How Math Becomes Difficult

This is a really scary looking equation with all these complex, terrifying symbols, weird operations, and bizarre notation. But is it actually difficult? Is it actually scary, or is it just notation?

Well, in order to understand that, we'll have to start with addition. I won't overcomplicate this too much. We all know what adding is, and if you take a number like three and add five to it, you will get an eight. That's simple.

But then we have multiplication, which technically would be kind of just repeated addition. But a big key of how mathematical tools work is the fact that if you complicate an operation like addition enough to make multiplication, in return for this complexity, you get some additional usefulness. For example, we can use multiplication to describe the area of a rectangle. Right here, we have a number representing the left side and the bottom side. Multiplying these two numbers gives us the area of this rectangle. And even though you can think of multiplication as repeated addition, you can't really describe this area using addition. That's because if we have, for example, 2.2 and 4, then this means 4 + 4 plus well, 0.2 over four. So once again, it's multiplication. And that's what makes complex operations different from their underlying parts: the fact that complexity gives rise to a completely new usage.

And speaking of repeating an operation in a way which gives it more usefulness, next we have exponentiation, so powers, which once again can be thought of as repeated modification. So M pressing once does the thing to the first power, twice second power, thrice third power, and so on. Why would that be useful? Well, it's because in real life, we really often encounter things which increase exponentially. So if you, for example, have an apple tree and each year that tree drops five apples, which then grow into new trees and then die, then your apples will increase exponentially. Another good example of exponentiation would be dimensions. So if you have a one-dimensional line with a length of five, then its length is equal to well, five. That's pretty simple enough. If it's a two-dimensional square, however, then its area would be equal to 25. If it's a three-dimensional cube, its volume is 125, and so on.

Early in the video, we're actually getting a pretty good idea of why tools exist in math in the first place. You see, tools aren't really just about what we can have in math, only about what's useful in math. That's why addition, multiplication, and exponentiation are so common that we all know what they are. But not that many people know about tetration. Tetration is the next step up: addition, repeated incrementation; multiplication, repeated addition; exponentiation, repeated multiplication; tetration, repeated exponentiation. So why do most people not know about tetration? Well, it's because tetration is kind of useless. The numbers get too big too fast, and there aren't really any good scenarios where tetration is good at describing something. It has so few use cases, in fact, that we don't even have tetration properly defined or notation for it. That's because no one can nothing to do it. There is no point. Tools are only recognized as tools when they have specific use, and they don't even have to be all that different for us to recognize them as different tools.

Take inverse operations, for example. What are inverse operations? Well, that would be subtraction, division, and roots. They technically exist on paper but aren't quite as separate as you might think. Subtraction can be thought of as a separate operation next to addition, or just addition of a positive number with a negative number. Division could be thought of as a separate operation next to multiplication, or just multiplication with a number to the negative power. Roots could be thought of as a separate operation next to exponentiation, or well, we don't have the tetration out properly to see if the pattern continues, but you get the idea. It's a power to the divided number.

The point is, when you start with a certain mathematical tool, if there is a specific case where there is a lot of potential use to be had, it's better to separate it off and create different rules for that spot to let it grow and potentially even become a separate operation. Going back to a rectangle, for example, if we freeze one of the sides and just move the other one, we get a pretty good idea of how multiplication works: increase the size, decrease the side, simple. But now imagine letting that side choose whatever size it wants and freezing the area. If we increase the width with that locked area, the other length has to get shorter. I'll just play it again to let it really sink in. So basically, the higher the number we multiply by, the higher the result. The higher the number we divide by, the lower the result.

But maybe there's a better way to display this relation. Well, imagine creating two axes like this and replacing some numbers in our typical boring equation with placeholders. So this axis is representing the right placeholder, this one representing the left placeholder. The way it works is pretty simple. Right here was out of zero. The more to the right, the higher the red number. The more to the left, the lower the red number. The more up, the higher the green number. The lower down, nonetheless, the green number. All right, got that? Okay, now let's pick a number like 1.5 and input it into our left placeholder. When we multiply it out, we get 0.6666666 and so on. So if we created dots on our first placeholder axis and we move it 1.5 units to the right to represent this number, we can show that it corresponds to 0.666 and so on in the second placeholder by moving it up. Now, yeah, it got a little bit confusing, but if you understand this operation, which, well, I think you should, it's just division, then you should understand why we move the dot 1.5 units right and 0.6666666 units up.

