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1: Discovering of negative numbers.

Assume a and b are positive integers

$x+a=b$ ----> if $b>a$ then $x$ is positive integer

$x+a=b$ ----> if $b=a$ then $x=0$

$x+a=b$ ----> if $b then $x=b-a$ is negative integer


2: Discovering of rational numbers.

$x+x+....+x=a.x=b$ ----> $b \equiv k\pmod a$ if $k$ is not zer0, $x=\frac{b}{a}$ is not integer .


3: Discovering of irrational numbers

$x+x+x......+x=x.x=x^2=2$ ----> $x=\sqrt{2}$ is not rational number


4: Discovering of complex numbers

$x^2=-1$ ----> $x$ is not irrational number


I wonder what 5th step can be for next generation numbers. Is there any known operator or equation to find next generation numbers?

or in other words, Are the complex numbers end of story for numbers to be found as an equation via an operator?

Could you please tell me your ideas and share your knowledge about this subject?

NOTE: I know the quaternions that are a number system that extends the complex numbers. Actually I wonder if possible or not to define next generation numbers via known operators or new operator such as previous numbers (negative numbers,rational numbers,irrational numbers, complex numbers) were defined as equation of $x$.

Thank you very much for answers and links.

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    @JyrkiLahtonen I've read the reals can be constructed purely algebraically as a wreath product, which of course is designed to capture the familiar digital expansions and their carrying properties. (Just tangential trivia.)2012-07-04

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Several extensions can be named. One of them is the set of "hypercomplex" numbers, see mathworld. They are somewhat like the quaternions in that you have additional elementary numbers $j$ and $k$, except the relations are chosen such that all numbers commute.

There are also "hyperreals", which is like a refining of real numbers. If you've heard about infinitesimals and infinities, this is how they're made rigorous. I'd recommend you read up on wikipedia. I'm confident the hyperreals can be extended to a 2D version, thus yielding a different "hypercomplex": this is not Davenport's algebra, but a 2D version which is, in a sense, more "dense" than the regular reals.

Though, as noted in the comments to the question, the real problem is Hurwitz' theorem, which says we cannot find very many algebras with interesting properties.

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    Now more clear general picture of algebra. If we define different rules for some operators, we have different algebra. I read the remarks in the link below and it mention my question's answer above that $x^2+1=0$ has infinitely many quaternion solutions. Now I feel more stable. I understood how important to define clearly which algebra we use to solve an equation. http://en.wikipedia.org/wiki/Quaternion#Remarks2012-07-04