# Is there is a number system which is extension of complex number system?

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?

• Do you mean something Hamilton's en.wikipedia.org/wiki/Quaternion system? Commented Mar 24, 2012 at 12:33
• There is also this question. Commented Mar 24, 2012 at 12:59

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 $$\mathbb{C}(x)=\left\{\,\frac{f}{g}\;\middle\vert\; f,g\in\mathbb{C}[x], g\neq0\right\}.$$