Euler’s identity proof for calculus 2 students!

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0:00 Proving Euler's Formula e^(iθ)=cos(θ)+i*sin(θ)
4:58 Check out Brilliant
5:52 Proving Euler's Identity e^(i*pi)+1=0
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[WestExplainsBest]

As a teacher, the ability to speed up like that would be amazing.

[RobinHillyard]

You forgot to mention that Euler’s identity also employs the four principal mathematical operators: addition, multiplication, exponentiation, and equality. ;)

[slavisastojakovic]

Whenever he says 'Ladies and gentlemen' I know magic is about to happen.

[leziehr]

i have been looking for this equation’s derivation for ages, thank you for this! this really is the most beautiful equation ever.

[Generalth]

If you're ever curious about where those series come from (or you don't feel like remembering them), just derive them yourself! Suppose cos(x) = a + bx + cx^2 + dx^4 + ... . Set x = 0, solve for a, then differentiate both sides with respect to x and solve for b. Differentiate yet again and keep going! Within four or five iterations, you should see a pattern emerge.

You can do the same thing for any function, like sin(x) or e^x or ln(1+x). It's much easier than having to remember them.

[okenyodong]

Amazing proof, thank you so much. I was struggling to wrap my head around Eulers identity for months.

[JSSTyger]

I agree that this is the most beautiful equation in math. I learned it in my fifth and final year at the university, taking a course i didnt even need. Needless to say, I am quite grateful I took that course.

[blackpenredpen]

See how we can integration e^(ax)*sin(bx) with Euler's formula and no integration by parts:

[valerium]

The second you wrote down "1-(1/2!)t^2" i immediately knew what was up, and it's beautiful.

[idjles]

Not only does it have the 5 most important numbers, it also has the 4 most important operations: equality, addition, multiplication and exponentiation.

[meiljohnbueno]

I really admire those people who really understand maths and its complexities. Wisdom is really a gift ❤️❤️

[jordimayorgisbert]

Amazing !!! I don’t know that development of e^z using Taylor Series.

Thank you !!!!

[stevenlin]

I’m a 10th grade student in Taiwan, and I chose pre-calculus as my elective course (it’s really PRE-calculus, we won’t even mention Epsilon-Delta definition of a limit this semester!)
Though I can’t fully understand most of your calculus videos, I still find them interesting! And I wish to have a teacher like you ：）

[bhavydugar]

What I like best is that he is one of the few Americans to correctly pronounce Euler’s name is

[sr.tarsaimsingh]

Everybody knows they are watching YouTube,
But I know I am watching a Legend. 😁

[Mark-nmsm]

I am watching your 100 infinite series little by little and the difference in hair style is so nuts compared to now. Also the mic is now a pokeball amazing

[GaryFerrao]

OMG the sudden close up of your face for the sponsor message was so creepy lol 😅😂

[dimastus]

Well, it is good and valid proof/derivation, but I think, that using infinite series in that case is like shooting at fly with a bazooka. Why not to use limit definition of e^x and geometric meaning of complex number operations?

[dr.rahulgupta]

Excellent presentation of the topics in a beautiful manner by an experienced teacher . Vow !

[idrisShiningTimes]

This equation id really beautiful, like no words to describe this masterpiece