Imaginary and Complex Numbers

Video Transcription

Hi and welcome. This is Anthony Varela and today I'm going to introduce imaginary and complex numbers. So we'll define complex numbers, take a look at how we can represent them on what's called the complex plane, and then we're going to look at an interesting pattern with the powers of i, the imaginary unit i. So first, a complex number is a number a plus bi, and it contains a real part a, an imaginary part b, and then we have this imaginary unit, the square root of negative 1. Now this is imaginary because any real number squared results in a positive number. So to take the square root of a negative number would be not real, so we call that imaginary.

So a complex number has a real part and an imaginary part. So 5 plus 4i is a single complex number. And when thinking about numbers and number relationships, when we learned about our real numbers, this consisted of whole numbers, natural numbers, integers, rational, and irrational, are all real numbers. Well, placing complex numbers within this relationship, we see that it represents a larger set of numbers because it includes real numbers but also imaginary numbers. So let's give a definition to imaginary numbers as well.

This is a non-real number, which is a multiple of i, and once again, i is the square root of negative 1. So now I'd like to talk about the complex plane, and it looks very much like the coordinate plane that you're used to for graphing curves, lines, or plotting points. It has a horizontal axis and a vertical axis, but we don't call them x and y. We call the horizontal axis the real axis, and the vertical axis is the imaginary axis. So we're going to plot some numbers on the complex plane.

And the first number that we're going to plot is 2 plus 4i. So this is a complex number, has a real component of 2, an imaginary component of 4 being multiplied by i. So we're going to locate 2 on the real axis. So here is the real number 2, and then we're going to locate 4 on the imaginary axis. So this would be 1, 2, 3, 4. So then our complex number 2 plus 4i is right there. Now we're going to plot 2i, but notice it doesn't have a real component at all, so starting at the origin here, we're going to find 0 on the real axis, and we're already at 0 on the real axis. So then we'll just find 2 on the imaginary axis, and this is 2i.

And lastly, here we have negative 7. It has no imaginary component, so we're not going to travel up or down here on this imaginary axis. We're just going to find negative 7 on the real axis. So that is how we can find negative 7 on our complex plane. So notice that our pure real numbers that have no imaginary components lie on the horizontal axis, and our purely imaginary numbers, so containing no real component, lie on the vertical axis.

Last thing I'd like to talk about the powers of i, and there's a really interesting pattern when you increase the powers of i. So we're going to start with just i to the first power. This is the square root of negative 1. Now if we increase that power, we would be then multiplying i by itself, so this would be the square root of negative 1 times the square root of negative 1. That is the real number negative 1. So i squared is a real number. Well, when we increase that power again, so we're going to multiply then negative 1 by the square root of negative 1.

So i cubed then is negative square root of negative 1. So we can call then i cubed negative i. And then when we increase the power of i once again, we can think of i to the fourth as being i squared squared, so this would be negative 1 times negative 1, and that is positive 1. So we see that i squared is a real number. i to the fourth power is a real number, but they have opposite signs. And this pattern actually continues.

When we increase the power of i again, i to the fifth is going to be i, because we're just taking 1 and multiplying it by i, and we're back to this cycle. If we were to increase the power of i again, we would have negative 1. Increase the power of i once again, we have negative i, and increasing the power again, we have positive 1. So the powers of i cycle through this pattern of i, negative 1, negative i, positive 1. Pretty interesting. So it's a cyclic pattern or powers of i, and they will rotate in this pattern of grouping as of four, i, negative 1, negative i, and 1.

So let's review our lesson on imaginary and complex numbers. Well, we talked about a complex number containing a real number and an imaginary number, so a plus bi is our general complex number. We plotted numbers on the complex plane. So the horizontal axis is the real axis, the vertical axis is the imaginary axis. And lastly, we talked about this pattern of the powers of i cycling between i, negative 1, negative i, and positive 1. Thanks for watching this tutorial on imaginary and complex numbers. Hope to see you next time.