Next, we can repeat the process for a one, or a two, or a three, or a half, and keep repeating it until we get the line like this. And this is a much bigger step that may seem. That's because I've actually been tricking and deceiving you. You see, these placeholders are not rectangles at all. They are, in fact, letters in the math. Yeah, we can basically replace this with an X and a Y, and we end up with something that most people absolutely despise: math. And yeah, okay, I get it. Doing equations with letters instead of numbers is more difficult. But the common rule of science: if there is something that's widely used and has easy alternatives, then it's either done because of tradition (I'm looking at moles, moles are not a unit), not done because they introduce a lot of functionality. And in this case, it's the second one.

You see, when we set up an equation like this, it's quite pointless. Both sides are equal to seven, simple. If we, however, change it up like this, adding two letters, A and B, then once we solve it, we get this. Or rather, an equation telling us exactly how large A has to be in relation to B, or how large B has to be in relation to A. This is extremely convenient and saves a crazy lot of time because it means that if you have to account for, let's say, 10 different scenarios where A may differ, then you can just solve the equation once and be done with it. Plug in the A and you're done. So as much as most people hate it when letters are introduced with the numbers, when you solve an equation with numbers, you just solve one. When you solve an equation with letters, you solve all of them. And when it comes to other characters like Greek letters or whatever, it's just the same thing, right? We just use them in different places for clarity, but it doesn't make any difference. Letters are just letters. Characters are just characters. Doesn't matter, Greek or any other.

And speaking of adding even more letters, now take a look at a function like this. You can see that it's going up, it's going down, it's going all around. But what if you wanted to quantify just how much it's going up? Well, then what we could do is we could introduce a slope. How do we define the slope? Well, imagine that we have two points which define a slope. First point with its X and Y coordinates, here is the X, here is the Y. And second one with its X and Y coordinates, here's the X, here's the Y. Now, based on these coordinates, we'd like to create a single number for our slope. If it's high, the number should be high. If it's low, the number should be low. If it's flat, it should be zero. And if it's negative, it's going down. How would we do it? Well, first, we can realize that if we move these points around together, the slope isn't changing, which means that we won't really care about the X and Y coordinates, only about the difference in the coordinates. In physics and math, we represent difference with delta most of the time, so don't get spooked by it. It just means difference, okay?

As I mentioned, characters are just characters. So let's see how that difference affects our slope. Well, you can see that increasing the difference in heights increases our slope, and decreasing it decreases the slope. So we know that our slope will be proportional to delta Y. All right, cool. Now, what about delta X? Well, as we increase delta X, our slope decreases, and as we decrease it, it increases. So it would be whatever the opposite of proportional would be, like I'm like inversely proportional. But what would be the inverse of multiplication? One by delta X. That's what I meant by saying that division is just an inverse of multiplication. And how to combine them? And now to combine them. Yeah, delta Y divided by delta X. That's our slope. And just to convince you that that's the case, here you can see the computation of the slope for a couple of values. You can pause if you want to make sure that it all makes sense.

For now, so here's the number detailing exactly how high the slope is. Except, you see, there is a problem. What if we have a function? We set the points and we get the results. Seems innocent enough, right? Well, almost. That's because this isn't the function we're measuring the slope for, only this. And right here, you can see that the function is actually going up and down, and the slope doesn't align at all. So what we need is not a slope, but the slope defined by a single point to avoid this whole in-between-two-points-going-up-and-down problem. Well, this presents an issue. You see, we can take these points and bring them closer and closer and closer, and each time we do so, we get a more and more accurate result, but it's never perfect. So you might say, "What if we just keep bringing it closer to zero?" Well, if the distance between two points is zero, then we run into another issue. We get a zero divided by zero, which is a big mathematical no-no.

So how do we fix it? Well, we need them to be as close together as possible, but zero breaks everything. So to that, you may say something like, "Make it, what if we just keep bringing it closer to zero?" Wait, didn't we just try that and didn't work? Well, no, we set it to zero last time. But what if we don't set it to zero, only get closer and closer to zero, like as close as possible, like um, like a limit as this X approaches zero? That's what limits are for. Limits are for representing a value that gets closer and closer to something but never quite reaches that point. And now, if we make our equation, it works. Here we have the position of the first dot and position of the second dot. And down here at the bottom, we can just cancel out the X's out. And there, that is our derivative. And that's how we can calculate the derivative of any function, like X squared, for example. See, it's simple.

