WEBVTT

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Okay, so this, all this review we just did.

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Using the square root principle, we ended up with a negative under the square root.

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There are no real solutions. There are no real numbers that when you square it, is

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negative one.

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We make up a number

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that is a solution. And we call that number i. Call the

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number i. Right. Technically, it's just a numeral that stands for the number. And

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what property does i have?

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i squared equals negative one. This is the definition of the number i. It is the

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number that when you square it is negative one.

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It is called an imaginary number. Okay. Because it's not a real number in the sense

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of, on our real number line. Does i exist?

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Yeah. Yes. Just because we call it imaginary, doesn't mean it doesn't exist. Okay.

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When we say real numbers,

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we're talking about numbers on the real number line.

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This is the definition of i.

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Sometimes we say i equals the square root of negative one, and that is okay. Like

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that.

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Now, this equation does have two solutions. Okay. By the square root principle, x is

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plus or minus the square root of negative one.

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That is i, so the solutions are plus and minus i. The solution set is negative i and

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positive i.

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So, it has two solutions.

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It has two solutions over the set of complex numbers. I'll explain that in a minute.

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Okay.

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So, this is the real number line. Right? Zero, one, two, negative one, negative two.

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This is called the real number line.

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So, let me show you what it really is.

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The complex numbers are represented on

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a plane. This is called the complex

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number plane.

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Okay.

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This looks like our x-axis. This is the real number line.

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Okay. This is zero.

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One, two, three, four, and so on.

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Okay. Okay. Those are the real numbers. Everything on that line. And then we have a

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whole axis of imaginary numbers. This right here is i.

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What's this right here? Two i. Two i.

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Three i.

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Four i, and so on. This would be negative i.

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Negative two i. Negative three i. And so on.

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This point right here would be one and a half i, or 1.5i. Pi i would be right here.

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So,

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what is this point right here?

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Two i's right here. Two i. Close.

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Okay. This is two plus i.

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Okay. It's a number. It's not an ordered pair.

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That point represents the number two plus i. Just like this point represents the

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number three.

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What is this point right here? One plus three i.

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What is this point right here? Negative three plus two i. Negative three plus two i.

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Okay.

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This point would be negative two minus two i.

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Okay. There are so many complex numbers.

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It fills up an entire plane.

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And again, every point just represents

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a complex number. So, let's talk about complex numbers.

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Every complex number can be written in this form: a+bi, where

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a and b are real numbers.

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Okay.

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So,

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the set of complex numbers

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is a bold face c to represent it.

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Is the set of a plus bi such that a and b are any real numbers.

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The a, the first number, like for example, three plus four i.

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This is called the real part

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and this is called the imaginary part.

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Are real numbers

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inside the set or we call the subset. Are the real numbers a subset of the complex

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numbers?

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They are. They're all these numbers on this line. So, all real numbers are also

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complex numbers.

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But anything not on that line is only a complex number. It's not a real number. It

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has a real part, an imaginary part,

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but it is not a real number. But every real number is a complex number.

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Let's do a quick subset.

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Venn diagram.

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Set of complex numbers. The set is represented by a boldface C.

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There is a small subset

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called the set of real numbers.

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It's much smaller than it looks in this graph. It's simply one line of numbers

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compared to the rest of the plane.

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We break up real numbers into two different types of numbers. How do we break those

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up?

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Rational and irrational.

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Rational.

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Which we use the letter Q for rational, bold face Q.

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The remaining real numbers form the complement of Q within R. The complement here

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means the real numbers that are not rational.

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Those are irrational.

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And then inside the set of rational numbers

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is the set of what?

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What are these numbers?

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The integers.

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In here we have a subset called integers.

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Which we use Z for the integers.

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A subset of the integers is the whole numbers. I didn't draw this very good.

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The set of whole numbers.

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What are the whole numbers?

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Close.

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Zero, one, two, three, four. The whole numbers include zero.

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Which is zero. One, two, three.

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And then inside of that is this set. One, two, three. What do we call that?

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Call that the natural numbers.

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Which is a subset of that.

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Which is one, two, three, four.

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Natural numbers also known as counting numbers. When counting, we naturally begin

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one, two, three.

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Okay, so that's the Venn diagram of the complex numbers. Real numbers are a subset.

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And then those have other subsets.

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How in the world do we add

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complex numbers?

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So this plus

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is the same as this plus.

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So we rearrange everything. Two plus negative four plus three i plus five i. Two

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plus negative four is negative two.

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Three i plus five i is eight i. You add the real parts and you add the imaginary

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parts. It's very simple.

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How do we subtract? The imaginary numbers.

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Start with 4i−(7+3i)+(4−6i).

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We have one subtract and one addition. So what am I going to do different? I'm going

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to distribute the negative. Over the addition.

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There's four i minus seven minus three i plus four minus six i.

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Which is what? Negative three plus

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negative five i.

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Negative five i. Adding and subtracting works as if we had an x instead of an i,

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when adding linear polynomials.

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Now let's see. How do we multiply

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complex numbers?

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Let's start with a very simple one.

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So let's do a simple one.

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This multiplication is the same as that multiplication.

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So what do we do?

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We just rearrange everything.

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So it is ten i squared.

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And what is i squared? Negative one. The answer is negative ten.

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Once we get to i squared, we change it to negative one. Which is the definition of

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i.

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i squared equals negative one. And we end up with a real number.

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You definitely end up with a complex number because every real number is a complex

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number. But sometimes you end up with a real number,

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sometimes you don't. Let's try another one.

