Euler's Theorem (2 Viewers)

[pine-apple01320]

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?

 

[D-BOSS]

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

 

[Drdusk]

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.

 

[lolzdj]

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

 

[Trebla]

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]

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.

 

[HeroWise]

*Intense Power series noises*

 

[Arrowshaft]

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]

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]

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]

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]

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]

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]

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]

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.

 

[Arrowshaft]

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.

I don’t remember it exactly as I haven’t covered complex analysis yet, but he’s fairly credible, I may just be recalling it bad

 

[Trebla]

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.

The syllabus treats Euler’s formula as if it is a definition rather than an actual result. That means the only benefit of introducing it is simply notation.

But doing so does not explain why you can suddenly exponentiate a complex number and why this suddenly relates to the polar form of complex numbers.

 

[stupid_girl]

I don’t remember it exactly as I haven’t covered complex analysis yet, but he’s fairly credible, I may just be recalling it bad

I'm not saying anyone is not credible. However, any attempt to "prove" this formula will require some "hidden" assumptions that are definitely out of reach for 4U students. I have seen various versions of sloppy "proof" that omits those assumptions.

 

[Arrowshaft]

I'm not saying anyone is not credible. However, any attempt to "prove" this formula will require some "hidden" assumptions that are definitely out of reach for 4U students. I have seen various versions of sloppy "proof" that omits those assumptions.

ok boomer.
jk I agree

 

[stupid_girl]

The syllabus treats Euler’s formula as if it is a definition rather than an actual result. That means the only benefit of introducing it is simply notation.

But doing so does not explain why you can suddenly exponentiate a complex number and why this suddenly relates to the polar form of complex numbers.

You need to define e^x in the complex plane before you can exponentiate a complex number. There are different ways to do it.

One approach is to define e^x as a complex power series, which would lead to a question why it converges in the same way as the real counterpart.

Another approach is to define y=e^x as the solution to the complex differential equation dy/dx=y, ie. assume e^x can be differentiated in the same manner as a real function. This is not trivial for 4U students.

Yet another approach is to define e^x as a complex limit, which would lead to a question why the limit exists in the same way as the real counterpart.

Taking this formula as the definition is a cleaner approach for me because it defines real part and imaginary part of e^x separately, which makes the derivation of other results straight forward.