I'm a computer scientist watching your videos who knows all of these concepts you are teaching, but I am still interested simply because you are an excellent teacher who seems to be passionate about the subject. Wish I had teachers like you.
Hello frozenbacon! please can you give me an advice about studying sofware ingeeniering or computer science ingeeniering which of them should i choose for my studies on computing?
Keep up the good virtuous hard work! You are an amazing teacher. You remind me, sadly, of just a couple I've had. Good teachers are few & far between. I'm so glad I stumbled upon your TH-cam channel. thank you.
I love everything about complex numbers, like entire functions, contour integrals and independence of path. It's as amazing and headache inducing as abstract algebra.
The use of the residue theorem in contour integration is the closest thing to wizard magic that I know in mathematics. Start with a seemingly intractable improper integral, move into the complex plane and just melt it instantly with a few steps of algebra.
For anybody who cares, you actually find a lot of complex (imaginary) numbers in quantum mechanics. They do exist in nature, just maybe not as obvious as others.
To "see" complex numbers in nature requires an appreciation of numbers as representing both a static element and a dynamic function simultaneously. Unfortunately we are somewhat mislead by favoring only the static aspect of number, whereby we mostly associate it with static images of collections of discrete elements. BUT, a number also has a dynamic functional aspect which gives us information on how they "interact" with other numbers. For example, if 2 is combined under multiplication with another number it doubles it, 3 triples, and in general every unique number has its own stretch or scaling factor built into its design when interacting with other numbers. Complex numbers not only stretch under multiplication, but also rotate. Both stretching and rotating are ubiquitous as physical processes in nature, and hence complex numbers can be "seen" as very naturally encoding the interactions of physical systems.
@@hehexd9892 Here's a specific example I encountered recently: When you rotate something in space around the origin -- say, in a 3D space with a Cartesian coordinate system of x, y, and z -- the quantity x^2 + y^2 + z^2 stays constant (as if you're moving around a sphere centered at the origin). However, in special relativity you deal with a lot of rotations in 3 plus 1-dimensional spacetime; that is, with three dimensions of space and one dimension of time, t. As it turns out, Einstein and others found that instead of x^2 + y^2 + z^2 + t^2 being conserved during rotations, x^2 + y^2 + z^2 - t^2 is conserved.* This minus sign means the math can get pretty nasty (involving hyperbolic geometry, if you feel like investigating further), so instead you can make the substitution t = iτ (where i = sqrt(-1)). Doing this means that now x^2 + y^2 + z^2 + τ^2 is conserved, and you can use Euclidean geometry (geometry on a flat plane, with sines and cosines and other things we like) like we normally do, except in four dimensions. *Actually, the last term should be c^2 * t^2 where c = the speed of light in a vacuum, but physicists usually just choose units such that c = 1, and I didn't want to unnecessarily muddle things up here when I just want to demonstrate the idea. I also suggest you check out the channel 3blue1brown -- specifically, his videos on e^(i*pi) and the Fourier transform. Things like that can really give a lot of insight into how powerful complex numbers can be.
Great explanation! I've always wondered what's the difference between imaginary and complex numbers. And also why it was chosen to be sqrt(-1) and not some other number
When I was a small child, my father (who had a math degree) had introduced me to negative numbers, and one day I came home from school upset, saying the teacher had LIED in math class: she said you couldn't subtract a larger number from a smaller one. I wasn't quite accustomed to (and of course she didn't mention) the fact that these sorts of statements are always relative to some domain of numbers: first- and second-grade arithmetic was operating in the natural-number domain, and I had already been given a peek beyond.
Every adult knows that negative numbers exist, and to tell a child that they don't exist is simply a lie. You may have been at school and they may have not been ready to teach you about them yet, but it does not mean that they don't exist. How ridiculous. I think that I would wanna be having some very serious words with her about that.
I thought the most beautiful identity was Euler's identity since it shows a profound connection between the most fundamental numbers in mathematics. Euler's identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants 0, 1, i, e and π. It is the most beautiful mathematical identity known to man. e^(iπ) + 1 = 0
wait, where did you did a math degree withough complex numbers in it?? In my university you start with them in the first year and then use them everywhere
you can't teach passion and personality and aptitude... those traits are innate... but you can teach storytelling and lesson planning and visual presentation skills... so that teachers can at least be 1/10th as talented as this eloquent scholar !
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
I'm a computer scientist watching your videos who knows all of these concepts you are teaching, but I am still interested simply because you are an excellent teacher who seems to be passionate about the subject. Wish I had teachers like you.
Exactly me, but I strive to be a computer scientist
Hello frozenbacon! please can you give me an advice about studying sofware ingeeniering or computer science ingeeniering which of them should i choose for my studies on computing?
Well said. My maths teachers were not as inspiring as this gentleman.
@@abdoulayebah215 you should study computer science. it is a great subject. i'm also a computer scientist BTW
@@deltavalley4020 hows the journey going? you still committed to becoming one? are you one now?
You are a gifted teacher. I really enjoy watching your lessons. You are the math teacher that everyone should have had.
Keep up the good virtuous hard work! You are an amazing teacher. You remind me, sadly, of just a couple I've had. Good teachers are few & far between. I'm so glad I stumbled upon your TH-cam channel. thank you.
I love everything about complex numbers, like entire functions, contour integrals and independence of path. It's as amazing and headache inducing as abstract algebra.
The use of the residue theorem in contour integration is the closest thing to wizard magic that I know in mathematics. Start with a seemingly intractable improper integral, move into the complex plane and just melt it instantly with a few steps of algebra.
For anybody who cares, you actually find a lot of complex (imaginary) numbers in quantum mechanics. They do exist in nature, just maybe not as obvious as others.
