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If I want to invent intresting new types of numbers like complex numbers with specific properties, for example $i^2=-1$ for the complex numbers, what properties must I have to be a extension of $\mathbb{R}$ that :

  • Doesn't make contradictions ? Example : see my last question
  • Is not "equivalent" to another number ? Example : $j$ is a number such as $kj = k$ and $k+j=k+1$
  • May be interesting ?
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PearlSek
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The algebraic closure $\Bbb C$ of $\Bbb R$ has degree $2$ over $\Bbb R$, so $\Bbb R$ has no algebraic extension except $\Bbb R$ itself and $\Bbb C$ (see here).

I'm not sure what you can do with transcendental extensions, but you can try to work with ring extensions. For instance, the dual numbers $\Bbb R[\epsilon] = \Bbb R[x]/(x^2)$ can be interesting.

But your conditions $kj=k$ and $k+j=k+1$ are very strange: $k+j=k+1 \implies j=1$, so $kj=k$ is trivially satisfied, whenever $k,j \in A$ and $A$ is a commutative ring such that $\Bbb R \subset A$, and $1$ (as the real number $1$) is the unity of $A$.

Watson
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  • Yes it was an example of the equivalence to another number – PearlSek Nov 30 '16 at 14:33
  • A nice way to "create" new numbers is to consider quotient rings of $\Bbb R[x_1,\dots,x_n]$, e.g. $\Bbb R[x]/(x^2+1)$ is just isomorphic to $\Bbb C$. – Watson Nov 30 '16 at 14:43