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

hello, first: excuse me if the question is stupid but im 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 couldnt 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). Im not sure but as far as I know this mas made mainly to make math more complete. I dont 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 wouldnt it make math more complete?

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

thanks.

 Quote by danov hello, first: excuse me if the question is stupid but im 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.

 Quote by Larne 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 cant 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.

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

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.
 Recognitions: Gold Member Homework Help Science Advisor 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..