Is the set of a plus bi such that a and b are any real numbers.
[00:06:44] The a, the first number, like for example, three plus four i.
[00:06:55] This is called the real part
[00:06:59] and this is called the imaginary part.
[00:07:02] Are real numbers
[00:07:06] inside the set or we call the subset. Are the real numbers a subset of the complex numbers?
[00:07:17] They are. They're all these numbers on this line. So, all real numbers are also complex numbers.
[00:07:25] But anything not on that line is only a complex number. It's not a real number. It has a real part, an imaginary part,
[00:07:33] but it is not a real number. But every real number is a complex number.
[00:07:40] Let's do a quick subset.
[00:07:44] Venn diagram.
[00:07:48] Set of complex numbers. The set is represented by a boldface C.
[00:07:59] There is a small subset
[00:08:04] called the set of real numbers.
[00:08:16] It's much smaller than it looks in this graph. It's simply one line of numbers compared to the rest of the plane.
[00:08:24] We break up real numbers into two different types of numbers. How do we break those up?
[00:08:34] Rational and irrational.
[00:08:40] Rational.
[00:08:42] Which we use the letter Q for rational, bold face Q.
[00:08:50] The remaining real numbers form the complement of Q within R. The complement here means the real numbers that are not rational.
[00:08:59] Those are irrational.
[00:09:07] And then inside the set of rational numbers
[00:09:12] is the set of what?
[00:09:20] What are these numbers?
[00:09:25] The integers.
[00:09:27] In here we have a subset called integers.
[00:09:37] Which we use Z for the integers.
[00:09:44] A subset of the integers is the whole numbers. I didn't draw this very good.
[00:09:52] The set of whole numbers.
[00:09:55] What are the whole numbers?
[00:10:00] Close.
[00:10:02] Zero, one, two, three, four. The whole numbers include zero.
[00:10:10] Which is zero. One, two, three.
[00:10:17] And then inside of that is this set. One, two, three. What do we call that?
[00:10:25] Call that the natural numbers.
[00:10:29] Which is a subset of that.
[00:10:32] Which is one, two, three, four.
[00:10:36] Natural numbers also known as counting numbers. When counting, we naturally begin one, two, three.
[00:10:42] Okay, so that's the Venn diagram of the complex numbers. Real numbers are a subset. And then those have other subsets.
[00:10:53] How in the world do we add
[00:10:58] complex numbers?
[00:11:03] So this plus
[00:11:07] is the same as this plus.
[00:11:11] So we rearrange everything. Two plus negative four plus three i plus five i. Two plus negative four is negative two.
[00:11:27] Three i plus five i is eight i. You add the real parts and you add the imaginary parts. It's very simple.
[00:11:45] How do we subtract? The imaginary numbers.
[00:11:52] Start with 4i−(7+3i)+(4−6i).
[00:12:09] We have one subtract and one addition. So what am I going to do different? I'm going to distribute the negative. Over the addition.
[00:12:18] There's four i minus seven minus three i plus four minus six i.
[00:12:34] Which is what? Negative three plus
[00:12:43] negative five i.
[00:12:47] Negative five i. Adding and subtracting works as if we had an x instead of an i, when adding linear polynomials.
[00:13:02] Now let's see. How do we multiply
[00:13:08] complex numbers?
[00:13:19] Let's start with a very simple one.
[00:13:25] So let's do a simple one.
[00:13:33] This multiplication is the same as that multiplication.
[00:13:39] So what do we do?
[00:13:44] We just rearrange everything.
[00:13:50] So it is ten i squared.
[00:13:55] And what is i squared? Negative one. The answer is negative ten.
[00:14:10] Once we get to i squared, we change it to negative one. Which is the definition of i.
[00:14:18] i squared equals negative one. And we end up with a real number.
[00:14:26] You definitely end up with a complex number because every real number is a complex number. But sometimes you end up with a real number,
[00:14:33] sometimes you don't. Let's try another one.
[00:14:37] What do you think we're going to do here?
[00:14:41] Distribute. Just like we distributed binomials, we could distribute the whole thing and distribute again. But we just use our quick.
[00:14:53] Here, here, here, here. We've got two times four. Plus two times negative five i. Plus three i times four.
[00:15:08] Plus three i times negative five i.
[00:15:15] Eight minus ten i. Plus twelve i.
[00:15:25] Minus fifteen i squared.
[00:15:32] And what is i squared? Negative one.
[00:15:40] Plus two i. This becomes positive fifteen. Eight plus fifteen is twenty-three plus two i.
[00:15:56] Multiply any two complex numbers. You get another complex number. It can always be written in the form a+bi. That's the standard way we write it.
[00:16:11] Two plus three i times four minus five i.
[00:16:19] So we get two plus three i. We just multiplied two plus three i. This number right here.
[00:16:25] What was the other number? Four minus five i.
[00:16:29] Times four minus five i.
[00:16:33] This number times this number turned out to be what was it? Twenty three plus two i.
[00:16:41] So what's right over here? Twenty three plus two i. It's kind of weird but
[00:16:49] what's that?
[00:16:54] It's just how it works.
[00:17:06] Multiplying these two complex numbers can give a point farther away in the plane. The result has a real part and an imaginary part.
[00:17:16] Try this out. Try this one on your own.
[00:17:19] The result is negative fourteen minus eighteen i. You guys not too bad?
[00:17:27] Let's try another one. Two plus three i times two minus three i.
[00:17:41] Distribute to check this product.
[00:17:48] The two middle terms add to zero, and we get thirteen.
