Number Space

[mackdaddy]

Lately I have been thinking about number systems as all lying in a 3D space, with x y and z axis. I set the x axis as the real number system, which to me represents distance or physical space. The imaginary number system lies on the y axis, and I see as a *measurement of time. And upon concluding this, I can not think of what would fit on the z axis. I don't even know if this is a good way of thinking about this.


* I actually am not quite sure on this, because we measure time with real numbers (minutes/hours)

I'd love to hear al of your opinions on this

 

[JeffM]

Lately I have been thinking about number systems as all lying in a 3D space, with x y and z axis. I set the x axis as the real number system, which to me represents distance or physical space. The imaginary number system lies on the y axis, and I see as a *measurement of time. And upon concluding this, I can not think of what would fit on the z axis. I don't even know if this is a good way of thinking about this.


* I actually am not quite sure on this, because we measure time with real numbers (minutes/hours)

I'd love to hear al of your opinions on this

You are on solid ground to think of the real numbers as mapping to a line and the complex numbers as mapping to a plane. I suppose quaternions map to a three-dimensional space, but, honestly, I know virtually nothing about quaternions. Here is the wiki article about them:
http://en.wikipedia.org/wiki/Quaternion

I know almost as little about physics as I do quaternions. But as far as I know, physical space is seldom (never?) represented by the real number line. Physical space is usually represented by the set of 3-tuples of real numbers (or a vector in three dimensions). Time then comes in as a fourth dimension. This should get you started http://en.wikipedia.org/wiki/Spacetime

 

[daon2]

You are on solid ground to think of the real numbers as mapping to a line and the complex numbers as mapping to a plane. I suppose quaternions map to a three-dimensional space, but, honestly, I know virtually nothing about quaternions. Here is the wiki article about them:
http://en.wikipedia.org/wiki/Quaternion

I know almost as little about physics as I do quaternions. But as far as I know, physical space is seldom (never?) represented by the real number line. Physical space is usually represented by the set of 3-tuples of real numbers (or a vector in three dimensions). Time then comes in as a fourth dimension. This should get you started http://en.wikipedia.org/wiki/Spacetime

Unfortunately Quaternions are not a true generalization, as they are not a field (but still a division ring). I think I remember reading that there is no structure on \(\displaystyle \mathbb{R}^3\) that turns it into a field, but I could be mistaken.

edit, right, I had forgotten the basics! If you have studied abstract algebra, taking the set of polynomials over a field \(\displaystyle \mathbb{F}[x]\) and quotienting by an irreducible polynomial: \(\displaystyle \mathbb{F}[x]/\langle f(x)\rangle \) (sending all multiples of f(x) to 0) will produce a vector space which is also a field over the base field \(\displaystyle \mathbb{F}\).

The way one can construct \(\displaystyle \mathbb{C}\) is by taking the quotient of \(\displaystyle \mathbb{R}[x]\) by the irreducible polynomial \(\displaystyle f(x)=x^2+1\), that is, all polynomials, but taking care to replace any multiples of \(\displaystyle x^2\) by \(\displaystyle -1\). This, as a set of polynomials ends up being just \(\displaystyle \{ax+b;\, a,b\in \mathbb{R}\}\), which looks exactly like the set of complex numbers (\(\displaystyle x\leftrightarrow i\)).

The reason why there is no such structure on \(\displaystyle \mathbb{R}^3\) is because all real polynomials of odd degree have a root (calculus lends us a hand here), and so there is no irreducible cubic to divide with.

 

[JeffM]

Daon

I am not arguing: I do not know enough to argue. I am trying to make sure I understand your post. One way to interpret it is as saying no more than that quaternions are not a complete generalization of the real and complex numbers. In other words, Hamilton was not completely successful in his goal.

That in itself educates me because as I said in my original post, I really know almost nothing about quaternions.

However, my guess was that quaternions could be put into a simple one-to-one correspondence with the set of 3-tuples of real numbers. I founded that guess on Hamilton's intent to generalize from the complex plane. So I am not sure whether his failure to make a perfect generalization means there is a simple and useful one-to-one correspondence, there is a one-to-one correspondence that is too complex to be of much use, or there is no one-to-one correspondence at all.

 

[daon2]

The Quaternions are a 4-dimensional vector space over the reals, the basis being 1,i,j,k. So there is a bijection with 4-tuples. There is a theorem what states the Quaternions are the best one can do, as far as having a division ring (Link, page 73).

 

[JeffM]

Thanks.