Complex Numbers: Are They The Ultimate?

In summary: So that means the constructions do not go on forever. So my memory was wrong.In summary, complex numbers are introduced to fill a gap in the real numbers and are algebraically closed. There are other number systems such as the quaternions and octonions, but they lose important properties. The Cayley-Dickson construction allows for the creation of algebras up to the sedenions, but beyond that, there is not much algebraic structure left.
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warrianty
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Complex numbers - are they the 'ultimate', or are there any "complex complex" numbers

When we try to calculate the root of a negative number, we come to the idea to introduce complex numbers. Is there any operation for which complex numbers wouldn't be enough, so there's a need to introduce complex complex numbers, complex ^ n numbers, etc?
 
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warrianty said:
When we try to calculate the root of a negative number, we come to the idea to introduce complex numbers. Is there any operation for which complex numbers wouldn't be enough, so there's a need to introduce complex complex numbers, complex ^ n numbers, etc?

Sure, how about the quaternions.
 
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... or the octonions, or sedenions, and so on. Every power of 2 yields a different algebraic structure. Something is lost with each step. The complex numbers are not orderable, the quaternions are not commutative, the octonions are not associative, the sedonions are not alternative, and so on.

The complex numbers fill an algebraic hole in the reals. The real numbers form a complete field, but they are not algebraically closed (there is no real solution to x2+1=0). The complex numbers fills that gap. Unlike the reals, complex numbers are algebraically closed. Like the reals, the complex numbers form a field. Moreover, the complex numbers are algebraically closed. Those larger algebraic structures, unlike the complex numbers, do not fill a hole. They are instead invented structures motivated by the complex numbers.
 
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warrianty said:
When we try to calculate the root of a negative number, we come to the idea to introduce complex numbers. Is there any operation for which complex numbers wouldn't be enough, so there's a need to introduce complex complex numbers, complex ^ n numbers, etc?
There are lots of different number systems we use, geared for different purposes.

You have learned the complexes, which contain the reals, which contain the rationals, which contains the integers, which contain the natural numbers... but don't be mislead by this: the number systems aren't arranged in a neat hierarchy like this. Instead, they branch off in all sorts of directions.

Probably the most common number systems in use, other than the ones you already explicitly know, are:

. The cardinal numbers
. The ordinal numbers
. The extended real numbers
. The projective real numbers
. The projective complex numbers

I would assert things like
. Polynomial rings
. Vector spaces
. The "algebra" of all matrices
. The "algebra" of all Abelian groups
are essentially number systems too, although I would probably get more disagreement about that.
 
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If you start with natural numbers and try adding properties, you end up at the complex numbers.
- Start with the natural numbers.
- If you want to be able to subtract any number from any other, you have to add negative numbers to get the set of integers.
- If you want to divide, you need the rationals.
- If you want to take limits, you need the reals.
- To take square roots (or solve general polynomials), then you have to go to the complex numbers.

The complex numbers are a complete and algebraically closed field, so there's not really much reason to go further. It is possible to use quaternions and octonions. I don't know if you can derive them by trying to complete the complex numbers under some operation, although you do lose properties.
With quaternions you lose commutativity, ab != ba.
With octonions you also lose associativity, (ab)c != a(bc).
 
  • #6


D H said:
... or the octonions, or sedenions, and so on. Every power of 2 yields a different algebraic structure.

No it doesn't, not in any real sense. It stops there I believe - only a few powers of two yield decent algebraic structures (a result of differential geometry I seem to rememeber to do with the possible space of vector fields).

EDIT - I think that the 'real sense' means something like you lose enough structure for it not satisfy some property - I can't find the result I'm thinking of. There are of course algebras of arbitrary degree over C.

EDIT EDIT I have it - proved by Bott Periodicity, the only division algebras over R are C, quarternions and octonions.
 
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  • #7


matt grime said:
No it doesn't, not in any real sense. It stops there I believe - only a few powers of two yield decent algebraic structures (a result of differential geometry I seem to rememeber to do with the possible space of vector fields).
The Cayley–Dickson does go on forever, but as you noted, the utility of the constructed algebras pretty much comes to an end because each of the first few steps takes away some very useful property. There isn't much algebraic structure left beyond the octonions. The sedenions aren't even alternate but are still power associative. Beyond that, about all that is left is the ability to construct a conjugate and a norm.
 
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D H said:
The sedenions aren't even alternate but are still power associative. Beyond that, about all that is left is the ability to construct a conjugate and a norm.

Wikipedia has an entry, http://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction" [Broken]. According to them, the algebras are all power associative, and not just up to the sedenions.
 
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1. What are complex numbers?

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. The imaginary part is represented by the letter "i", which stands for the square root of -1. Complex numbers are written in the form a + bi, where a is the real part and bi is the imaginary part.

2. Why are complex numbers important?

Complex numbers are important because they allow us to solve equations that cannot be solved with real numbers alone. They also have many applications in science and engineering, such as in electrical engineering, quantum mechanics, and signal processing.

3. Are complex numbers considered "the ultimate" in mathematics?

This is a matter of opinion, but many mathematicians would argue that there is no ultimate concept in mathematics. Complex numbers have proven to be very useful and powerful, but there are still many unsolved problems and open questions in mathematics.

4. Can complex numbers be graphed on a number line?

No, complex numbers cannot be graphed on a number line because they have two components (real and imaginary) whereas a number line only has one. Complex numbers are typically graphed on the complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

5. How are complex numbers used in real life?

Complex numbers are used in a variety of real-life applications, such as in electronics, physics, and computer graphics. They are also used in everyday calculations, such as calculating the square root of a negative number. In addition, complex numbers have important applications in number theory and abstract algebra.

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