Number Theory

[Aethelwulf]

I have a question... now this is based on memory, if I have anything wrong, please correct me...

 

 

We have natural numbers, then it is imaginary numbers, then we work with quarternions, then it's... Octonians and then it is the Cayley numbers? Is that right? I hope there is nothing inbetween...

 

 

... anyway, I understand imaginary numbers and quarternions, I am a bit vague on Octonians and how many numbers they involve... but why does everything stop at the Cayley Number... is it something to do with the Division Ring?

[D H]

You missed quite a few steps from naturals to the imaginary numbers. Natural numbers are 0,1,2,... (or 1,2,3,... depending on who you're reading). This has addition and multiplication as two base operations, and multiplication can be defined in terms of addition for the natural numbers. There's obviously an inverse to addition. 3+4=7, so define "-" such that 7-4=3, 7-3=4. But what's 3-4? Arithmetic completion leads to the integers, the first set you omitted. Doing the same with multiplication leads to division and the rationals, the second set you omitted. Cauchy sequences or Dedekind cuts lets to the reals the third (and arguably most important) set you omitted.

 

The complex numbers add something very important to the reals: They are algebraically closed. All of the roots of any polynomial with complex coefficients are complex numbers. The complex numbers also remove something very important, which is the operation "<". Which is smaller, 1+i, or 1-i?

 

The quaternions removes even more. Multiplication is not commutative: a*b is not necessarily equal to b*a. Each step up the Cayley–Dickson hierarchy adds a little, subtracts a little. The octonions aren't even commutative, but they are associative. The next step up, the sedenions, aren't even associative. You can go on forever creating new structures, but by the time you get to the octonions and beyond they aren't very useful.

[Aethelwulf]

Sedenions>??

 

 

Argghhh...this is something I have not heard of...will be back soon

Edited June 15, 2012 by Aethelwulf

[Aethelwulf]

Ok, had a read about Sedenions.

 

Now you say that you can go on forever, but from a different source I heard you can't get any higher than the Cayley Numbers --- which I was informed, had something to do with the Division Ring, so I am guessing something is a bit wrong here?

 

Thanks DH for helping me understand this.

[D H]

The Cayley numbers are just another name for the octonions (to within an isomorphism).

 

One shortcoming of the sedenions S is that the problem of zero divisors. There are non-zero sedenions whose product is zero. One consequence: [imath]|a|\,|b| = |ab|[/imath] is not necessarily true. Another consequence is that there's no way to define division. As I mentioned earlier, each step up the Cayley-Dickson hierarchy loses something. Tossing out the concept of division and tossing out the identity [imath]|a|\,|b| = |ab|[/imath] : You've lost a whole lot. This tosses a whole lot of mathematical structure out the door.

 

The loss of that identity [imath]|a|\,|b| = |ab|[/imath] is quite deep. I think what you are talking about is Hurwitz' theorem, which says that the only finite dimensional normed division algebras over the reals are the reals themselves, the complex numbers, the quaternions, and the octonions (to within an isomorphism). This is in a sense an extension of the Frobenius theorem, which says that the only finite dimensional associative division algebras over the reals are the reals themselves, the complex numbers, and the quaternions (to within an isomorphism).

 

Hurwitz' theorem is a bit different from Frobenius theorem, however. It talks about sums of squares. For example, if a,b,c,d are real,

 

[math](a^2+b^2)(c^2+d^2) = (ac-bd)^2 + (ad+bc)^2[/math]

 

Note that the left hand side is the product of a two sums of a pair squares, the right hand side is the sum of two squares. There are similar identities involving the product of two sums of four squares on the left and the sums of four squares on the right, and also for the product of two sums of eight squares on the left and the sums of eight squares on the right. The case n=1 is trivial. Are there any others besides 1, 2, 4, and 8? The answer is no. Hurwitz' theorem is also called the 1,2,4,8 theorem because of this.

 

This can be brought back to the realm of algebras over a field by looking at the identity [imath]|a|\,|b| = |ab|[/imath]. Square both sides and you get [imath]|a|^2\,|b|^2 = |ab|^2[/imath]. Treat a and b as n-dimensional vectors with some kind of multiplication defined two vectors (i.e., an algebra over a field), expand out the terms and you get the product of two sums of n squares on the right, the sum of n squares on the left. Hurwitz' theorem says that the only n for which this holds are 1,2,4, and 8: The reals, the complex numbers, the quaternions, and the octonions.

 

 

One final note: If you want to study this stuff in detail, I suggest you get a book (or take a course) on modern algebra or abstract algebra. Note well: I'm not talking kiddie algebra here. I'm talking about the algebra class (classes) that math majors take in college after they've taken multiple calculus courses and a class or two in analysis.

Edited June 15, 2012 by D H

[Aethelwulf]

 

One final note: If you want to study this stuff in detail, I suggest you get a book (or take a course) on modern algebra or abstract algebra. Note well: I'm not talking kiddie algebra here. I'm talking about the algebra class (classes) that math majors take in college after they've taken multiple calculus courses and a class or two in analysis.

 

I think I could handle it, so whenever there is a chance for my to study absract algebra covering these topics,I will

 

 

Thank you again.