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What do you think we're going to do here?

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Distribute. Just like we distributed binomials, we could distribute the whole thing

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and distribute again. But we just use our quick.

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Here, here, here, here. We've got two times four. Plus two times negative five i.

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Plus three i times four.

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Plus three i times negative five i.

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Eight minus ten i. Plus twelve i.

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Minus fifteen i squared.

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And what is i squared? Negative one.

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Plus two i. This becomes positive fifteen. Eight plus fifteen is twenty-three plus

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two i.

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Multiply any two complex numbers. You get another complex number. It can always be

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written in the form a+bi. That's the standard way we write it.

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Two plus three i times four minus five i.

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So we get two plus three i. We just multiplied two plus three i. This number right

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here.

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What was the other number? Four minus five i.

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Times four minus five i.

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This number times this number turned out to be what was it? Twenty three plus two i.

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So what's right over here? Twenty three plus two i. It's kind of weird but

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what's that?

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It's just how it works.

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Multiplying these two complex numbers can give a point farther away in the plane.

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The result has a real part and an imaginary part.

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Try this out. Try this one on your own.

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The result is negative fourteen minus eighteen i. You guys not too bad?

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Let's try another one. Two plus three i times two minus three i.

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Distribute to check this product.

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The two middle terms add to zero, and we get thirteen.

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We end up with a real number. Which is also a complex number.

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What about this causes to just be a real number?

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It is the difference of two squares.

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What do you mean by that? It's like the same.

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The only difference

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is the sign in the middle.

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In fact we have a name for that.

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When we were

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rationalizing

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denominator, how did we divide radicals? We really didn't divide, we multiplied by

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something. What did we multiply by? The conjugate.

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Two minus rad three.

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And the two terms in the middle add to be zero.

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The conjugate product is real. So these are called conjugates.

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They are called complex conjugates.

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Complex conjugates. A plus B i and A minus B i are complex conjugates.

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And what happens when you multiply them?

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You get A squared

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minus A B i plus A B i minus B squared i squared.

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These add to be zero.

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This becomes negative B squared times negative one, which becomes plus B squared,

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which is a real number.

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You multiply any two complex conjugates, you end up with a real number.

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And this is going to be very important. What we are about to do

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is divide.

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So to divide two complex numbers, just like we did with radicals, we are not really

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going to divide.

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We are going to multiply.

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And what are we going to multiply by?

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The conjugate. And the conjugate of two minus five i is?

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Two plus five i. Two plus five i. The fraction (2+5i)/(2+5i) equals one.

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And then we start distributing.

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Six i. plus fifteen i squared.

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Four

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plus ten i minus ten i minus twenty five i squared.

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This is negative one.

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This is negative one.

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So we get

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negative fifteen plus six i.

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divided by

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the denominator. Four plus twenty-five.

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Which is twenty nine.

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Once the denominator is real, divide each term of the numerator by it. So this is

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going to be negative fifteen divided by twenty nine.

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plus six twenty-ninths i.

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And there is our how you divide two complex numbers.

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And we always want the final answer to be a plus b i. So if you have fractions,

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that's okay.

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Alright, let's try another one.

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Now let's change it to three.

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Which is called the what of that denominator?

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The conjugate, or the complex conjugate. And now he's got a lot of distributing. Be

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real careful.

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So

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Distribute carefully. These i-squared terms turn into

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negative ones.

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So the numerator is

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Three plus two plus five i, divided by, we get nine plus one. That gives five plus

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five i, divided by ten.

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Does that look right?

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Five.

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Now write the result with real coefficients: divide both terms by ten.

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And then one more step.

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One half plus one half i. That is cool.

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So, find the zeros. We set Y equal to zero. We solve.

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And we use the quadratic formula.

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Four minus eight is negative four. Over two.

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Okay. Negative two plus or minus the square root of four times the square root of

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negative one.

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And what is the square root of negative one?

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Negative two plus or minus two i, all over two.

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Divide the whole numerator: negative one plus or minus i. This quadratic

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has two zeros. They're not real.

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We call them imaginary or we call them complex.

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Last thing. Let's just show that when you plug it in here, you end up with zero.

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Here we go: negative one minus i times negative one minus i.

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Expanding gives one plus i plus i plus i squared, then negative two

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minus two i plus two.

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This is negative one.

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One plus two i minus one, plus negative two

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minus two i plus two.

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These terms add to zero. These terms cancel, and these terms cancel. We get zero.

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We will not often substitute complex numbers into functions in this class.

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This is a function in one complex variable.

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We typically would use the letter z.

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z squared plus two z

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Plus two? Yes.

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The domain would be all complex numbers, instead of just real numbers. We will not

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study complex-function graphs in this class.

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The problem is how do we graph this?

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This plane represents just the input domain.

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What's the range going to be? All complex

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numbers.

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It's not in three dimensions.

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We need four dimensions.

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The input has two real coordinates, and the output has two real coordinates.

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A full graph requires four real coordinates.

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We need four dimensions. It has lots of properties.

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You can graph what's called slices of it. You fix one

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real or imaginary input coordinate. You end up with a three dimensional graph.

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It's just one slice of the graph. It's crazy stuff.

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Complex numbers allow every quadratic to have two zeros, counting multiplicity.

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Guaranteed. Two complex zeros, counting multiplicity. Later on we get into higher

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degree polynomials. There are important algebraic

00:30:10.560 --> 00:30:13.680
theorems about how many

00:30:15.220 --> 00:30:16.760
zeros a polynomial has.