Transcription
In this video, I want to introduce you to the number "i," which is called the imaginary unit. As you will see, and what may be a little difficult to appreciate at first, this number is more bizarre than other stray numbers we learn about in math, like pi or e. It is more bizarre because it doesn't have a tangible value in the sense we're used to when defining numbers. "i" is defined as the number whose square is equal to -1. This is the definition of the number "i," which leads to many interesting properties.
In some places, you will see "i" defined in this way: "i" is equal to the square root of minus one. I want to emphasize that this is not wrong, and it may make sense to you: if the square of something is equal to -1, then maybe that something is the square root of -1. It seems, therefore, that this is almost the same statement, but I want you to be careful when you do this. Some people go further and say that this is wrong, and it turns out they are wrong in saying that it is wrong. But when you do this, you have to take care of what it means to take the square root of a negative, imaginary, and as we will see in the future, complex number. But for your understanding now, you don't have to distinguish between these definitions. There's no point in splitting hairs now, distinguishing these definitions.
Using this definition now, let's consider what these different powers of "i" are. Because if something raised to the second power is equal to -1, you can imagine that if I calculate different powers of this number, I can get strange results. We will soon see that the powers of the number "i" arrange themselves quite nicely, creating a cycle. They go around and around a certain set of values. Let's start with "i" to the zeroth power. You can say that anything to the power of zero is equal to one, so "i" to the zeroth power is also equal to one. That's true. You could derive this directly from this definition, but this is more direct: anything (including "i") to the power of zero is equal to one. Okay, and how much is "i" to the first power? Well, anything raised to the first power is equal to simply that something. This is therefore simply equal to "i." According to the definition of what power means. That makes sense.
Next, we have "i" to the second power. "i" to the second power, by definition, "i" to the second power is equal to minus one. Let's try to calculate "i" to the third power. I'll do this in a color I haven't used yet. "i" to the third power, well, that will be equal to "i" squared, times "i". And we know that "i" squared is equal to minus one, so this will be -1 times "i". I will explain this again. This is the same as this, which is the same as this. "i" squared is minus one. So we multiply: minus one times "i" equals "-i".
What happens if we take "i" to the fourth power? I'll do this up here. Again, this will be equal to "i" times "i" to the third. That is, "i" times "i" to the third. How much was "i" to the third? "i" to the third was equal to minus "i". This here is equal to "-i," and "i" times "i" gives us minus one, but we have a minus here, so "i" times "i" is equal to minus one, and we have a minus at the beginning, so we get plus one. I'll write that down. This is the same as "i" times "-i", which in turn is the same as minus one times, remember that multiplication is commutative, when we multiply several numbers, we can change the order. So this is the same as minus one times "i" times "i". "i" times "i" is equal to one, by definition. Minus one times minus one is equal to plus one. Therefore, "i" to the fourth is equal to "i" to the zeroth.
Let's try "i" to the fifth. "i" to the fifth. This will simply be "i" to the fourth... times "i". And we know that "i" to the fourth is equal to one. So we have one times "i". So this is again "i". So once again we have the same thing as for "i" to the first. Let's try again to see that the pattern continues. Let's try "i" to the seventh power. I'm sorry, "i" to the sixth power. This is "i" times "i" to the fifth. "i" to the fifth, as we calculated, is equal to "i", so "i" times "i" is by definition equal to minus one. Let's stop here. We could go on doing this. We could calculate higher and higher powers of "i" here. And we will see that the values will repeat themselves in a circle. In the next video, I will show you how to calculate any power of "i", how to calculate how much it is. But let's check that the circle keeps turning. "i" to the seventh power is equal to "i" times "i" to the sixth. "i" to the sixth power is equal to minus one. "i" times minus one is equal to minus "i". If you take "i" to the eighth, it will be one again, "i" to the ninth will be equal to "i" again, and so on.