Learn about i and Imaginary numbers Video

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This is an awsome video! It really made things alot more clear to me because I came into this not knowing much on this subject so it took it from the very basics to a bit more advanced stuff

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Learn about i and Imaginary numbers

Learn about i and Imaginary numbers Welcome to the presentation on i and imaginary numbers. So let me just start out with the definition. i = √-1 or to view it in the other way, you could say that i2 = -1. Now, why is this special? Well, we knew that—or we’ve learned that any number when you square it is equal to the positive number, right? If I had -1 × -1, that equals positive 1. You don’t have to write the positive every time but that equals +1 and so does 1 × 1; that equals +1. So, if we think about the square root of a number, so far all we’ve learned is taking the square roots of positive numbers and that make sense to us because the notion of a square root of a negative number didn’t really exist until now. So, what we’ve done is if we set this definition that the number i and i isn’t a variable, it’s an actual number. Its value is equal to the square root of -1. Now, I won’t go into all of the philosophical musings about whether i—the number i or any imaginary numbers actually exist. I mean maybe I’ll make another presentation on that but they exist enough to be very useful to many engineers and physicist. So, I’ll leave you with that and I’ll also—well, I’ll just hint at—well, I won’t go into the whole, either the iπ = -1. But that pulls my mind but I won’t go to that. And when you think about whether I really exist, you should also think about whether anything really exists. Well anyway, so I’ve diverged for too long. So let’s back to what I was saying before. i = √-1 and i2 = -1. So let’s think about the implications of this. If I were to say—well i1, just like anything else is equal to itself, right? i2, I’ve already said using this definition. i2 = -1. i3, well that would just equal i2 × i, right? And i2 is -1. So, it would be -1 × i and that just equals negative i, right? An i4 = i3 × i, right? I am just using my exponent rules here. Well, i3 is -1 × I, right? i3 is -i and then we skip that i. Well, what’s -i × i? Well, that is the same thing as -1 × i × i, right? And what’s i × i? Well, definition. i × i; i2 = -1. So that equals -1 × -1 which equals 1. Interesting. Let me clean this up a little bit. So actually, let me start with i0. i0, well we know anything to the 0 of power is equal to 1. So, we’ll just keep that. That’s still equals one. i1 = i. i2 by definition is equal to -1. i3; I just showed you is equal to -1 and that makes sense because that’s just i2 × i. An i4 = 1 again and if I did i5; well that is just equal to i4 × i, right? I’ll write that down. i4 × I; that’s times. i4 = 1, right? This is equal to one. So, 1 × i = i. Did you see a pattern here? i0 = 1, i-1 = i, i2 = -1, i3 = -i, i4 = 1 again. So i4 = i0 and i1 = i5. I think you’ll find out that i0 and you could try this out if you don’t believe me. i0 = i4 which equals i8 which equals i12. I think you see the pattern. Any multiple of 4 = 1 and i1 = i5 = i9 = i13 = i. So that’s i to any power that is a multiple 4 + 1, right? Because 5 = 4 + 1, 9 = 8 + 1 and we could do a similar pattern; i2 = i6 = i10 and so on and that equals -1 and finally i3 = i7 = i11 and so on equals -i. So why is this useful? We see a pattern. It’s a cycle of 4 and so in this pattern, if we look at this, we can use this to determine what i to any power is. So if I were to ask you what i100 is. Well, you could just work it out. You could say, well that’s just equal to i × i99 and so far so down. But if we use the cycle, we see that 100 is a multiple of 4, right? 4 × 25 is a 100. So, i100 will fall into this category; this first one. It’s a multiple 4. So, we know that i100 = 1. Similarly, if I said i101, that’s going to equal i, right? It is because that equals a 100 + 1. So it puts you into this category. It equals a multiple 4. 100 is a multiple of 4 and a 101 is a multiple + 1. i102 power? Similarly would equal -1. i103 would equal -i. I hope you understand what I am doing here and all I did is I defined i as a square roots of -1. And then I kept multiplying i to figure out a pattern. I said i0 = 1, i1 = i, i2 = -1, i3 = -i and then i4 = 1 again and then the pattern repeated itself. And then I use that pattern to be able to figure out i to any power even it’s a very high number. So, a very simple way to think about it is, if I had i323, what I do is I say, if I were to divide 4 into 323, what is the remainder. Well, I know 4 goes into 320, right? 4 × 80 = 320. So I know that when I divide 4 into 323. So, 4 goes into 323, 80 times with the remainder of three, right? And the remainder is what we care about. And this number is actually called the modulus and maybe I’ll do another module on modulus. It’s very important and actually in computer programming. But since we knew that when you divide this exponent by four, the remainder is 3. We can say that this is the same thing as i3 which we’ve learned is –i similarly, if I said i502. Well, I know 500 is divisible by 4, right? 4 × 125 is 500. So the remainder is 2 if I were to divide it by 4. So, I could say that this is the same thing as i2. In i2, we learned by definition is -1. If I were to ask you i37, we know 36 is divisible by 4, so the remainder is 1. So, it would i1 which equals i. Hopefully that gives you an indication of what I—it might have been confusing the first time because we are dealing with a number that’s “imaginary.” And I am teaching you the cycle property of it. What you might want to do is review the video again but then after that, you could just try the module on i which essentially just keeps working you through this type of problem. I hope you have fun. Bye.

