So real numbers are 1-dimensional, and complex numbers are 2-dimensional. Going along the same lines, we also have Quaternions, 4-dimensional numbers: http://en.wikipedia.org/wiki/Quaternion Further again we have 8-dimensional Octonions, and 16-dimensional Sedenions.
I'm curious as to why we don't have any useful numbers for the non-power-of-2 dimensions. E.g. 3-dimensional numbers.
There are no three dimensional numbers because it's impossible to construct such a system that behaves like 'numbers'. The real, complex, quaternion and octonion numbers are the only 'normed division algebras'. This basically means they are the only spaces with a notion of length in which mutliplication is invertible (division). These properties are needed for anything that we want to call 'numbers'. The fact that there are only 4 such spaces is quite a profound result and leads to a lot of other important results. See http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_%28normed_d... for more information.
Why are there no 16, 32, 64, and so on, dimensions of numbers? Would multiplication not be invertible with them? Is there a possibility of things that are distinct from our concept of numbers but are none-the-less useful being discovered in the future?
> Why are there no 16, 32, 64, and so on, dimensions of numbers? Would multiplication not be invertible with them?
Yes, this is exactly why the 16 dimensional sedenions are not in the same category as the others. For it to be a division algebra, you must be able to reverse any multiplication that isn't by zero. This is just the normal concept of division in the real numbers. It also allows us to define division in the complex, quaternion and octonion numbers. However, it is possible to multiply two non-zero sedenions together and get zero. This breaks this property completely.
> Is there a possibility of things that are distinct from our concept of numbers but are none-the-less useful being discovered in the future?
Absolutely! These four numbers systems are a tiny portion of spaces studied by mathematicians. There are a lot of interesting things that can be said just by generalising a few properties of numbers (e.g. without requiring that we can divide, or without requiring that we have both addition and multiplication). There are groups, fields, rings, algebras, topological spaces, and all sorts of interesting things that aren't 'numbers' as such.
This is pretty fascinating, do you know of any good sites or books which will get me up to scratch on different number systems? I haven't studied math in a long time, so do you think it would be worth going over the basics, like calculus and geometry, again to build a foundation before venturing so far out?
As others have noted, we don't have 3D numbers because there is a theorem saying that they don't exist (at least with norm). Interestingly, Hamilton's discovery of the quaternions was the result of a long and fruitless search for a three-dimensional analog to the complex numbers.
Thank you for posting this. I remember trying to understand quaternions a few years ago and didn't get it. Your comment combined with this article made it click.
I wonder if there's anything of numbers in infinite dimensions and, more interestingly, numbers in different degrees of infinite dimensions (yup, infinity comes in different degrees too - http://sites.google.com/site/degreesofinfinity/). It seems that math is only limited by the imagination.
You can define any dimensional number that you like, including infinite. What you can't do is get certain properties out of them. But infinite dimensional numbers are actually frequently used, under the term "n-dimensional vector".
(Infinite is often confused for "really big", but a better way of understanding it is often to use the term "unbounded". When we deal with n-dimensional vectors, we're not saying that we are always literally dealing with vectors with a millionybillionytertrillion dimensions, such that we can't even represent one in a real computer, we're saying that there isn't a firm upper limit on the number of dimensions we may encounter. It is often more the "unbounded" aspect of infinity that we are concerned with, rather than the "larger than anything else" aspect.)
From what I gather from jpallen's response to my other message the properties you're talking about are things like being able to add, multiply, subtract and divide, right? What's to stop one from coming up with novel properties which are unique to these systems? Is our commonly used system basically an arbitrary set of conditions? Do you think that some systems correlate better to how nature does things than others, and if so, which ones would they be?
Ah, so it's unbounded like how Haskell's lazy evaluation treats infinity.
> What's to stop one from coming up with novel properties which are unique to these systems?
Interestingness. Usefulness, perhaps, if you have a specific definition of 'useful' in mind.
> Is our commonly used system basically an arbitrary set of conditions?
It's an arbitrary set of conditions with interesting properties, and it's not always obvious which arbitrary conditions will have interesting properties.
> Do you think that some systems correlate better to how nature does things than others, and if so, which ones would they be?
Well, that is an interesting question, which means anyone who claims to know an absolute answer to it is a moron. We do know, for example, that vectors are a very useful tool to model a lot of what happens in physics, and that complex numbers are a compact way to talk about rotation, especially complex exponentiation (taking a real number, such as e, to complex powers).
> Ah, so it's unbounded like how Haskell's lazy evaluation treats infinity.
Yes. For example, a definition (the Peano axioms) of the set of the natural numbers (either the positive or non-negative integers; the set may or may not include zero, as conventions vary) states that the set contains 0 (or 1) and that for every number x it contains, it also contains x+1.
I wish they taught this back in school, math would have been so much more interesting. Anyway, I lack the knowledge to ask any more meaningful questions so I'm gonna go do some digging. Thanks for taking the time to respond.
I wonder why mathematicians use this funny i-notation? You could write complex numbers as vectors, e.g. 2+3i <=> (2,3). Of course, it is more terse to write i instead of (0,1) and this seems to matter a lot to mathematicians.
i is commonly used as the unit vector in the x-dimension (a basically meaningless term without more context). 3-dimensional vectors are often written as:
5i + 2j + 3k.
Since you can assume that 2 is getting multiplied by the unit real vector (equal to 1), you can think of 2+3i as
I think it'd be awesome to have some sort of "decimal" notation where 2 + 3i was encapsulated as a single entity, something like 2_3. We don't write 2 + .3 when we mean 2.3, and this distinction makes it seem like a complex number is "less put together" than a real one.
in a sense it is - it involves magnitudes along two orthogonal directions. there is no qualitative difference between 2 and 0.3 in this particular context.
Thanks for mentioning this. I gauge the value of an explanation by the follow-up questions... if you really understand imaginary numbers, then suddenly you start asking about other dimensions, etc. and get to awesome discussions like this one :). And even better, people are enthusiastic to explore these new avenues.
I'm curious as to why we don't have any useful numbers for the non-power-of-2 dimensions. E.g. 3-dimensional numbers.