B Getting from complex domain to real domain

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The discussion focuses on the transition between the expressions Acosx and Ae^(jx), highlighting the discomfort with taking the real part of complex numbers as a mathematical operation. While Euler's formula is understood, the leap to using the real part raises questions about its normalcy in mathematical practice. The explanation provided emphasizes that taking the real part is standard in linear algebra, as it relates to the independence of vectors in complex space. The conversation also touches on the historical context of these operations, asserting that they are now accepted as standard mathematics. Understanding this transition is essential for grasping the relationship between complex and real numbers.
jaydnul
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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?
 
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jaydnul said:
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.
 
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.##
 
jaydnul said:
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.
jaydnul said:
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.)
 
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