Introduction to Complex Numbers. I'll begin by saying, one property that fascinates mathematicians is closure, whether a particular set of number is closed. Now this is a word you do not have to know for the ACT. For example, the real numbers, the numbers on the number line are closed under addition. Show Transcript
They have closure under addition. In other words, if we add any two numbers we get another real number. We can pick any two numbers on the number line, no matter what ones we pick. When we add, we'll land somewhere else on the number line. We never leave the number line by the process of adding two numbers, it's a closed system.
Similarly, real numbers are closed under subtraction and multiplication. We could subtract any two numbers, multiply any two numbers, the result will still be somewhere on the number line. We can also divide with the stipulation that we not divide by zero. Now that's not a problem for closure, if we just have an exception of a single number.
Can't divide by zero, but other than that, we could divide any two numbers. We'd still be on the number line. So in other words, in terms of addition, subtraction, multiplication, division, the real numbers are a closed system. They have closure. Real numbers are also closed under exponentiation as long as all the exponents are integers.
So we can pick any number on the number line, raise it to any integer power. Now again, the exception is with zero we can't raise zero to the power of negative integers cuz that would be like taking a reciprocal of it. But other than that single exception, the real numbers are closed under exponentiation. We can pick any real number, raise it to an integer power, we'd get another real number.
It's still a closed system. We're not leaving the number line. With roots though, we come to a huge lack of closure. There is no real number answer for the square root or any even root of a negative number. So we can't find the square root of a negative, we can't find the fourth root or the sixth root of a negative.
All of those leave the number line, all of those do not have solutions that are on the number line. Among other things, this means that a very simple algebraic equation such as x squared plus 4 equls 0 have no solution. There is absolutely no number on the number line that satisfies that particular equation.
Now that's kinda crazy if you think about it. Such an easy equation. How could such an easy equation have no solution? But nothing on the number line satisfies that particular equation. So this kind of thing bothers mathematicians. Mathematicians in the 16th century realized we could solve a great many closure problems simply by allowing the square root of a negative.
So this sounds odd because you probably have always heard you can't take the square root of a negative. It is perfectly true that the square root of negative does not exist anywhere on the real number line. So now what we're doing is we're leaving the real number line. We're expanding our concept of what counts as a number.
So we'd begin by defining this odd number i, which is the square root of negative 1. So this number does not exist on the number line. It is off the number line. We define this in the equation x squared equals negative 1, would have two solutions positive i and negative i. In analogy to the two solutions of an equation such as x squared equals positive 4.
That would add the solutions positive 2 and negative 2. A positive root and a negative root. So much in the same way, positive i and negative i. This i and its multiples have the unfortunate name imaginary. So this is the word that everyone uses to refer to these. So you see, earlier mathematicians such as Mr.
Rene Descartes, he's the guy who invented the xy graphing plan. He was quite a great mathematician. He used this term because he didn't believe that these numbers really counted as the equal of numbers on the number line. He really thought the numbers on the number line, those were really real numbers.
Those were real solid bona fide numbers. And these other things, it was kinda like cheating to use them. So he used the word imaginary. We now, understand that these numbers are just as legitimate as any other numbers. Turns out we measure things in the real world with these numbers. We actually measure electricity.
And an electrical current with an imaginary amplitude could kill you. It could have very real results in your life. So in other words there are real things in the real world that we measure with this. So the name imaginary is a misnomer. It's an earlier misunderstanding. Unfortunately, this sobriquet this nickname has stuck even though it's not accurate.
So I just want to emphasize even though I'll be calling these imaginary, cuz that's what people call them. I want you to appreciate they're not in fact imaginary. They're just as bona fide and legitimate as the numbers on the number line. This number i makes it possible to find the square root of any negative number. So for example, suppose we have to find the square root of negative 9.
Well, we can express negative 9 as the product of 9 times negative 1. Separate out the square roots. Square root of 9 of course, is 3. Square root of negative 1 is i, so we get 3i. Similarly, the square root of any negative would be i times the square root of the absolute value.
This number i also makes it possible to solve algebraic equations that previously had no solution on the real number line. So for example, x squared equals negative 25. There is no number on the number line that satisfies that equation, but we can solve it using imaginary numbers. And in fact, the solution would be positive 5i or negative 5i.
Consider this unfactorable quadratic equation. So this is an equation. There's no way to factor it. If we graph this, we'd find that It's a parabola that is entirely above the x axis. It never even intersects the x axis.
