I agree that this is the most beautiful equation in math. I saw it before, but it was done by Maclaurin series, which actually is a special case of Taylor series. Good explanation as usual. 👍
been trying 2 think of an intuitive way to understand this without taylor series. I think it's because, if you have a point whose x and y coordinate's second time derivatives are minus themselves, it will create a circle (at least with some set of initial conditions). Just for geometrical reasons(?). So that gives you a pi. And differentiating something to get itself times a constant introduces e. Differentiating *twice* gives you that constant squared, and if that constant squared is minus 1, then the constant is i. So that's the link between e, i and pi
Imaginary numbers frustrate me because they're like the only thing that's not well-represented in physical phenomena (other than relativity, but even then that's i^2 and not i). The best I can think of it is, normal exponentiation causes a number to build upon itself, but an imaginary argument causes the numbers to get knocked 90% to the side, so things spiral rather than build. Maybe there is no way for imaginary numbers to be intuitive, because they really do stand in defiance of the world as we know it.
@@kingbeauregard i personally find imaginary numbers purely understandable. In life we are taught that numbers exist on a number line, that you can count from 0 to 51 for example, and square any number. However, the real numbers are only a small subset of actual numbers. Complex numbers are nothing more than just simply 2d numbers. And i think that most education systems make a mistake teaching it so late. After you learn to count and gain an intuition that the number line are the numbers, imaginary numbers merely are just “hypothetical additions” and are often seen as well, imaginary. Even though in quantum mechanics, they are very real. And yes, even though every result you’ll ever get will only be a real number, its only due to how we use the number system we as humans created. Maybe some alien species out there uses a 2d number system like us? (But more officially).
I completely agree with you on the idea that the educational system teaches complex numbers too late. By the time complex numbers come into play, most students are already decided on whether they want to learn it or not. I think complex numbers should be introduced as soon as the real number line is taught. This way, students know that there are other numbers outside the 'real world'.
True, the Taylor series expansion was what led Euler to discover e^(ix) = cos(x) + i·sin(x). The way that you worked it out was exactly the same way that Euler himself worked it out, but did you know that you can prove e^(ix) = cos(x) + i·sin(x) without appealing to the Taylor series? Any two functions that satisfy the same differential equation are equivalent to each other. With that in mind, let f(x) = e^(ix) and let g(x) = cos(x) + i·sin(x). Now, df/dx = i·e^(ix) = i·f(x), and dg/dx = - sin(x) + i·cos(x) = i·[cos(x) + i·sin(x)] = i·g(x). Since df/dx = i·f(x) and dg/dx = i·g(x), it follows that f(x) = g(x) ◼
TREMENDOUS VIDEO NOTE: 💥TIME STAMP: 3:20 - 3:43:💥 eˣ = 1 + x + ½x² + ⅙x³ + ... + xⁿ/n! + ... Greetings From Curaçao, An Island Nation in The Caribbean.
Prime Newtons is in a class by himself! Excellent job explaining this. ❤😊🎉
Wow, thank you!
Keep posting we will support 100 %
I agree that this is the most beautiful equation in math. I saw it before, but it was done by Maclaurin series, which actually is a special case of Taylor series.
Good explanation as usual. 👍
been trying 2 think of an intuitive way to understand this without taylor series. I think it's because, if you have a point whose x and y coordinate's second time derivatives are minus themselves, it will create a circle (at least with some set of initial conditions). Just for geometrical reasons(?). So that gives you a pi. And differentiating something to get itself times a constant introduces e. Differentiating *twice* gives you that constant squared, and if that constant squared is minus 1, then the constant is i. So that's the link between e, i and pi
Imaginary numbers frustrate me because they're like the only thing that's not well-represented in physical phenomena (other than relativity, but even then that's i^2 and not i). The best I can think of it is, normal exponentiation causes a number to build upon itself, but an imaginary argument causes the numbers to get knocked 90% to the side, so things spiral rather than build.
Maybe there is no way for imaginary numbers to be intuitive, because they really do stand in defiance of the world as we know it.
@@kingbeauregard i personally find imaginary numbers purely understandable. In life we are taught that numbers exist on a number line, that you can count from 0 to 51 for example, and square any number. However, the real numbers are only a small subset of actual numbers. Complex numbers are nothing more than just simply 2d numbers. And i think that most education systems make a mistake teaching it so late. After you learn to count and gain an intuition that the number line are the numbers, imaginary numbers merely are just “hypothetical additions” and are often seen as well, imaginary. Even though in quantum mechanics, they are very real. And yes, even though every result you’ll ever get will only be a real number, its only due to how we use the number system we as humans created. Maybe some alien species out there uses a 2d number system like us? (But more officially).
I completely agree with you on the idea that the educational system teaches complex numbers too late. By the time complex numbers come into play, most students are already decided on whether they want to learn it or not. I think complex numbers should be introduced as soon as the real number line is taught. This way, students know that there are other numbers outside the 'real world'.
True, the Taylor series expansion was what led Euler to discover e^(ix) = cos(x) + i·sin(x). The way that you worked it out was exactly the same way that Euler himself worked it out, but did you know that you can prove e^(ix) = cos(x) + i·sin(x) without appealing to the Taylor series?
Any two functions that satisfy the same differential equation are equivalent to each other.
With that in mind, let f(x) = e^(ix) and let g(x) = cos(x) + i·sin(x).
Now, df/dx = i·e^(ix) = i·f(x), and dg/dx = - sin(x) + i·cos(x) = i·[cos(x) + i·sin(x)] = i·g(x). Since df/dx = i·f(x) and dg/dx = i·g(x), it follows that f(x) = g(x) ◼
You have five important constants , addition , multiplication and exponentiation
TREMENDOUS VIDEO
NOTE:
💥TIME STAMP: 3:20 - 3:43:💥 eˣ = 1 + x + ½x² + ⅙x³ + ... + xⁿ/n! + ...
Greetings From Curaçao, An Island Nation in The Caribbean.
Yeah!
Such a beautiful identity
very true cool adam
What a nice vedio...
Understanding complex number makes it easy 😋
Beautiful!
Magical.
hey man im 16 years old and i realy like your videos
Welcome to this world of Math
،،،،،،very very very well and thank you for video,,,,,,,,sr
Awesome,sir 🎉❤😮
Uderrated af