[00:17:54] We end up with a real number. Which is also a complex number.
[00:18:03] What about this causes to just be a real number?
[00:18:11] It is the difference of two squares.
[00:18:18] What do you mean by that? It's like the same.
[00:18:25] The only difference
[00:18:30] is the sign in the middle.
[00:18:34] In fact we have a name for that.
[00:18:41] When we were
[00:18:45] rationalizing
[00:18:49] denominator, how did we divide radicals? We really didn't divide, we multiplied by something. What did we multiply by? The conjugate.
[00:19:01] Two minus rad three.
[00:19:05] And the two terms in the middle add to be zero.
[00:19:13] The conjugate product is real. So these are called conjugates.
[00:19:25] They are called complex conjugates.
[00:19:29] Complex conjugates. A plus B i and A minus B i are complex conjugates.
[00:19:48] And what happens when you multiply them?
[00:19:54] You get A squared
[00:19:58] minus A B i plus A B i minus B squared i squared.
[00:20:12] These add to be zero.
[00:20:16] This becomes negative B squared times negative one, which becomes plus B squared, which is a real number.
[00:20:28] You multiply any two complex conjugates, you end up with a real number.
[00:20:35] And this is going to be very important. What we are about to do
[00:20:42] is divide.
[00:20:44] So to divide two complex numbers, just like we did with radicals, we are not really going to divide.
[00:20:52] We are going to multiply.
[00:20:55] And what are we going to multiply by?
[00:20:59] The conjugate. And the conjugate of two minus five i is?
[00:21:04] Two plus five i. Two plus five i. The fraction (2+5i)/(2+5i) equals one.
[00:21:14] And then we start distributing.
[00:21:27] Six i. plus fifteen i squared.
[00:21:35] Four
[00:21:37] plus ten i minus ten i minus twenty five i squared.
[00:21:52] This is negative one.
[00:21:57] This is negative one.
[00:22:02] So we get
[00:22:06] negative fifteen plus six i.
[00:22:11] divided by
[00:22:14] the denominator. Four plus twenty-five.
[00:22:20] Which is twenty nine.
[00:22:23] Once the denominator is real, divide each term of the numerator by it. So this is going to be negative fifteen divided by twenty nine.
[00:22:36] plus six twenty-ninths i.
[00:22:40] And there is our how you divide two complex numbers.
[00:22:50] And we always want the final answer to be a plus b i. So if you have fractions, that's okay.
[00:22:58] Alright, let's try another one.
[00:23:13] Now let's change it to three.
[00:23:17] Which is called the what of that denominator?
[00:23:22] The conjugate, or the complex conjugate. And now he's got a lot of distributing. Be real careful.
[00:23:58] So
[00:24:02] Distribute carefully. These i-squared terms turn into
[00:24:18] negative ones.
[00:24:22] So the numerator is
[00:24:26] Three plus two plus five i, divided by, we get nine plus one. That gives five plus five i, divided by ten.
[00:24:41] Does that look right?
[00:24:45] Five.
[00:24:51] Now write the result with real coefficients: divide both terms by ten.
[00:25:06] And then one more step.
[00:25:10] One half plus one half i. That is cool.
[00:25:16] So, find the zeros. We set Y equal to zero. We solve.
[00:25:22] And we use the quadratic formula.
[00:25:57] Four minus eight is negative four. Over two.
[00:26:06] Okay. Negative two plus or minus the square root of four times the square root of negative one.
[00:26:20] And what is the square root of negative one?
[00:26:24] Negative two plus or minus two i, all over two.
[00:26:32] Divide the whole numerator: negative one plus or minus i. This quadratic
[00:26:41] has two zeros. They're not real.
[00:26:46] We call them imaginary or we call them complex.
[00:26:49] Last thing. Let's just show that when you plug it in here, you end up with zero.
[00:26:55] Here we go: negative one minus i times negative one minus i.
[00:27:04] Expanding gives one plus i plus i plus i squared, then negative two
[00:27:17] minus two i plus two.
[00:27:23] This is negative one.
[00:27:27] One plus two i minus one, plus negative two
[00:27:35] minus two i plus two.
[00:27:40] These terms add to zero. These terms cancel, and these terms cancel. We get zero.
[00:27:58] We will not often substitute complex numbers into functions in this class.
[00:28:05] This is a function in one complex variable.
[00:28:10] We typically would use the letter z.
[00:28:20] z squared plus two z
[00:28:25] Plus two? Yes.
[00:28:33] The domain would be all complex numbers, instead of just real numbers. We will not study complex-function graphs in this class.
[00:28:44] The problem is how do we graph this?
[00:28:51] This plane represents just the input domain.
[00:28:55] What's the range going to be? All complex
[00:29:01] numbers.
[00:29:06] It's not in three dimensions.
[00:29:11] We need four dimensions.
[00:29:14] The input has two real coordinates, and the output has two real coordinates.
[00:29:19] A full graph requires four real coordinates.
[00:29:24] We need four dimensions. It has lots of properties.
[00:29:29] You can graph what's called slices of it. You fix one
[00:29:37] real or imaginary input coordinate. You end up with a three dimensional graph.
[00:29:45] It's just one slice of the graph. It's crazy stuff.
[00:29:48] Complex numbers allow every quadratic to have two zeros, counting multiplicity.
[00:29:56] Guaranteed. Two complex zeros, counting multiplicity. Later on we get into higher degree polynomials. There are important algebraic
[00:30:10] theorems about how many
[00:30:15] zeros a polynomial has.