Just working in three dimensions, one can note that cross products don’t behave the same way as the original vectors under discrete transformations. If, for example, you reverse the direction of every vector in your original space, the directions of any cross products will remain unchanged. So if A, B, and C are vectors, and A cross B is parallel to C in the original space, in the inverted space, they’ll be antiparallel. For this reason, cross products are sometimes referred to as pseudovectors, and it can be important to keep track of what’s a vector and what’s a pseudovector in some calculations (I especially find this to be so in electromagnetism).
If you look in a little more detail, though, and see the generalization of cross products to other numbers of dimensions, it turns out that cross products are actually antisymmetric second-rank tensors. It so happens that a 3-dimensional antisymmetric second-rank tensor has 3 independant components, but that’s just a coincidence. In four dimensions, for instance, a cross product would have 6 independant components. So when you look at cross products in detail, the structure is all different.
I was reading a biography of Paul Erdos the other day (this one) and it noted that Erdos came up with a single equation which expressed e, pi, i, 1, and 0 (sorry can’t do the greek pi):
> . . . Erdos came up with a single equation which expressed e, pi, i, 1, and 0 . . .
Not only is that a wrong attribution, but that’s not what Paul Hoffman wrote in The Man Who Loved Only Numbers. You’re talking about page 228. (At least in my paperback copy that’s the page.) Hoffman says that it was Euler that came up with the formula. I presume that you missed that he wrote “Euler” and not “Erdos”.
As for integers, what’s real about 0 or negative numbers. Some mathematician said (memory quote here), “God invented the natural numbers, man the rest.”
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But if God created man, then didn’t God create all numbers too?
No, that’s inhomogeneous. Although, come to think of it, multiplication on the reals is still anisotropic, since x*x = x is true for x=1, but not for x=-1. And I don’t think I’ll get very far trying to argue that multiplication on the reals is not natural…
OK, so I’ll concede that multiplication on the complex numbers is just as natural as on the reals, and on the quaternions, almost as natural (commutivity is nice, when you can get it).
Actually, there’s a view that I haven’t yet fully internalized that parallels the tower of real division algebras with higher-categorical constructions. Basically, commutativity in general arises when there’s a sort of auxilliary multiplication one can use to “move around” the main multiplication. Think of the proof that all higher homotopy groups are Abelian for the basic idea. Of course, for algebras, commutativity is all you can get, but at the 1-category level you can get braiding before you get full symmetry. With 2-categories you get even more choices, and so on.
I presume that braiding would be something like abc = bca = cab, but not necessarily ab = ba? Situations like that show up not infrequently in physics, but I’m not sure what sort of mathematical structure they’re considered to be.
No, that’s the cyclic property of a trace function, which is indeed very important.
Braiding is roughly that the two factors can be moved around each other, and that if you move one all the way around the other it’s not the same as the first compostition.
More specifically: in a monoidal category (think vector spaces or modules over a fixed field or ring) there is a tensor product sending objects A and B to AB ( should be \otimes in TeX) with various properties turning the class of objects into a monoid. There may be a “braiding”, which is an isomorphism b[sub]AB[/sub] from AB to BA for each pair (A,B). Obviously there is also a morphism b[sub]BA[/sub][sup]-1[/sup] from AB to BA. Schematically, b[sub]AB[/sub] moves B around A one way in the plane and b[sub]BA[/sub][sup]-1[/sup] the other way. If the two are the same morphism, then the braiding is called symmetric.
At the level of basic abstract algebra, one decategorifies: one forgets about how two objects are isomorphic and only that they are. This means that when talking about real division algebras, braided monoidal structures and symmetric ones are no different, and it’s just called “commutative”.
That would be Euler, not Erdos and it is, I guess, about 250 years old.
How do you like this one? i to the power i is 1 divided by the square root of e to the power pi. Among other things. Complex exponents are multi-valued but all the values of this one are real numbers.