Florian Koenig can u give an example of imaginary numbers in nature? Sadly we dont do that stuff at school so im not really that familiar with it
Yes, I'm interested too, for the same reason :)
To "see" complex numbers in nature requires an appreciation of numbers as representing both a static element and a dynamic function simultaneously. Unfortunately we are somewhat mislead by favoring only the static aspect of number, whereby we mostly associate it with static images of collections of discrete elements. BUT, a number also has a dynamic functional aspect which gives us information on how they "interact" with other numbers. For example, if 2 is combined under multiplication with another number it doubles it, 3 triples, and in general every unique number has its own stretch or scaling factor built into its design when interacting with other numbers. Complex numbers not only stretch under multiplication, but also rotate. Both stretching and rotating are ubiquitous as physical processes in nature, and hence complex numbers can be "seen" as very naturally encoding the interactions of physical systems.
@@hehexd9892 Here's a specific example I encountered recently: When you rotate something in space around the origin -- say, in a 3D space with a Cartesian coordinate system of x, y, and z -- the quantity x^2 + y^2 + z^2 stays constant (as if you're moving around a sphere centered at the origin). However, in special relativity you deal with a lot of rotations in 3 plus 1-dimensional spacetime; that is, with three dimensions of space and one dimension of time, t. As it turns out, Einstein and others found that instead of x^2 + y^2 + z^2 + t^2 being conserved during rotations, x^2 + y^2 + z^2 - t^2 is conserved.* This minus sign means the math can get pretty nasty (involving hyperbolic geometry, if you feel like investigating further), so instead you can make the substitution t = iτ (where i = sqrt(-1)). Doing this means that now x^2 + y^2 + z^2 + τ^2 is conserved, and you can use Euclidean geometry (geometry on a flat plane, with sines and cosines and other things we like) like we normally do, except in four dimensions.
*Actually, the last term should be c^2 * t^2 where c = the speed of light in a vacuum, but physicists usually just choose units such that c = 1, and I didn't want to unnecessarily muddle things up here when I just want to demonstrate the idea.
I also suggest you check out the channel 3blue1brown -- specifically, his videos on e^(i*pi) and the Fourier transform. Things like that can really give a lot of insight into how powerful complex numbers can be.
You're the best math teacher I've ever seen
Grant Sanderson
Leged a legend!! ~ ✨Such a great teacher 🌟💫✨
Man.....,
I wish we had such teachers at our school,I would love going to school
Great explanation! I've always wondered what's the difference between imaginary and complex numbers. And also why it was chosen to be sqrt(-1) and not some other number
He explains both in the video.
Difference between imaginary and complex: 10:57 to 11:05
Why it's chosen to be sqrt(-1): 9:33 to 10:07
Complex numbers are also used in Jolias/Mandelbrot formula, used to create colorful patterns in computer graphics.
When I was a small child, my father (who had a math degree) had introduced me to negative numbers, and one day I came home from school upset, saying the teacher had LIED in math class: she said you couldn't subtract a larger number from a smaller one. I wasn't quite accustomed to (and of course she didn't mention) the fact that these sorts of statements are always relative to some domain of numbers: first- and second-grade arithmetic was operating in the natural-number domain, and I had already been given a peek beyond.
Every adult knows that negative numbers exist, and to tell a child that they don't exist is simply a lie. You may have been at school and they may have not been ready to teach you about them yet, but it does not mean that they don't exist. How ridiculous. I think that I would wanna be having some very serious words with her about that.
00:00
I thought the most beautiful identity was Euler's identity since it shows a profound connection between the most fundamental numbers in mathematics. Euler's identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants 0, 1, i, e and π. It is the most beautiful mathematical identity known to man.
e^(iπ) + 1 = 0
This video series is basically the history of maths
wait, where did you did a math degree withough complex numbers in it?? In my university you start with them in the first year and then use them everywhere
Eddie please teach differential equations
trey poirrier hes teaching math at a high school level lol...
Differential equations is high school math. In denmark we have it if we pick math A level at our high schools.
Eddie, how may I contact you in person?
Do you have lectures about Laplace transform and Fourier series ??
3b1b
great video!
what is the practical use of cauchy integral theorem and analytic function.. pls explain
I also at my childhood use to think 100 as the biggest number😂😂
Ananjay Gurjar my daughter is set on 188 for some reason
Has any child ever been shown why there is no smallest number?
I am worried because I pay more attention to this than the students themselves
Postulate 1: Square root of 5 is irrational.
Postulate 2: Square root of -1 is imaginary.
Conclusion: Surds are absurd!
F
F
DidJewNaziMe You're sick minded to have a fake name like that.
-1=e^i(pi)
Sir this is content of which class
Can u prove (v+u)(v-u) is smaller than v2
Please mention what are U and V
@@pahularora9642 great!! Look if u=0 then both r equal and other cases it is smaller as you are subtracting u2 term from v2 in LHS< V2
he shouldn’t teach maths...
....he should teach teachers how to teach maths....
Just pay him for teaching I reckon
you can't teach passion and personality and aptitude... those traits are innate... but you can teach storytelling and lesson planning and visual presentation skills... so that teachers can at least be 1/10th as talented as this eloquent scholar !
Why this video not getting viewers
I think the your browser is broken. There is views on there.
The five dislikes was maybe by mistakes
His accent confuses me. Where the hell is he from?
China. Or least he has a family from China.
That noise in the background is annoying. Just be quiet, let him speak.
When i was a kid i would say that 3-5=2 im smart
He should write a song: "i can only imagine..."
Mercy me!
even better than albert einstien
Westeruras.Westaara.Westeraaa.Westeyesh.Westaara.Westresy.Westaara.Westaktf.Westeraaa.Westomer.Westaara.Westomeisy.