A new set of numbers as a z-Axis to imaginary and real numbers?

[danov]

hello, first: excuse me if the question is stupid but I am still at school.
my question: some days ago i came across imaginary numbers. You know what I mean - the imaginary number i^2=-1 and the imaginary numberline is not on the the same line as the other numbers. The imaginary numberline is alligned verticaly to the horizontal numberline and both lines have the common point 0. But I guess you already know this. To me this is interesting because this numberlines together form a 2-Dimensional "Field". The real numbers line is the x-axis and the imaginary number line is the y-axis. Now I wonder: Can there be a third axis, the z-axis, which woud make a 3.
-dimensional "field" out of this? Ok, I guess there may not be a mathematical need for such a third axis, at least I couldn't find one. But I think this way: The imaginary numbers together with imaginary axis were introduced because mathematicians wanted to have an solution for sqr(-1). I am not sure but as far as I know this mas made mainly to make math more complete. I don't think that people from beginning had a use for imaginary numbers BUT it made math more complete. So I think it may be the same with a third axis for another set of numbers. We may not have a use for this now but wouldn't it make math more complete?

What do you think about it? Maybe its complete nonsense but could you tell me why?

thanks.

 

[Larne]

hello, first: excuse me if the question is stupid but I am still at school.
my question: some days ago i came across imaginary numbers.

[...]

Now I wonder: Can there be a third axis, the z-axis, which woud make a 3.
-dimensional "field" out of this?

Not a stupid question at all!

Around 1843 the famous mathematician, Sir William Rowan Hamilton, was wondering exactly the same thing. Eventually he realized that it is impossible to extend the complex numbers into three dimensions in a consistent way... but that it is possible in four dimensions! The result is called the "quaternions."

Here's a neat article about quaternions by John Baez.

 

[danov]

Not a stupid question at all!

Around 1843 the famous mathematician, Sir William Rowan Hamilton, was wondering exactly the same thing. Eventually he realized that it is impossible to extend the complex numbers into three dimensions in a consistent way... but that it is possible in four dimensions! The result is called the "quaternions."

Here's a neat article about quaternions by John Baez.

ok thanks. An article I just read bout quaternions says that there are even 8-Dimensional hypercomplex numbers (octonions), 16-Dimensional (sedenions) and new higher dimension-systems can be constructed with the Cayley-Dickson-Method (creates new systems with twice as much dimensions as the source-system).

Thats really interesting. Does that mean that you can create systems with infinite dimensions as long as you can divide the dimension by 2? Why is that so?

I am wondering: why can't you add a z-axis (3-D.) to the existing system (yet) which, at least seen from a geometrical point of view, would be much easier than a 4-Dimensional axis? I mean you can even imagina such a 3-Dimensional system easyer than a 4-Dimensional. In fact: How does a 8-Dimensional or 16-D. System even looks like? Cant imagine that.

Thats seems very strange to me.

 

[D H]

It's a power of 2, not factor of 2. You lose something each step up the ladder. You can compare two reals to determine which is less than the other. There is no meaningful way to compare compex numbers. Multiplication is commutative for the real and complex numbers (i.e., a*b=b*a). Quaternion multiplication is not commutative. Multiplication is associative for the reals, complex numbers, and quaternions (i.e., a*(b*c)=(a*b)*c). Octonion multiplication is neither commutative nor associative.

 

[arildno]

The really fascinating thing about the complex numbers is that they are "algebraically closed":
What that means, for example, is that any polynomial equation formulated solely in terms of the complex numbers ALSO have their solutions WITHIN the realm of complex numbers!

This is not true of "lower-order" number systems:

Take the naturals:

The equation x+7=5 is expressed only be means of naturals (1, 7 and 5), but the solution, x=-2 is NOT a natural number.

Take the integers:
The equation 4*x=2 is expressed by means of integers, but the solution, x=1/2 is not an integer.

And so on with the rationals and reals as well..