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Learn about i and Imaginary numbers

Welcome to the presentation on i and imaginary numbers. So let me just start out with the definition. i = √-1 or to view it in the other way, you could say that i2 = -1. Now, why is this special? Well, we knew that—or we’ve learned that any number when you square it is equal to the positive number, right? If I had -1 × -1, that equals positive 1. You don’t have to write the positive every time but that equals +1 and so does 1 × 1; that equals +1.

So, if we think about the square root of a number, so far all we’ve learned is taking the square roots of positive numbers and that make sense to us because the notion of a square root of a negative number didn’t really exist until now. So, what we’ve done is if we set this definition that the number i and i isn’t a variable, it’s an actual number. Its value is equal to the square root of -1.

Now, I won’t go into all of the philosophical musings about whether i—the number i or any imaginary numbers actually exist. I mean maybe I’ll make another presentation on that but they exist enough to be very useful to many engineers and physicist. So, I’ll leave you with that and I’ll also—well, I’ll just hint at—well, I won’t go into the whole, either the iπ = -1. But that pulls my mind but I won’t go to that. And when you think about whether I really exist, you should also think about whether anything really exists.

Well anyway, so I’ve diverged for too long. So let’s back to what I was saying before. i = √-1 and i2 = -1. So let’s think about the implications of this. If I were to say—well i1, just like anything else is equal to itself, right? i2, I’ve already said using this definition. i2 = -1. i3, well that would just equal i2 × i, right? And i2 is -1. So, it would be -1 × i and that just equals negative i, right? An i4 = i3 × i, right? I am just using my exponent rules here. Well, i3 is -1 × I, right? i3 is -i and then we skip that i. Well, what’s -i × i? Well, that is the same thing as -1 × i × i, right? And what’s i × i? Well, definition. i × i; i2 = -1. So that equals -1 × -1 which equals 1. Interesting.

Let me clean this up a little bit. So actually, let me start with i0. i0, well we know anything to the 0 of power is equal to 1. So, we’ll just keep that. That’s still equals one. i1 = i. i2 by definition is equal to -1. i3; I just showed you is equal to -1 and that makes sense because that’s just i2 × i. An i4 = 1 again and if I did i5; well that is just equal to i4 × i, right? I’ll write that down. i4 × I; that’s times. i4 = 1, right? This is equal to one. So, 1 × i = i.

Did you see a pattern here? i0 = 1, i-1 = i, i2 = -1, i3 = -i, i4 = 1 again. So i4 = i0 and i1 = i5. I think you’ll find out that i0 and you could try this out if you don’t believe me. i0 = i4 which equals i8 which equals i12. I think you see the pattern. Any multiple of 4 = 1 and i1 = i5 = i9 = i13 = i. So that’s i to any power that is a multiple 4 + 1, right? Because 5 = 4 + 1, 9 = 8 + 1 and we could do a similar pattern; i2 = i6 = i10 and so on and that equals -1 and finally i3 = i7 = i11 and so on equals -i.

So why is this useful? We see a pattern. It’s a cycle of 4 and so in this pattern, if we look at this, we can use this to determine what i to any power is. So if I were to ask you what i100 is. Well, you could just work it out. You could say, well that’s just equal to i × i99 and so far so down. But if we use the cycle, we see that 100 is a multiple of 4, right? 4 × 25 is a 100. So, i100 will fall into this category; this first one. It’s a multiple 4. So, we know that i100 = 1. Similarly, if I said i101, that’s going to equal i, right? It is because that equals a 100 + 1. So it puts you into this category. It equals a multiple 4. 100 is a multiple of 4 and a 101 is a multiple + 1. i102 power? Similarly would equal -1. i103 would equal -i.

I hope you understand what I am doing here and all I did is I defined i as a square roots of -1. And then I kept multiplying i to figure out a pattern. I said i0 = 1, i1 = i, i2 = -1, i3 = -i and then i4 = 1 again and then the pattern repeated itself. And then I use that pattern to be able to figure out i to any power even it’s a very high number.

So, a very simple way to think about it is, if I had i323, what I do is I say, if I were to divide 4 into 323, what is the remainder. Well, I know 4 goes into 320, right? 4 × 80 = 320. So I know that when I divide 4 into 323. So, 4 goes into 323, 80 times with the remainder of three, right? And the remainder is what we care about. And this number is actually called the modulus and maybe I’ll do another module on modulus. It’s very important and actually in computer programming.

But since we knew that when you divide this exponent by four, the remainder is 3. We can say that this is the same thing as i3 which we’ve learned is –i similarly, if I said i502. Well, I know 500 is divisible by 4, right? 4 × 125 is 500. So the remainder is 2 if I were to divide it by 4. So, I could say that this is the same thing as i2. In i2, we learned by definition is -1. If I were to ask you i37, we know 36 is divisible by 4, so the remainder is 1. So, it would i1 which equals i.

Hopefully that gives you an indication of what I—it might have been confusing the first time because we are dealing with a number that’s “imaginary.” And I am teaching you the cycle property of it. What you might want to do is review the video again but then after that, you could just try the module on i which essentially just keeps working you through this type of problem. I hope you have fun. Bye.

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