My name is Ali Al-Kharagouli. I am a postdoctoral fellow at the NASA Jet Propulsion Lab. And in this video, I'm going to show you how imaginary numbers are not imaginary. In fact, the name imaginary is really stupid, and we should probably stop using it.
And I'm going to show you the math perspective on imaginary numbers, and I'm going to show you the physics engineering perspective, which is kind of the math perspective, but with the application in mind. So what I have over here is the way I is usually thought of, which is I equals the square root of negative 1. And then I have something else over here that says I equals 90 degrees. Now what does that mean?
So for the rest of this video I'm going to first go over this, what it means, how we came to this, and then I'm going to show you what this means, i equals 90 degrees, and I'm going to erase this, I'm going to draw a little graph that's going to make things a lot more intuitive so you can finally visually understand what imaginary numbers are, and going forward you can hopefully think of them a little bit different. So where did this come from, this i equals square root of negative one? Why do we have it?
And why are people thinking about it all the time? Well, this is basically if you've taken any type of high school math or college math or anything of that nature, you've probably learned that when you're trying to solve for an equation, like let's say, I don't know, y equals, I don't know, like x squared minus 4. Well, the way you try to solve for the roots or you try to find a solution for this, for it to be true, you set it equal to 0. So you have 0 equals to x squared minus 4. So then you have x squared equals 4. You have x equals plus or minus 2, right? Because you took square root of each. And then you just go ahead and plot this. And then what that basically looks like is you have negative one, two, three, four over here.
You have four, you have a minus two, you have a two. And then you basically have a parabola that looks like this, where these are the two solutions or roots, right? And that basically your algebra teacher in high school tells you, well, that's basically how you do it. And then something terrifying happens, right?
You are faced with an equation that does not look like this. You're handed an equation at some point. that says instead of x squared plus four, you'll get one that says like, I don't know, plus one. And you start freaking out because you try to do the same thing and you're like, okay, x squared equals to minus one.
Well, take the square root of both sides, square root of x squared, square root of minus one. This becomes an x and this becomes the square root of minus one. And like, what is that? What does that even mean?
What does the square root of minus one mean? And then you're basically just told, oh, that is i. And then you just get like an equal sign and it's like i and then going forward You know that i is the square root of negative one And like everyone kind of claps and moves on and like no one understands what the hell i is so this kind of the math perspective this for the people who are like Only doing like pen and paper math and and kind of have no sense of what's happening in the real physical world Now i'm going to show you the physics engineering perspective something way cooler and way more intuitive But before I explain to you what an imaginary number is in terms of physics and engineering I'm going to first ask you, well, what even is a number, right?
Before we talk about imaginary numbers, let's talk about numbers, right? Numbers are things that we basically use to count things. Like, I have one eraser in my hand right here.
There's one board. There's multiple chalks. There's a bunch of, well, there's like a couch, two couches. So you basically use the number to count physical quantities. At least that's how we use it in engineering and physics in the real world.
Now, basically, numbers can be represented on a number line, right? We can basically say... We have a number line over here and it starts out like one, two, three, and it goes all the way to infinity, right? And you've probably heard of like the different kinds of numbers, like the rational numbers and whatnot, the natural numbers and whatnot.
And then some people thought, okay, well, what happens when you have no numbers? What happens when you have zero quantity of, no quantity of things? Well, that's what zero is, right?
So we basically added zero to that number. Now, another... Beyond that, we thought, okay, well what about if you start going the opposite way?
What if you're going in the opposite direction and you start having things like negative 1, negative 2, negative 3? And this is where things get tricky because immediately, philosophically and intuitively, you're like, well, what the hell is a negative number? What does that mean when a number is negative?
We know what it means when a number is positive. That's basically how many things there are that you can count. We know what zero means.
Zero means there's the absence of it. And well, what does a negative number mean? Well, negative number does not imply that the quantity itself is negative.
The quantity remains the same, but it just implies that now we are going in a different direction. And this can be really useful in an example such as like banking or money, for example, where let's say if I give you $20, you earn $20. And then if I take away $20 from you, you lose $20. And we realize that negative numbers are something that's very useful that teach us like how to do this. values that are bi-directional, that can go one way or the other.
And for example, if you're a student and you have student loans, you can say that you're walking around and you have negative money, right? Like you have... negative money tied to your name and that's basically money that you owe and that's basically one way we understand negative numbers now here's where things get really really cool is instead of thinking the negative number like when you multiply something by negative one let's say i don't know i have a value which is five and let's say i apply it and multiply it by negative one what exactly am i going to do to it i'm basically going to take that sign i'm going to add it so it becomes minus five right and you've probably learned this in school where like you apply anything by a and you get the negative of whatever that is.
Now here's what's actually happening, is whenever you're multiplying on this same line, all you're doing is you're doing 180 degree rotation. You're basically going from going in this direction, you're like oh nope we're going to start going in this direction. So if I have this 3 over here and I multiply it by a negative 1, well what's going to happen? Now I'm going to have negative 3, which means I'm taking this arrow over here, this value that is 3, rotating it 180 degrees, Right?
And then I'm going to end up with this negative 3 over here. So in a way, you can think of a negative number multiplication as 180 degree rotation. Okay?
And it makes sense because if I'm going this way, and now I'm going like negative, the complete opposite of it, now I'm going complete opposite way. Right? Makes sense. Now here's where things get really cool. Well, you might ask, well, okay, Ali, you just, this is happening on a single dimensional number line.
What happens if we have two dimensions? Right? And I'll get a little bit why numbers have to be two-dimensional sometimes or why most numbers are one-dimensional.
