# Euler's Theorem (1 Viewer)

#### pine-apple01320

##### New Member
hey guys,
just wondering in ext 2 do you ever have to prove/derive/understand where Euler's Theorem in complex numbers come from, or is it sufficient to understand how to use the theorem?

#### blyatman

##### Well-Known Member
You don't need to know where it comes from. There are several proofs, the most direct and popular one is using Taylor series, which is beyond the scope of the 4u syllabus. There's also another proof that involves integrating an expression where they treat i as a constant, but it's not popular since you need to be careful when you mix complex numbers and integration.

Here's a relatively simple proof if you're interested:
$\bg_white \cos x+i\sin x = \left(\cos\frac{x}{n}+i\sin\frac{x}{n}\right)^n$
Take limit as n->infinity:
$\bg_white \cos x + i\sin x = \lim_{n\rightarrow\infty} \left(\cos\frac{x}{n}+i\sin\frac{x}{n}\right)^n$
cos X -> 1 and sin X -> X if X is small, so
$\bg_white \cos x + i\sin x = \lim_{n\rightarrow\infty} \left(1+\frac{ix}{n}\right)^n$
The RHS is the definition of the exponential function, hence
$\bg_white \cos x + i\sin x = e^{ix}$

#### D-BOSS

##### Member
wtf they use euler's method in 4u. DAMN

#### blyatman

##### Well-Known Member
It should've been taught that way from the start I reckon. Nobody uses cis theta lol.

#### Drdusk

##### π
Moderator
Nobody uses cis theta lol.
I always get told off by my Complex analysis lecturers/tutors for using cis when explaining questions ;-;

Which is so goddam annoying like who tf cares if I use cis. As long as it's a well sustained proof. Pisses me off so much because I'm so used to writing and using cis.

#### blyatman

##### Well-Known Member
I always get told off by my Complex analysis lecturers/tutors for using cis when explaining questions ;-;

Which is so goddam annoying like who tf cares if I use cis. As long as it's a well sustained proof. Pisses me off so much because I'm so used to writing and using cis.
Lol yeah it's because cis isn't common notation. I've never seen it anywhere outside of the HSC, and most academics would have no idea what it was if they saw that. I hate using cis theta, since the algebra is so much nicer when with the exponential. E.g. The proof for deMoivres theorem becomes trivial in exponential notation. Complex analysis also largely centers on the exponential function, which is another reason why its preferred over cis.

Nowadays when I hear the word cis, I think of cisgender haha.

#### lolzdj

##### New Member
My 4U teacher taught and made us prove it using a Taylor or Maclaurin series lol.

#### Trebla

##### Administrator
Administrator
It should've been taught that way from the start I reckon. Nobody uses cis theta lol.
The cis notation is actually not explicitly in the old syllabus. It seems to have been introduced by teachers and textbooks as their own abbreviation.

The problem I have using Euler’s formula in the HSC is that you have to introduce it without proof - otherwise you have to go beyond the syllabus. Hence, for most students it seems like this really random result that was pulled out of thin air. I reckon it should’ve been left to uni level maths where it is properly proven and used.

Unless the new syllabus intends to involve logarithms with complex numbers, I don’t see any other benefit using Euler’s formula in the HSC course other than for abbreviation purposes.

#### Arrowshaft

##### Well-Known Member
Lol yeah it's because cis isn't common notation. I've never seen it anywhere outside of the HSC, and most academics would have no idea what it was if they saw that. I hate using cis theta, since the algebra is so much nicer when with the exponential. E.g. The proof for deMoivres theorem becomes trivial in exponential notation. Complex analysis also largely centers on the exponential function, which is another reason why its preferred over cis.

Nowadays when I hear the word cis, I think of cisgender haha.
No one should be required to prove De Moivre’s theorem algebraically, its just a mathematical description for the ‘stretching’ of the modulus and the rotation of complex numbers. My 4u teacher hated De Moivre’s theorem as it stripped the intuition brought by visualizing rotations.