So why do we think of derivatives as hard, difficult, and complex? Well, it's because of how we solve them with more difficult functions. Okay, so let's just say that for any arbitrary function, we have this equation, and we'd like to solve it. The right side is already as tight packed as possible, so we only have a limit to solve. Now, in order to solve this limit, we can use something called L'Hôpital's Rule, which I won't really explain because it won't be relevant later on, and this is a video about more main mathematical tools, not solutions to them. And so, in the end, we end up with this equation, which in order to solve, we need to take care of this derivative. Uh, so once again, we bring up the equation for the limit. Okay, there's still a limit to solve. So L'Hôpital's Rule, and um, all right, that's, that's the derivative again. Uh, I think you get the idea. The point is, we don't have the tools to algorithmically solve all derivatives, and with some of them, you do have to get clever. And whilst we do have this equation, we can't just rely on it for all cases. This is more of a definition rather than a method for solving them.

So there, if you take a function, in order to check how it's changing, you need the derivative. But now there's another question: what if you have the derivative and you would like to get the original function? Right here, we enter the domain of the integral. You might have seen it before with notation like this. It's scary, really often used in movies or games to represent complex math equations. But how scary is it in reality? What does it mean? Well, the secret to understanding integrals is right on your screen right now, actually, in the notation. First, we have this f(x), or in other words, a function. We have it graphed right here. Or actually, let's simplify it a bit for the sake of understanding it better. Now, on a completely unrelated note, how would we get the area under this function? Well, we could imagine a bunch of rectangles like this. The area of the rectangle, which we could then get using multiplication. So this side times this side. So for each rectangle, its height would be equal to the coordinate of the function plugged into the function. Now, what about the width? Well, we don't know exactly what the width would be, so let's just say that if we start with this X and end with this X, this would be the difference in X, or like, um, dx for short. Now, the total area under the function would be just all of these added together. So rectangle one plus two plus three, and so on. Okay, this is a bit of a, it's a bit too infinite for me to actually put it on the screen right now. And that's why we can use the alternate notation. So this. And now, yeah, it may seem complex, but this is just this. All right, with different notation. Yeah, this capital sigma notation is just us adding stuff together. It's the exact same thing. Okay, this capital sigma just means add all of them. I'm just going to leave it on the screen for a second to make sure you understand that.

And with that in mind, there. So just to reiterate, this right here is the height of a rectangle. This is the width of the rectangle. If you multiply them together, you get the area. Now we just add together all rectangles, and this gives us the approximation of the area under the function. But the keyword here is approximation, not the actual area. That's because, as you can clearly see right here, we have a gap in between the rectangle and the function, which we're not counting. So how will we fix it? Well, a solid way to do it would be just add more rectangles. Then the gap gets smaller and smaller, eventually disappearing. So we want the area under the function. We'd want this, except instead of adding a certain discrete amount of rectangles with a certain thickness, we want an infinite amount of infinitely thin rectangles. Now, infinitely thin is simple enough, just like with our derivative segment. DX is getting closer and closer to zero. But what about the adding infinite amount of rectangles junk? Well, we can just create a whole new character. So whilst capital sigma here means add together for every integer between A and B, this squiggly line would mean add for every number A and B. And that's actually, that's the integral. That, that's what it is. That's the exact notation for the integral, and that's what integrals are. And that's why the secret to understanding integrals is right here on your screen right now, in the notation.

But there is a little bit more I'd like to mention. That's the fact that integrals are opposites of derivatives. So if you have a function, you take a derivative, and you end up with the derivative. But if you take that derivative and integrate, then you get back the original function. How? Well, we can start with a function of X squared and analyze what happens when we take the derivative of it. We had this. And right here, I don't want you just compute the derivative or look it up. I want you to understand why the derivative of X squared looks like this. Right, pause if you want. Okay, got it. Well, the answer is that their slope is going up. So right here it's zero because the line is flat. Right here it's going up and up and up. Now, what I like to focus on here is that derivatives are blind to whatever happened before with the function. And I'll skip the appreciation of the poetic aspect of derivatives, how they tell you how the function changes whilst caring only about the single point, which cannot tell you anything about how the is changing. And you know what? It's just absolutely beautiful. Math is amazing. But I'll skip up for now and just accent the fact that we can take this function, move it up, move it down, and the derivative doesn't change. It's always the same. That's because this arrow is always pointing up, doesn't matter where it is. We don't care about the position of this arrow, we care about where it's pointing.