But sometimes we need two dimensions to describe things. I'm going to explain that in a little bit. But let's just stick with this example. So suppose, again, I have my 0, 1, 2, 3, negative 1, negative 2, negative 3. Well, what happens if instead of doing 180 degrees, what if I only want to rotate 90 degrees?
Well guess what? I can multiply by i. Because whenever you take a number such as 3 and you multiply it by i and you get like 3i, all you're getting is this axis over here which is like 1i, 2i, 3i and so on.
This is like minus 1i, minus 2i, minus 3i. So basically if I'm over here, and again I'm going to use a different color to explain it, if I again have my number 3 at the arrow line, and instead of rotating 180 degrees, I only want to rotate 90 degrees, I'm not gonna multiply it by negative one, I'm gonna multiply it by I. And then I do this rotation 90 degrees, and then I end up with three I.
But now here's where it gets cooler. What if I want to do another 90 degree rotation, and I multiply it by another I, because we said that I is basically a 90 degree angle. Then I take this three I times another I, and then guess what? We know that I times I is basically negative one, because I is the square root of negative one.
and basically I'm rotating all the way over here. So I could either take this 3 and multiply it by 1, and that's basically 180 degree rotation, or I can take the 3 multiply by i twice, and then I'll get the exact same thing, which is 180 degree rotation. So you can think of i as just something that allows us to visualize math as two-dimensional numbers, uh, numbers as two-dimensional rather than uni-dimensional, and by basically multiplying something with i, we're being we're able to rotate rather than just like go back and forth between positive and negative numbers now i know you might what you're thinking well like like why the hell would a number be two-dimensional what does that even mean in which cases do we need numbers to be two-dimensional and why is there another axis over here right and and here's and here's here's the reality is that even though this is called real and this is called imaginary when you're taking this value and you're rotating it by 90 degrees it's not any less real than it was when it was over here it has just rotated right so one example you can think about this is if you're looking at a signal at certain uh as a signal goes by so like let's say for example sine wave and actually i'm gonna give you a very simple example let's compare this eraser and this chalk over here and let's say i'm trying to plot how they vary in size over time so let's say that i have this Eraser and as time goes by let's say I don't know like the length of the eraser We're gonna call this L and if we were to measure this I don't know like this probably like 10 centimeters or so It would just hang around at 10 centimeter mark, right?
Or like we say if it stays like at 10 centimeters or if it's even or if it's odd or not Now let's take a look at this chalk. If I measure how this chalk varies over time and let's say again, this is length Let's say this is time obviously the more I use it Let's say I'm using it like constantly like at a linear in a linear rate is gonna start out with a certain length And it's going to keep going, keep going, keep going, keep going. And it's going to keep going down linearly in a single line like that. Right? But now here's what's cool.
Let's suppose I do a different kind of function where I say instead of plotting how much the chalk is getting shorter, let's say I want to oscillate from, I want to say that the chalk, like I want to oscillate between when it goes from odd, from even to odd. So let's say it starts out at 10 centimeters and then it goes to 9. and then it goes to eight and then it goes seven and then it goes to six and i want to model that i want to model that every time it's an even number like i get one value and every time it's an odd number i get another value what i'm going to get as a plot over here is something that looks like this like starts out even and then it goes odd and then it's even and then it's odd and then it's even and then it's odd and depending on where i am looking depending on Whenever I take a snapshot in time, because this goes on in time, all I'm basically doing is the value is basically rotating in a circle. And let's say here is even, and then here is odd. Okay, and then it goes from like 10 to nine to like eight, seven. Now, a very simple way to think about this.
Well, what about what happens in between? What about like 8.5? And then what about like 7.5?
And so on. So you can think of the reason we include imaginary numbers is we're interested in beyond just like what happens at the two extremes. We're interested beyond of what just happens, let's say, between summer and winter.
We're curious about what happens in fall and what happens in spring and what happens in between. What about all these values over here that exist somewhere between the real and the imaginary number line? Well, guess what? Those are called complex values. And That's actually another stupid name because they are not really complex.
Complex does not mean they're complicated. Complex just means that if you were to plot them and draw them into an axis over here in x and then in y, they are made of one component here, which again we're going to call real, and then there's another component here which is called imaginary, and then these guys are basically modeled. One way you can think of them is using this equation, e to the j theta.
It's usually it's i, I use j because I'm an electrical engineer. And then that can be modeled as a cosine theta plus j sine theta, which basically is telling you that whenever you have a signal at any point on this domain, and you can model it by its magnitude and phase or degree, or you could basically break that. Oh, this kind of looks like a bigness. So basically whenever I have like a signal or like some value that I'm representing, I can either represent it by its angle over here and its length, and hence this guy, or I could just break it down into its cosine and sine components.
And I basically say, it's this guy and this guy added together, and you get some value that's quote real plus j, some other value. And so this is the real component. This is the imaginary component. Now, this is actually something that's very counterintuitive, and something that's used a lot in Fourier transforms and Fourier analyses and things of that nature. So if you don't have a complete grasp of complex numbers or imaginary numbers, Or while we need them to do some kind of rotations.
It's something you definitely want to take a deeper dive into But in the next video i'm most likely going to cover the fourier transform Or before that I might dive a little bit deeper into complex numbers So you can leave a comment. Let me know if this explanation Uh gave you like a good solid into intuition of complex number or if you want a much deeper Uh delve into it before we jump into the fourier transform because that one's going to be very very important And before we learn about Fourier transforms, you definitely want to understand phasors and complex numbers and oscillation and rotation And all that cool stuff. But anyway, that being said i'll see you guys in the next video. Peace