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Complex number system is extension of real number system. Is there is a number system which is extension of complex number system in which algebra is well defined?

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Do you mean something Hamilton's en.wikipedia.org/wiki/Quaternion system? –  martini Mar 24 '12 at 12:33
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There is also this question. –  Asaf Karagila Mar 24 '12 at 12:59
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up vote 13 down vote accepted

The quaternions and octonions are number systems that extend the complex numbers. Together with the complex numbers and the real numbers themselves, these form the only normed division algebras over the real numbers. In a normed division algebra, there is a notion of "size" (given by the norm), and every non-zero element has a left and right multiplicative inverse (this is part of the definition of "division algebra"). However, in the quaternions, multiplication is associative, but not commutative; and in the octonions, both associativity and commutativity of multiplication are lost. So, algebra is perfectly well-defined in these number systems, but they don't behave like you might expect.

The word "number" has no actual mathematical definition; so if you are willing to interpret "number" abstractly, there is a very good case for considering polynomials to be "numbers" as well. They certainly have well-defined algebraic operations (e.g. adding, multiplying) we can do to them; in other words, they form a ring. So you can also extend the complex numbers by forming polynomial rings, and quotients of such rings. For example, we have the polynomial ring $$\mathbb{C}[x]=\{a_nx^n+\cdots+a_1x+a_0\mid a_i\in\mathbb{C}\},$$ or the ring of so-called "dual numbers" over $\mathbb{C}$: $$\mathbb{C}[x]/(x^2)=\{a+bx\mid a,b\in\mathbb{C}\}$$ (in this ring, $x$ has the property that $x^2=0$). You can also make the field of rational functions $$\mathbb{C}(x)=\left\{\,\frac{f}{g}\;\middle\vert\; f,g\in\mathbb{C}[x], g\neq0\right\}.$$

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You're not going to tell OP about sedenions? ;) –  J. M. Mar 24 '12 at 13:06
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It all depends on what you mean by "algebra". It isn't really very fruitful to just create extensions of things without having some reason for doing so.

We create the complex numbers for a reason...so that all real polynomials have solutions. It just so happens that this is a field so has nice structure and is algebraically closed.

If you want to make an extension of the complex numbers you can do so in many ways but what properties would you like the given extension to have?

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