Getting from complex domain to real domain

[jaydnul]

Hi!

I am ok with understanding Euler's formula and how its proven. It is basic mathematic operations that are made possible by the characteristics of i, cos, sin, and exp.

What still makes me uncomfortable is the jump we make at the very beginning or end of calculations, basically Acosx <==> Ae^(jx). The explanations are usually the "real" part of the exponential, and Euler's formula is used to help with this.

But for my complete understanding, taking the real part of something just ins't a "normal" mathematical operation if that makes sense (it is invented for dealing with complex numbers). Is there any other explaination for the transistion we make Acosx <==> Ae^(jx) and why we can do that?

 

[fresh_42]

Hi!

I am ok with understanding Euler's formula and how its proven. It is basic mathematic operations that are made possible by the characteristics of i, cos, sin, and exp.

What still makes me uncomfortable is the jump we make at the very beginning or end of calculations, basically Acosx <==> Ae^(jx). The explanations are usually the "real" part of the exponential, and Euler's formula is used to help with this.

But for my complete understanding, taking the real part of something just ins't a "normal" mathematical operation if that makes sense (it is invented for dealing with complex numbers). Is there any other explaination for the transistion we make Acosx <==> Ae^(jx) and why we can do that?

It is linear algebra. The vectors $\vec{1}$ and $\vec{\mathrm{i}}$ are linear independent over the real numbers. That means that any real expression
$$
\alpha \vec{1} + \beta \vec{\mathrm{i}} = \alpha' \vec{1} +\beta' \vec{\mathrm{i}}
$$
implies
$$
(\alpha-\alpha')\cdot \vec{1} + (\beta-\beta')\cdot \vec{\mathrm{i}}=\vec{0}
$$
and therefore $\alpha=\alpha'$ and $\beta=\beta'$ by linear independence.

 

[fresh_42]

Another picture of looking at the complex numbers is $\mathbb{C}=\mathbb{R}[T]/\langle T^2-1 \rangle$ which is a quotient ring of the polynomials over the real numbers in one variable $T.$ A complex number is thus a polynomial $\alpha+\beta\cdot \vec{\mathrm{i}} =\alpha +\beta \cdot T$ where we identify $T^2$ with $-1.$ Since $0 \neq T \neq 1,$ we can conclude from $\alpha+\beta\cdot \vec{\mathrm{i}}=\alpha+\beta\cdot T=0$ that $\alpha = \beta=0.$

 

[FactChecker]

Hi!

I am ok with understanding Euler's formula and how its proven. It is basic mathematic operations that are made possible by the characteristics of i, cos, sin, and exp.

Good. That is the hard part.

What still makes me uncomfortable is the jump we make at the very beginning or end of calculations, basically Acosx <==> Ae^(jx). The explanations are usually the "real" part of the exponential, and Euler's formula is used to help with this.

But for my complete understanding, taking the real part of something just ins't a "normal" mathematical operation if that makes sense (it is invented for dealing with complex numbers).

It is very normal. If you have a point in two dimensional space, $(x,y) \in \mathbb{R}$ X $\mathbb{R}$ ,it is completely normal to look at its $x$ value. So looking at the real part of $Ae^{(jx)} = (A\cos(x), A\sin(x))$ is normal.
(The question of how and why it was invented is a historical question. It is now standard mathematics, which is all that matters for this discussion.)