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I know that it is similar to a vector. A complex number is "just a number" except that isn't necessarily a real number. I see the similarity with a vector, but my question still holds. I argue that it just can't be the same as a vector no matter how similar. I think the "proof" I posted in the image above is something similar to what I'm looking for. Was my point clear in the image I posted? (I'm sorry if I'm being annoying).

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- #28

olivermsun

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Your diagram is perfectly clear. The radial and tangential parts really do come straight out of the chain rule in polar coordinates.

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- #30

olivermsun

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Well, it isn't so bad considering the usefulness of the polar form. Also, once you've derived it, you have the formula in-hand and don't need to do it again.

If your interest is in elliptical planetary orbits, I also recommend looking through the wikipedia entry on Kepler's[/PLAIN] [Broken] laws.

If it's a more general motion you're looking at, then I'd probably start with vector paths in cartesian coordinates, [x(t), y(t), z(t)], which have very easy derivatives. Or perhaps the TNB frame I linked to earlier.

If your interest is in elliptical planetary orbits, I also recommend looking through the wikipedia entry on Kepler's[/PLAIN] [Broken] laws.

If it's a more general motion you're looking at, then I'd probably start with vector paths in cartesian coordinates, [x(t), y(t), z(t)], which have very easy derivatives. Or perhaps the TNB frame I linked to earlier.

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Your argument is invalid. A complex numberI know that it is similar to a vector. A complex number is "just a number" except that isn't necessarily a real number. I see the similarity with a vector, but my question still holds. I argue that it just can't be the same as a vector no matter how similar..

- #32

sophiecentaur

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Those two statements contradict each other.Your argument is invalid. A complex numberisa vector, not just similar to one.That is why we need a plane to visualize them. That is why we can use complex numbers to derive statements on 2D vectors, like it was done in your first message. If you really want to argue further, look up the definitions of a complex number and a vector, and show that a complex number does not satisfy the definition of a vector.

I would rather put it that a complex number is two dimensional - which can be

- #33

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The second one is not a statement - not my statement at least. So no contradiction as far as I can tell.Those two statements contradict each other.

A complex number is a vector because all of the axioms of a vector space are satisfied by the set of complex numbers. It is also a vector in the sense that one can be used to represent little arrows on a flat sheet of paper and vice versa, which may be what you meant by "spatial".I would rather put it that a complex number is two dimensional - which can berepresentedas a vector and this is common usage. But nothing 'spacial' is implied by complex numbers.

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sophiecentaur

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I agree.

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"" Of course you can represent the real and imaginary parts of a complex number as a point on a plane (the Argand diagram) and you can do the same for the components of a 2-dimensional vector. Therefore complex numbers and 2-dimensional vectors will have some "geometrical" properties that are similar. But as you go further into using complex numbers in calculus (for example "analytic functions"), and study things like infinite-dimensional vector spaces where the elements of the vectors are not even numbers at all, you will find there are many more differences than similarities. ""

Apart from that I thought of another argument for my REAL question : If complex numbers add like vectors, then it makes sense that those two terms are their actual components in that direction.

- #36

olivermsun

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Y + Z = Re(Y) + Re(Z) + i(Im(Y) + Im(Z)),

and so on, which allows you to use complex numbers interchangeably with 2-vectors for adding, rotating, etc. As you've already pointed out, however, some of the operations are not the same.

- #37

sophiecentaur

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We look at vectors (2 and 3D) in spatial terms but, once you get more than 3D, you are back to abstractions. No 'direction' at all, even if there is a magnitude.

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Well yeah, but my question was not philosophical. I think what I wrote on my diagram with the square shows why the complex plane analysis actually does represent the tangential and radial components. I thought (and still do) there was a more general way of proceeding than my diagram analysis.

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- #39

sophiecentaur

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To return to my beans, if you follow your approach, you could say that, because two apples plus two apples gives four apples, there is some inherent connection between beans and apples.

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