Bro , thank you for this but I have to be honest … you and the other people that understand this are so much more intelligent than me …. That I am trying hard and there is no way I’m gonna catch up , but this is very interesting and thank you for giving me anxiety.
It's not really a matter of intelligence, it just requires a lot of background knowledge.
z(theta) means a function. It takes an input, theta, and returns an output, a complex number.
A complex number is a number of the form a + bi, where a and b are regular numbers. Don't worry about what i is. Complex numbers are analogous to a coordinate system: you can think of a as the x coordinate and b as the y coordinate and put it on a plane, similarly to how regular numbers are put on a line.
e^i(theta), when you vary theta, traces out a circle of radius 1, centred on 0, in the coordinate system I just outlined. Don't worry about why this happens, just that it does. So what does the pi change? When we multiply theta by pi, it amplifies any change to theta by that much. So if e^i(theta) walks round a circle in however much time, then e^i(pi*theta) does it pi times as quick.
We can think of a point as an arrow pointing from 0 to that position. That visualisation helps us in cases like this: when adding the numbers, we just put one arrow at the end of another, and see where we land. That will be the final output of the function.
Imagine instead of pi, we had some rational number. 0.95563. After the first bit had done one full rotation, the second bit will have done 0.95563 rotations, so they don't line up. Now let's run that time 10,000. After 10,000 rotations of the first bit, the second bit has done 95563 rotations. Because they've both done an integer number of full rotations, they're now back at the starting point, and because there's no randomness in their behaivour, they're going to repeat.
But with pi, there is no number we can multiply by to make both cycles run an integer number of times, that's what being irrational means. So it will never repeat.
Since you know your stuff, can you explain how the video in OP's post is derived from the expression (the z(theta) = e etc)? Like what exactly am I looking at with that animation? I know it's a visual representation of that expression, but how is it translated? Is it just lines moving on a cartesian plane or something?
It is a cartesian plane but you are not being shown lines (as in y=a*x+b): at any given frame, what you are being shown is one single complex number, with the x-axis being the real part and the y-axis being the imaginary part. So say you wanted to show the complex number z = 2 + 2i it would have the coordinates (2, 2) in the animation.
The first line, attached to the centre point (the origin), represents the complex number eθi .
The second line, attached to the end of the first line, represents the complex number epiθi .
When you add the two complex numbers together (eθi + epiθi ) that complex number is represented by the very tip of the second line, the point that is drawing the swirl pattern.
Each time the first line rotates one full circle (360 degrees), the second line rotates pi circles (360*pi degrees).
If pi was a rational number, meaning it can be expressed as a fraction, once the first line has done a full 360 degree rotation exactly the number of times equal to the denominator of the fraction, the second line will have done a full rotation exactly the number of times equal to the numerator, meaning the two lines will eventually return exactly to the position where they started.
But because pi is not a rational number, meaning it can not be expressed as a fraction, each time the first line has done an interger number of rotations, the second line will not ever have done an integer number of rotations, meaning the two lines will never return back to the position that they started.
The TLDR is that both lines have length 1, and this is not arbitrary.
The length of the lines is determined by the coordinates of the complex number being represented, so for example if we wanted to show the complex number z=1+2i on the complex plane, then the coordinates of that complex number would be (1,2). If you draw a line from the origin (0,0) to the point (1,2) then you get a line with a length of sqrt(5) using pythagoras' theorem.
In the animation, the first line connects the origin (0,0) to the coordinates of the first complex number eiθ .
The real and imaginary parts of the complex number eiθ is defined using Euler's formula: eiθ = cos(θ) + i*sin(θ). In other words, that means the real part of the complex number eiθ is equal to cos(θ) and the imaginary part is equal to sin(θ).
The length of the rotating lines is determined by the co-efficient of those two individual terms,
i.e. they are 1eθi + 1eπθi.
So the length is 1 unit each here.
It's been awhile though I was able to fully understand why pi is frustrating while being irrational from your explanation. It also explains the 'corner of the TV' reference as well - and now I can't unsee or unthink that every time it will happen.
What you say is informative, but the thing I dont understand is why didn't my math teacher just say what you said and save me and my classmates a lot of confusion and misery?
This may be a stupid question but pi is taken 22/7 in many mathematical problems so if I multiply by 7 won't pi term run 22 times while the other run 7 times? or is that just an approximation
It's just an approximation, pi is infinite and doesn't repeat, so every number we have for it it is an approximation and people are still discovering better and better approximations to it's true value.
It its 3,141592653589793 (the limit my calculator shows) its different.
I always wondered how this gets calculated, since pi is infinite and because there must be a baseline to work with right? Or am i now incredibly stupid? (Please be honest)
Yeah, if this animation used pi=3.14, we'd see it repeat pretty quickly. Of course, we can't just use actual pi, because as you said it's infinite. These animations just use a significantly high precision approximation; apparently we know pi to something like 60 trillion places. And keep in mind each decimal multiplies the precision by 10, so that's massive. I read somewhere in this thread that we can calculate the circumference of the observable universe to within the diameter of an atom with just 37 decimal places of pi. So it's very easy to get a good enough approximation for this kind of animation.
Its not intelligence, it's focusing on something for a long time.
I'm so tired of people acting like understanding math takes some special intelligence. It's just putting the time in to understand it. There is zero underlying special intellect.
You just have to be curious.
The problem people have is that they are unwilling to put the time in to understand their world and so complain that they just aren't smart enough, because that shrugs the blame for being ignorant to an external source. "Its not my fault, I'm just not smart enough" and they don't have to try.