Okay, now let's forget about the original function. Take the derivative and modify a bit. Modify it by flattening it on the right, like so. Now, what would the original function look like? Okay, let's think about what the integration means this time. It means taking the area under this function. So let's go to the point before we flatten and let's start here. Okay, so if we take the integral from this point onward, we get some value. Whatever. We don't care about the value exactly. What we care about is what happens as we go on. You see, the derivative stays the same. It's constant. It's flat, just the same old number. Whilst the derivative is constant, the area is increasing. The line may be flat, but this area is still growing. That's why our original function will grow at a constant rate. And the basic idea is: if you have a function and you want to get its rate of change, you derive. If you have something telling you the rate of change and you want to get the original value, integrate. And that's why we need this plus C right here. It's because no matter if we move our function up or down, the derivative stays the same. Oh, also, I do believe I should mention integrals at everything above zero and subtract everything below zero. Now, I just thought I should mention that. It seems a bit obvious, but yeah, still important.

And there, now we have access to tools to work with different functions. But what functions should we use them on? And so here's where oriented the domain of trigonometric functions. And actually, what are those functions like? What is sine? Well, you have the sine, the cosine, tangent, secant, cotangent, and arcsine. So what do all of these functions have to do with each other? No, honestly, it's a genuine question. I'm really asking you, what is one thing all these functions have in common? I'll give you a second. Okay, got it. Well, the answer is they have something that none of the functions we've seen today have, and that's the fact that they are cyclic. As you can see, they occur in cycles forever, except for arcsine, but more on that later. Let's take sine, for example. Right here, you can clearly see this segment which is repeating forever. So why is that useful? Well, that's because with the functions we've seen so far, they can only go up or go down or well, stay in the middle. They can't just repeat forever. But why would that be a problem? Well, consider the following. You have a circle, and you're walking along it. After a certain distance, you end up in the same spot. That's why if you'd like to describe the position of this ball, you can't use something that will always go up or always go down. You need to use something that will go up and down and up and down and oscillate like that. And so that's what our trigonometric functions will be useful for: describing things going in circles, like angles.

Okay, so for angles, there are two ways you can describe them: with degrees or radians. Degrees are simple enough, and everyone's familiar with them. They measure this angle in a way where this is 0 degrees, this 45, this is 90, this 180, this is 270, and this is 360. Radians are different, however. Radians measure how far you've gone along a circle with radius one. And so right here, we've moved one unit, here two units, three units, pi units, 2 pi units. There we go. So degrees tell you the angle. Radians tell you the distance you've walked on a circle, which in turn tells you the angle. Now, we'll disregard the degrees for the rest of this video because they are a bit too arbitrary, and we'll stick to radians. That's because they are just so rad.

So what's the connection between radians and sine? In order to understand that, we'll need to add one more piece to this puzzle, and that's two lines like this. What is the angle between these two lines? Well, just like with derivatives, we can start with a point right here and a point right here, check the difference in their positions. And when you believe it, it turns out that the slope that we talked about in the derivatives chapter is proportional to the angle between these two lines, kind of. You see, it's not quite proportional. And to get what I mean, let's take our slope and start at zero. When our slope is equal to zero, we get an angle of zero radians. Now, as our slope increases, so does the angle, but the angle is lagging behind a bit. More notably, in order to get an angle of half pi radians, we need this side to be equal to zero and this one to be infinite. So yeah, in order to get the right angle, we need an infinite slope. But what if we were to go back? Well, going back, we need a negative slope, and it follows the exact same pattern. So there, this is what the slope looks like in relation to the angle. And would you know it? It just so happens that this is a trigonometric function called the tangent.

And so that's what trigonometric functions are. They tie together the angle rotating around with the values going in a straight line. If we create a triangle like this, and this is the angle we'd like to guess, then if we name these sides opposite, adjacent, and hypotenuse, then you can remember what each function does by remembering SOH CAH TOA, or in other words, sine is equal to opposite divided by hypotenuse, cosine is equal to adjacent divided by hypotenuse, and tangent is equal to opposite divided by adjacent. There, that's what trigonometric functions are used for. And I'd really like to pause this here for a second so that you understand their usage, because that's really kind of it. When you have an angle, you draw it around a circle with radius one and check how long this distance is until you reach your angle. There, that's your angle in radians. Then take that number, plug it into one of these functions, and you can get the ratio between this side and this side, ratio between this side and this side, ratio between this side and this side. There, that is trigonometric functions, at least the basic ones.