#### blyatman

##### Well-Known Member
No one should be required to prove De Moivre’s theorem algebraically, its just a mathematical description for the ‘stretching’ of the modulus and the rotation of complex numbers. My 4u teacher hated De Moivre’s theorem as it stripped the intuition brought by visualizing rotations.
Yeh but how do you justify that z^2 rotates the vector by doubling the argument? There doesn't seem to be a geometrical basis for that. Isn't the mathematical description (that's brought about using deMoivres theorem) required to show the geometrical result?

#### HeroWise

##### Active Member
*Intense Power series noises*

#### blyatman

##### Well-Known Member
The problem I have using Euler’s formula in the HSC is that you have to introduce it without proof - otherwise you have to go beyond the syllabus. Hence, for most students it seems like this really random result that was pulled out of thin air. I reckon it should’ve been left to uni level maths where it is properly proven and used.
Yeh I try to provide a proof of it (like the one above) to my students, so they can see the connection and where it comes from.

#### Arrowshaft

##### Well-Known Member
Yeh but how do you justify that z^2 rotates the vector by doubling the argument? There doesn't seem to be a geometrical basis for that. Isn't the mathematical description (that's brought about using deMoivres theorem) required to show the geometrical result?
My teacher actually explained this by redefining the concept of addition and multiplication in the complex plane through a process known as ‘sliding’ for addition and ‘stretching’ for multiplication. By defining it as such, you can think of z^2 as purely addition of the arguments

#### Arrowshaft

##### Well-Known Member
I’ve also seen it derived by my physics teacher (can’t remember exactly how) but by using a differential operator on e^ix, i dont remember the exact details but this was when he was teaching us about the fourier series for intro to quantum mech

#### Drdusk

##### π
Moderator
Lol yeah it's because cis isn't common notation. I've never seen it anywhere outside of the HSC, and most academics would have no idea what it was if they saw that. I hate using cis theta, since the algebra is so much nicer when with the exponential. E.g. The proof for deMoivres theorem becomes trivial in exponential notation. Complex analysis also largely centers on the exponential function, which is another reason why its preferred over cis.

Nowadays when I hear the word cis, I think of cisgender haha.
NESA would like to: Know your location

#### Drdusk

##### π
Moderator
My teacher actually explained this by redefining the concept of addition and multiplication in the complex plane through a process known as ‘sliding’ for addition and ‘stretching’ for multiplication. By defining it as such, you can think of z^2 as purely addition of the arguments
I think that teacher is 3B1B amirite

#### Arrowshaft

##### Well-Known Member
I think that teacher is 3B1B amirite
Dude I was so lucky for 4u as I had a teacher who loved 3b1b and literally made identical presentations as him, albeit he didn’t use python (tho you can access his code from his resources!).

#### stupid_girl

##### Active Member
The cis notation is actually not explicitly in the old syllabus. It seems to have been introduced by teachers and textbooks as their own abbreviation.

The problem I have using Euler’s formula in the HSC is that you have to introduce it without proof - otherwise you have to go beyond the syllabus. Hence, for most students it seems like this really random result that was pulled out of thin air. I reckon it should’ve been left to uni level maths where it is properly proven and used.

Unless the new syllabus intends to involve logarithms with complex numbers, I don’t see any other benefit using Euler’s formula in the HSC course other than for abbreviation purposes.
I would rather call it a definition.

I don't think there exists a rigirous "proof" unless you make various assumptions that complex analysis behaves similarly to real analysis.

#### stupid_girl

##### Active Member
My 4U teacher taught and made us prove it using a Taylor or Maclaurin series lol.
Well, I guess your teacher assumes that there exists a Taylor series for e^(ix) in the complex plane first...which is not trivial.

#### stupid_girl

##### Active Member
I’ve also seen it derived by my physics teacher (can’t remember exactly how) but by using a differential operator on e^ix, i dont remember the exact details but this was when he was teaching us about the fourier series for intro to quantum mech
This approach would have to establish that the differential operator applies to a complex function e^(ix) in the same way as a real function e^(kx)...which is not trivial.