 

[masnevets]

There are very good questions, but they have rather lengthy answers.hello, first: excuse me if

You're touching on a number of ideas that you will encounter later in your studies if you continue with mathematics. First, you mentioned the fact that the axes of the real and imaginary numbers forms a 2-dimensional "field." The term normally used here is that the complex numbers are a 2-dimensional vector space over the real numbers. In general, a vector space doesn't have a multiplication defined on it, but in the case of the complex numbers, this is an extra bonus. If you continue with only being concerned with vector spaces, then yes, we can add an extra axis and get a 3-dimensional real vector space, or any dimension we like. There is always a multiplication here. For example, in the 3-dimensional case, we can take a triple of numbers and multiply them componentwise: (a,b,c) * (d,e,f) = (ad, be, cf). The problem with this is that you can't divide-- because, for example, (0,0,1) * (1,0,0) = (0,0,0).

So what the others are talking about with quaternions and octonions, in 4 and 8 dimensions, respectively, is that there is a way to define multiplication so that we can still divide. The problem is, as was pointed out, that the more dimensions you add, the less properties you get. For example, the real numbers are ordered in a compatible way with multiplication and addition (i.e., a < b means that a + c < b + c and ac < bc if c > 0). But you can't do this with the complex numbers-- should i be positive or negative? Neither works because i*i < 1, but morally, the square a nonzero number _should_ be positive. And then in the quaterion case, you lose commutativity, meaning that a*b is not necessarily equal to b*a. In the octonion case, you lose associativity, meaning that (a*b)*c is not necessarily equal to a*(b*c). But we can still divide in these cases at least!

So what happens in higher dimensions? Say for example 16. Well, if we're talking about extensions of the real numbers, then the next property you lose is the ability to divide. It's not so obvious that it's impossible to define a multiplication on a 16-dimensional real vector space and not be able to divide. One way that I know this is proven is using the Hopf invariant from algebraic topology (don't worry about what this is), although I am sure that there are other simpler ways. I believe that this fact was first proven by Frobenius.

Finally, to address your question about "completeness" in terms of why we need the complex numbers. You're right that we want to consider the complex numbers because for example the equation X^2 + 1 = 0 has no solutions if we only ever knew about real numbers. You might ask if adding the imaginary axis fixes all of our problems in that all polynomial equations (with complex coefficients) will now have solutions. The answer is surprisingly yes. This is known usually as the fundamental theorem of algebra. It says that the complex numbers are "algebraically closed," which means exactly that whenever you have a polynomial equation with complex coefficients, you can find a solution in the complex numbers. This is a fairly interesting subject and leads to Galois theory if you pursue it more.

In short, it's good that you're asking these questions. As you can see, they're good questions!

 

[Hurkyl]

If you're interested in fields (everything nonzero is invertible, multiplication is commutative), then I think there are essentially only two ways to extend a field:

(1) Algebraic extension.
You choose some polynomial that doesn't have a root in your field, and you make a new field that does contain a root of your polynomial.

(2) Transcendental extension.
You form the field of all rational functions in your field.



The complex numbers are algebraically closed; every polynomial has a solution. So, the smallest extension of the complex numbers is the field C(x) -- its elements are complex rational functions in one variable. (i.e. ratios of complex polynomials) This is an infinite degree extension of C.


Similarly, the two minimal field extensions of R are C and R(x).

 

[danov]

many thanks to everybody for the answers. Some answers are quite mathematical and hard to understand for me (17y-old student, 12th-grade) so I will need some time to understand and think about that. (I only had Integration and Differentiation (Analysis?) and no other higher mathematics yet)

But, even though I don't fully understand, my question was answered and I now know that a z-axis is not possible (yet?).

thanks!

 

[masnevets]

But, even though I don't fully understand, my question was answered and I now know that a z-axis is not possible (yet?).

It's not a matter of if someone has discovered it yet. It just depends on what you're asking for. If you want a 3-dimensional space with multiplication where you can divide, it cannot happen. It is proven not to exist. If you don't care about division, then you can make up infinitely many different types of multiplication.

 

[R1ckr()11]

Very interesting question.

I had a similar thought that has been clarified (though i still need confirmation): it must then be impossible to imagine a trigonal planar arrangement of the number line. It's simply one line, but there is no way to have two lines with less than 4 directions. Is it this sort of property that denies the possibility of 3-D extra complex numbers? The fact that the bi-directionality is reserved for number lines instead of number 'rays' as such?