And this is really important because even though the usage might be a bit more complex, and I get it, it may be hard to grasp at times, the basic idea is they connect together rotation with linear stuff. They tie together rotating circles with linear lines, at least the basic ones. Then we can expand them some more, like for example, what if you have the ratio but would like to get the angle? Well, then you can use the arcsine, arccosine, and arctangent functions, which all do the exact same thing for the respective sides. Then you also have things like cosecant, secant, cotangent, which is all inverse of the sides, and so on. So yeah, there are a lot of different trigonometric functions, and it may be a bit overwhelming to talk about all of them in these. But all you have to remember is that all they are used for is just to talk about angles, describing rotation using simple linear numbers. Except if this is now your first time getting into trigonometry, then first of all, I'm, I'm glad you checked my channel to learn math. I really appreciate it. I'm really thankful, and I hope that you've already learned something. Thank you so much for watching the video. But at the same time, if it is really your first time getting into trigonometry, then there is probably something that's not clicking yet, right? I mean, something feels off, like the connection between angles and numbers. Okay, but there is something missing, and that is the question of what is sine. Wait, didn't I just answer this question? Well, no, we answered the question of what sine is used for, but not what it is. And in order to answer that question, we have to do a complete 180 and go back to exponents to ask the question of what is the square root of -1. Huh? How is that in any way related to angles and rotation?

Well, let's go through it step by step. First things first, why is the square root of -1 interesting? Well, it's because if you take a number like one, you multiply it by one, you get one. If you take a number like one, multiply it by negative one, you get negative one. Same with -1 times 1, but then -1 times -1, you get 1. And so the interesting thing to note here is that this is just a square, and this is just a square. So both 1 squared and -1 squared are equal. They are both positive. So what square root of -1 is asking, basically, is what number multiplied by itself once would give us a negative number? And one thing that's really interesting is that nothing fits here. You can try it yourself if you want, like plug in a bunch of different numbers, multiply them by yourself, none of them would give you a negative number. That's because these minuses cancel each other out. No real number would ever give you a negative number if you square it. But if no real number would work, then how about we just imagine one? And if the real axis doesn't work, just add an imaginary axis. There. This right here is called a complex plane, and can represent any complex number, a complex number being a number that's made up of a real part and an imaginary part, like this. And that's where imaginary numbers come into play. There isn't really much more to say here. The history of imaginary numbers basically boils down to mathematicians realizing that square root of -1 doesn't make sense, they invent imaginary numbers out of thin air. Four stages of grief, accepting, and finally realizing that imaginary numbers could be useful.

How could they possibly be useful? Well, going back to the sine example, you see, one problem with sine is that it's going in cycles, up and down, and up and down, and up and down. But that's basically impossible to achieve using real numbers. Try it for yourself using the addition, multiplication, exponentiation, the inverse operations. Try making a function which isn't approaching a certain value as time goes on. It's impossible. But with imaginary numbers, it's a different story. Take i to the power of n, for example. We start with i to the power of zero, which is one. Okay, I guess I haven't re-explained the zero power, but it's kind of irrelevant, pretty simple. You think about it for yourself, just trust me, it's one. Next, if we bring it up to the first power, it's 1 times i, which is i. Next to the second power, it's i times i, and since i is just the square root of -1, that would be -1. Third power is -1 times i, which is i. And then the fourth power is i times i, which is -1 times -1, which is one. And then the cycle repeats. And that's why imaginary numbers are so useful. They have a certain property which real numbers just straight up lack. They can describe rotation. How? Well, they all this formula. And right here, unfortunately, can't go in depth about the formula, about the identity, about all the lovely ways in which it ties in together pi, e, imaginary numbers, exponents, and so on and so forth. But the basic idea looks like this. You have a function with this definition, then you simplify it, you take the derivative, and after the derivative, you realize that it's equal to zero, which means that nothing is changing. It's not changing, it's constant. And since the function at zero is equal to one, it being constant means it's always equal to one. And if we go back to the original function, you can see that if this is always equal to one, then the top of this fraction has to be equal to the bottom, which means e to the power of i theta is equal to cosine theta plus i sine theta. Ah, just read about it for yourself, or I would need to make whole separate videos about this.