So in other words, there is no point on the number line of all the infinity numbers on the number line, there's not a single one that satisfies that particular equation. So we know that the answer is not a real number. But we could still solve it, so what we're gonna do is we're gonna subtract 4 from both sides and when we do that we get the square of a difference pattern on the left.
So that's a pattern we talked about in the algebra lessons, if that's not familiar to you, it's worthwhile going back and watching those videos in the algebra module about square of a sum, square of a difference, those are important patterns to know. Turns out that, that expression on the left can be factored into x minus 3 squared and so what we get is x equals 3 squared equals negative 4.
Take a square root of both sides. We get x minus 3 equals plus or minus 2i. Add 3 to both sides, we get x equals 3 plus or minus 2i. And those numbers, 3 plus 2i and 3 minus 2i, those are the solutions to that equation. So it has no real solution, but it has a solution if we involve imaginary numbers.
Such a sum or difference of a real number plus an imaginary number is called a complex number. A general complex number is of the form a plus bi. If it helps you to visualize this, you can think of the complex numbers as lying in a plane. So the horizontal axis, that's the ordinary real number line.
That's the number line that we've already known. We've always known and loved that. The vertical axis is another number line. This is the imaginary axis, so we see going up on that we have i, 2i, 3i, 4i. If we went down that axis we'd get negative i, negative 2i, negative 3i. So that's the imaginary axis.
And then a complex number such as 3 plus 2i. Well we can imagine that as a point in the plane, drawn here. So we go over 3 on the real axis, we go up 2i on the imaginary axis and that's where we plot the point. So every complex number would rely on a different point in that plane. I wanna emphasize the ACT does not test this.
You do not need to know about the complex plane. In order to do anything that the ACT is gonna ask, I simply share this, because for some people especially visual thinkers, it kind of helps you to visualize, where do the complex numbers live? I've been saying that they don't live on the real number line. Well if they're not on the number line, where do they live?
Now, we can see the real number line is just one part of the larger complex plane, and these numbers live in other places on the plane off the real number line. Notice that, when we solve both of the algebra equations above, we got two solutions in which the imaginary parts had opposite plus or minus signs. For example, we had 3 plus 2i and 3 minus 2i. This is not a coincidence.
Two complex numbers with equal real parts and opposite signed imaginary parts are called complex conjugates. A plus bi and a minus bi are complex conjugates of each other. We will see some uses of complex conjugates in the next video on operations with complex numbers. So right now, just hold onto that thought.
We'll talk about it more in the next video. Finally, in this video, I'm gonna talk about powers of i. The ACT expects students to understand the pattern governing the powers of i. So of course, i to the first is just i. I squared by definition is negative 1. What about i cubed?
Well i cubed, that would be i squared times i. So that's negative 1 times i, so that would be negative i. And finally, what about i to the fourth? Well i to the 4th, one way to think of it is i cubed times i, so that would be negative i times positive i. Well the i times i is negative 1, so we have a negative of negative 1, that would give us positive 1.
Another way to think about that, we could also express i to the 4th as i squared times i squared. So that would be negative 1 times negative 1 which is positive 1. So either way we do it, i to the 4th is negative 1. So first of all, it's very important to know that pattern i to the 1st is i, i to the 2nd is negative 1, i to the 3rd is negative i, and i to the 4th is positive 1.
That's the basic pattern. Because the pattern returns to positive 1, means the next four powers will follow the same pattern. So i to the 5th would be i to the 4th times 1, i to the 6th would be i to the 2nd times i to the 4th, and so forth. So that 1 at i to the 4th means the pattern just gonna repeat.
It's gonna repeat again, and again. And you're just gonna get this repetition like wallpaper. It's just gonna go to infinity. So that's really important to appreciate, that's the pattern. We know that i to the power of any multiple of four has to equal positive 1. This allows us to evaluate i to the power of any large number.
So for example, the ACT might ask us what is i to the power of 91? So it seems like a ridiculously large number, how are we gonna calculate this? Well turns out, we could just write it this way. I to the 91 equals i to the 88, times i to the 3, because 88 plus 3 is 91. I to the 88, 88 is divisible by 4, so i to the 88 has to be positive 1. So it's just positive 1 times i cubed, you may remember i cubed is negative i, so we just get negative i.
If you remember this simple pattern, you can evaluate any power of i. In summary, we talked about the imaginary number i, which is the square root of negative 1. We also talked about why the word imaginary's not the best word for this, but it's the word that people use. We discussed how to use i to write the square root of any negative number and noted how i can be used to solve previously unsolvable algebraic equations.
So algebraic equations that had no real number solution, now we can find solutions in the complex plane. We introduced the idea of complex conjugates and we discussed the patterns of powers of i.