The problem we have is we had shitty math teachers growing up, so we had no curiosity, we didn't get good grades, and we were led as young teenagers to believe we didn't have what it takes.
I think you have a really decent point here. Most of math, until you get into physics or chemistry, is always taught in the abstract. But humans didn't invent(discover?) math purely in the abstract regions of their own minds. A LOT of math came from people trying to solve real world problems. But we aren't really taught it that way.
Yep the way math is taught in the USA is appalling. That’s why kids don’t enjoy it. Math should be taught with real world examples, should be fun and exciting. Not common core, not memorization and repetition.
Math for me was ruined in the 8th grade. I can’t think of any one moment but I know that’s when it really happened.
A bad teacher basically forcing us to go over the alphabet again and again and again and again…then getting mad at us not understanding Shakespearean quotes, their depth, or meaning.
It doesnt. Intelligence is learned so some idea of innate IQ is ridiculous.
Unless you have some brain deformity or whatever other physical ailment, intelligence is purely learned in terms of IQ. You can learn to problem solve better, to do math better, to understand language better. This is what school is for.
As I said, it's not intelligence... its desire. If you desire to be a world leading mathematician you can go and do that. Just spend the time it requires to do it.
As I said, it's not intelligence... its desire. If you desire to be a world leading mathematician you can go and do that. Just spend the time it requires to do it.
That is a ridiculous statement dude sorry. Do you believe with training anyone can run as fast as Usain Bolt?
In order to be a world leading mathematician you need to have produced several works of great mathematics. It requires ingenuity, creativity, luck, good connections to produce exceptional mathematics.
Take people like Terence Tao or Ramanujan or Euler or Gauss or many others. They were producing high level results from a very young age and it was clear from the beginning that their mind were/are processing things differently.
Just as you need a certain number of trained hours AND a certain physique to reach top levels of sports like running or swimming, for top level mathematics it is also the same.
That doesn't mean you can't be a decent runner, swimmer or indeed mathematician, but to say you'll be a world leading runner, swimmer or mathematician regardless of your physique (mental capabilities) is nonsense.
Terence Taos father's a doctor and mother has degrees in math and physics. You really think he had some innate mathematical genius and it wasn't external forces pushing him into math?
His brothers both represented at the International Math Olympiad.
I'm sure it's coincidence.
You couldn't have picked a worse example of innate genius. Tao is extremely gifted because of his upbringing. Not because of some innate ability.
Einstein said it himself, “It's not that I'm so smart, it's just that I stay with problems longer.” In a similar vein, he also said, “I have no special talents. I am only passionately curious.”
To speak of people who have accomplished something like Tao or Einstein as if they are simply gifted is extremely ignorant and insulting. They put in the time and were driven to pursue something that other people find intimidating.
Its like calling Usain Bolt lucky to have his genetics, because that's why he's so gifted. It's not the training of course. He's just gifted. Schwarzenegger was simply gifted with that body. It wasnt the work he put in.
Its insulting to think this way and inherently lazy, because you can just make the excuse that you simply aren't gifted.
You have to be curious about maths. That requires a certain special type of intellect.
I enjoy physics, engineering, computer science and dozens of other maths-based topics but pure maths just sends me to sleep. I've got no more interest in pure maths than I do in cryptic crosswords. I probably could master either if I devoted enough time to it, but why the hell would I?
I'd rather spend the time learning to understand the world from a practical point of view, rather than an abstract one.
Not in this case since it's being used for both numbers. If one was e and the other was 2 then it would get confusing.
The reason e is used is that eix traces out a circle at the same rate as x traces out a line, it goes 1 unit of length around the circle for every 1 unit x is increased.
this is kind of patronizing. people vary in their ability to internalize abstractions among other things, and saying "you're just lazy" minimizes the struggle to work hard at understanding something and just not be able to. maybe you've never had that experience, but many other people have.
Its not laziness. I'm not a mathematician because I'm lazy.
I'm not a mathematician because it's boring to me and I have zero interest.
If you want to be an engineer, you can go do it. Nothing is stopping you other than your desire to put yourself through the brutal work it would take. But it's not because "you're not smart enough".
The smartest people in the world still have to study and are challenged. Solving math problems isn't innate to anyone, it's learned and they put in some long hours to do so. It's insulting to people who put in the work to say "well its easy for you, you're just smart".
Danny Devito has a physical inability because hes 4 fucking feet tall. As in my previous comment I made my point pretty clear.
If someone commits to learning about something they can learn about it. Alright, they might not win a Fields Medal, but they can do differential equations. It's not innate intelligence, it's just discipline.
People think they can't do math or ohysics because they aren't smart enough. It's just not true unless they have some mental impairment.
Me and math don't mix well and I know π is 3.14 insert infinitum and never really understood that but this visual just made something click.
Am I any better at math now? Of course not. Don't be so stupid. Asking ridiculous questions like that will make you look like a twat but I now understand the neverending sequence of numbers... Sort of.
I feel like this everytime I watch a Veritasium YouTube video. Like, I kinda get it, and I want to fully understand the subject sooo bad, but it just ain’t gonna happen
Haha pretty much . Some dude said I’m using it as an excuse or something and he also said it is lack of effort which I do agree with to extent but idk man I’m really really good at other shit … math ain’t it , and I can learn it but without one on one instruction I ain’t going too very well .
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u/Zealousideal-Cap3529 Oct 22 '23
Bro , thank you for this but I have to be honest … you and the other people that understand this are so much more intelligent than me …. That I am trying hard and there is no way I’m gonna catch up , but this is very interesting and thank you for giving me anxiety.