Now, why would that be useful? Well, going back to the circle and putting a point on it, going around, you can see that as we move it around, we can use the cosine function to represent the x-coordinate and the sine to represent the y-coordinate. And then we get the rotating point, which means that cosine of theta would give us a point going like this. Sine of theta times an imaginary number would give us a point going like this on a complex plane. And so if we add them both together, we get a point going around a circle based on theta. And wouldn't you know it? It just so happens that this is equal to e to the power of i theta. Okay, a lot has happened. Let's go over it step by step. Why is cosine here? Because it describes how the point is changing on the real axis. Why is sine here? Because it describes how the point is changing on the imaginary axis. Why is sine multiplied by i? Well, it's because this is sine, and when you multiply by i, it moves to the imaginary axis. Why are we adding them both together? Because when you add them together, you end up with a point going around the circle. Why is this equal to e to the power of i theta? Because of the proof I skipped over. Why is that useful? It's because it lets you switch from polar coordinates to Cartesian.

And what are polar coordinates? Okay, so we're already familiar with Cartesian, that's the X and Y. So how much to the right and how much up. The polar coordinates, however, tell you the angle at which point is and then the distance from the origin. Now, in order to represent Cartesian coordinates, you just need to use a vector. Okay, I haven't mentioned vectors, but just I just mean multiple numbers. I made a whole video about dimensions and linear algebra and all the things like this. But if you want a second video explaining linear algebra in this sort of step-by-step from addition way, then you just need to wait because I have a lot of my plan, if I'm being honest. So for Cartesian coordinates, we use vectors. For polar, however, we use R times e to the power of i theta. So once again, why would that be useful? Well, imagine that you have a distance detector, so a device which fires a laser and measures the distance based on the time it took to reflect. And you point it at the wall like this. If you know the distance between the detector and the reflection point, and if you know the angle at which it is, then you can get the exact X and Y coordinates of this point. You can get them like this.

And so it has been a journey, but we can finally answer the question of what is sine. That's because from Euler's identity, we can derive the equation for complex sine and cosine. There, if you'd like to, you can solve it step by step for any number and see that it is, in fact, sine and cosine. And that's what trigonometric functions actually are. It's actually a real shame that they're almost never taught this way. Uh, that most of the time they are not introduced as functions. But you have what you have to remember is that they are functions at the end of the day. They are just equations with numbers in them. We just simplify them to sine or cosine because, well, this is a pretty big equation that's pretty difficult to solve.

And now there's just one last thing I'd like to mention, and it's this equation. In this equation, we have two numbers: the phase and the wave number. And honestly, right here, I can just show you what they do instead of going in there because it's pretty self-explanatory. Phase moves it around, and wave number stretches it and shrinks it. Now, one thing to note here is that wave number is not the wavelength. If you want the wavelength, then you need to divide X by the wavelength. Wave number represents frequency rather than wavelength. This is a technical difference at the end of the day since there's an equation binding them together because, you know, wavelength is just an invisible frequency. Uh, but yeah, it's, I still feel like I should mention it. And so there, that's all the relevant information about trigonometry and rotations. And yes, it's been a lot. If you're still sticking to it, that's really impressive, to be honest. And feel free to rewind the chapter because that was a lot of really advanced information.

But for now, with all this information in mind, we can finally move on to the final equation. It is the Fourier transform. And so that's that. Simple? Or rather, since we're studying complex, let's simplify it at the very least. Throw away the constant 2 pi and replace the Greek letter with a W for the sake of simplicity. Okay, now this may seem a bit overwhelming, but let's tackle it all one by one using every single tool we've explored so far. So first, we start with the function. And this function is just a function of some wave. So for example, cosine of 7x plus cosine of 3x. Now, what do these two functions mean? Well, if you start with cosine, it's simple. It's just the cosine. We're below it, we just talked about this. Now, if you multiply X by seven and three, you change the wavelength, and now both these functions look a bit different. Once again, up to this point, this should be simple enough. It's just cosines. It's literally just the last chapter.

