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I'd like to make up a definition of cosine with certain properties of "naturalness". Here is a sketch of the line I wish to follow and I'm hoping someone can state this in a mathematically formal way and fill the holes in the definition. It would be great to know a list of minimal conditions to make my approach work.

First, without any prior assumption about numbers, I want to define an algebra with elements $I(\theta)=A\exp(i\theta)$. It should be defined by requiring commutativity, associativity, division(?) for a multiplication operation. Because many ideas in physics are commutative and associative and reversible.

Next I want introduce an addition operation with the distributive law. Now, if you want to "split" the numbers into a "vector-like" sum to enable easy addition of components, you will naturally arrive at a definition for cosine (with some additional conditions?)?!

How can I phrase/extend this into a mathematical set of conditions yields the cosine function?

Gere
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  • I'm not quite sure what you mean by "an algebra with elements $I(\theta) = \exp (i\theta)$". – GPerez Jan 22 '15 at 17:09
  • I'd like to collect some axioms which are likely to occur in the real world and which single out the a "cyclic addition algebra" like $\exp(i\theta)$. Not sure what the correct term is. Next I assume this "real-world case" also happens to have a structure like the distributive law. This way I want to introduce cosine. – Gere Jan 22 '15 at 21:56
  • Axioms are very unlikely to occur in the real world, but I think I know what you mean. I still am not clear on this algebra you describe; do you sum the entire numbers $\exp(i\theta)$, or just the argument $\theta$? – GPerez Jan 23 '15 at 08:37
  • Indeed, I prefer axioms that are likely to match real-world cases. I think commutativity, associativity, invertibility (division) are a good fit. My thought was to continue this line and use the Frobensius theorem to circle in on complex number and their multiplication property. The main ingredient here is their cyclicality which isn't too abstract and is also likely to occur in natural processes. My exp() notation is just a sketch. I basically mean defining the multiplication of complex numbers. – Gere Jan 23 '15 at 12:06

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