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Imaginary and Complex Numbers

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Today we're going to talk about imaginary and complex numbers. So we're going to start by reviewing what a complex number is. We'll look at what it means to graph a complex number, and then we'll look a little bit more at imaginary numbers and the powers of i.

So let's start by reviewing complex numbers. Complex numbers are in the form "a plus bi," where a is the real part of the complex number, b is the imaginary part of the complex number, and i is the imaginary unit. We define i to be equal to the square root of negative 1, and we say that it's an imaginary number. And the reason that we say it's an imaginary number is because no real number squared can equal a negative number.

So let's see how we can graph complex numbers. I have here what looks like a traditional coordinate plane, except for this time, my horizontal axis is my real axis. So I'll plot the real number component of my complex number on the horizontal axis. And my vertical axis is called the imaginary axis. So I'll use the numbers on the vertical axis to plot the imaginary number component of my complex number.

So for example, if I want to plot the complex number "2 plus 3i," I see that the real number component is 2 and the imaginary number component is 3, so I'm going to plot a point where 2 and 3 meet. So this point represents the complex number 2 plus 3i.

Let's say I want to graph the complex number negative i. So now this number does not have a real number component, or we can think of it as a real number component of 0. So I'll go to 0 on my horizontal axis, and since it's negative i, I know that there's like a negative 1 here. So I'll go to negative 1 on my imaginary axis and place a point. So this point represents the complex number negative i.

Finally, if I want to plot the point, or the complex number 4, I see that this one does not have an imaginary number component, or we can think of the imaginary number component of being 0. So I'm going to go to 4 on my horizontal axis. 4 is my real number component, and I don't need to go up or down because my imaginary number component is 0. So this point represents the number 4.

So lastly, let's look at our powers of i, which will be useful when simplifying expressions with imaginary numbers in it. So first, if we just look at i, or i to the first power, we know that that's equal to the square root of negative 1. So if I wanted to figure out what i squared is, I know that I would just be multiplying the square root of negative 1 by itself. And now when I multiply two square roots together, I know that they undo each other, so this is just equal to negative 1.

If I wanted, then, to find the i to the third power, I know that I would just be multiplying i squared by i one more time. So i to the third power is going to be equal to what i squared is. So negative 1 times another i, negative 1 times i will give me negative i. So I've found that i squared is negative 1, and i to the third power is negative i.

To find i to the fourth, I know that I can multiply i to the third by another i. So I'm going to take negative i, because that's the same as i to the third, and I'm going to multiply that by another i, and that will give me a negative i times i, which would give me negative i squared. And I know that negative i-- I know that i squared is negative 1, so this is negative negative 1, which is equal to positive 1. So I've found that i to the fourth is equal to positive 1.

To find i to the fifth, I multiply i to the fourth, which is 1, by another i. So 1 times i will give me i, which we already know is the square root of negative 1. So I'm back to the square root of negative 1.

And if I find i to the sixth power, I'm going to be multiplying the square root of negative 1 by i, or a square root of negative 1, or-- yeah, square root of negative 1, which is going to give me a value again of negative 1.

So you can see that we have a pattern, when we continue with our-- continue increasing our powers of i. The pattern goes from the square root of negative 1, to negative 1, to negative i, to positive 1. Then we have the square root of negative 1 again, and we have negative 1. So if we were to find i to the seventh, we know that that would already equal negative i, just continuing our pattern.

And to find i to the eighth power, we know that it would equal 1, again just by continuing our pattern.

So let's go over our key points from today. Complex numbers consist of a real part and an imaginary part. Complex numbers encompass a larger set of numbers than real numbers, because they include imaginary numbers. The square root of negative 1 is imaginary, because no real number squared results in negative number. And when looking at increasing powers of i, the solutions follow a pattern of i, negative 1, negative i, and 1, continuously.

So I hope that these key points and examples helped you understand a little bit more about imaginary and complex numbers. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.

- Complex number:
- A number, a+bi, containing a real part, a, and an imaginary part, b, where i is the imaginary unit, .
- Imaginary number:
- A non-real number which is a multiple of i, .