Now, let's ask a question: what happens when we integrate one of these cosines from negative infinity to positive infinity? Well, think about it for a second. What would an integral from negative infinity to positive infinity give you? Well, nothing. That's because for every bump into the positive, we're getting also a bump to the negative, and they just cancel each other out. Okay, so that's simple enough. But what happens if we add them together? Adding them together won't really change the result. That's because there isn't really much of a difference between adding them and taking the integrals, or taking the integrals and then adding them. And so if the integral of the first one is zero, and the integral of the second one is zero, both of them added together, it's, it's zero. It's still nothing. So if our Fourier transform looked like this, it would have been simple enough. It would be nothing. But we just had to throw in that mess of an equation. I mean, what is that even supposed to be? No, no, really. Once again, try answering, what is that supposed to be? What does that remind you of? Well, that does look like the Euler's formula. And yeah, it, it is. And so right here, I'm going to modify this Fourier transform a bit. So first, we'll replace it with cosine minus i sine. The minus is there because e was the negative power, not the positive. And then I'll just drop the i sine. The reason why is for the sake of representing it with real numbers, to make things easier for us.

So why are we multiplying our function by a cosine? Well, let's take a look at what happens when you multiply it by cosine. It is, I mean, it's nothing crazy. It changed a bit, but I'd hardly call that useful. I mean, what's the point of multiplying by cosine? Well, the point lies in the fact that right here we have one more variable, a variable we have not yet touched. And so let's see what happens when we move it around. Well, the function is changing, and like a vaguely wave-like way. But hold on, did you see that? Something weird happens when you say to this value. When we set it right here, the function just moves up. I mean, I suppose it makes sense. You see, when you have two cosines and you multiply them together, then on each peak where they are positive, they just end up making a positive number. Or on each peak when they are negative, they just cancel each other around to positive again. So if you multiply two cosines, you will get a purely positive function. Makes sense. Oh, so I guess one of our cosines we added together, you know, one of them has a wave number equal to seven. So I guess it makes sense our equation is now mostly positive, right? They just align.

But hold on, remember how previously whenever we integrated our function from negative infinity to infinity, we wouldn't get anything useful because the function oscillated between negative and positive side equally? Well, right now, this is clearly more positive. So if we integrate now, the further you go, the larger the number will get on average. We'll get an infinity. And that's actually kind of it for a transform. So we got a transform which takes a function like a bunch of different cosines, adds it together, and gives you back another function which tells you the information about the individual cosines. Like in our simplified case, for example, it tells you the wave number of all our cosines. And you can see right here how only three and seven are clearly in the positive, and the other ones oscillate. Which means that if you were to subtract all of the red areas from all of the blue areas, all of these would give you zero, and all of these would give you infinity. Huh? I mean, the way I introduced it, it may seem like a minor thing, kind of obvious, but it's actually huge. Take a look at this function. What cosines is it made out of? Well, you probably can't tell, and that's because this looks like an insanely complex function. But with the Fourier transform, you can just get the underlying information. Fourier transforms are unbelievably useful, especially if humans one day hypothetically decided that they would like to, I don't know, wirelessly transfer information using waves. That's because all of these waves would end up interfering together, and you need a Fourier transform to detangle them into separate frequencies. So in that hypothetical scenario in which humans would like to use wireless modes of transferring information with waves, I'm set, you could tell why it would be useful.

And there is much more to say about them. There is still a whole, like, complex definition to talk about. There's all these things are left out, all the simplifications. But that's not the point of this video. The point of this video is that this is a really scary looking equation with all these complex, terrifying symbols, weird operations, and bizarre notation, but it's not actually that scary, and it is just notation. And so that would be the lesson of this video. In the wild, you see plenty of equations like these terrifying monsters of incomprehensible characters, but at the end of the day, that's all they are: equations. And you can understand them because this is how math becomes difficult.

Okay, so here's where I would normally uh end the videos. If you see in any of my recent videos, then you know that this normally where I end them. But recently, a lot of people have been mentioning that they don't actually know that I have other things, not only videos, but also live streams. I have a Discord server, I have a Patreon, I have all them things. So, um, if you'd like to chat with me, I would recommend Discord or join in on daily live streams, which I have been streaming every single day for the past over 600 days. It, it's, it's a lot. Uh, and also, if you happened have some, you know, disposable income on hands, then I would really appreciate it if you just, if you throw me some on some money on Patreon. Um, I'm not saying I'm struggling financially, but let's just say that I have not getting all that many views on my videos. They normally take tens of hours, sometimes hundreds of hours to complete. And, um, great. So if you have some money, I would really appreciate it. But anyhow, for now, that'll be it. Thank you for one so much for watching and have a great day. Bye.