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Are there numbers other than complex numbers? for example, \begin{eqnarray} |x| = -1 \end{eqnarray} Surely, the equation does not make much sense initially since by definition magnitude is positive. But in the previous times when people knew only positive integers they extended the field by assuming (something strange at that time probably) $a = -2$ , where $a$ is now negative integer.

I believe this and many other equations cannot be satisfied with complex numbers. Do we not need numbers other than complex numbers or other numbers are expressible in terms of the complex numbers some how?

chatur
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    there might always be equations which can't be solved . i bet at some point imaginary numbers were not even imagined. you can always device a new system. – avz2611 Feb 19 '15 at 16:30
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    See this question. We only create new numbers when they are useful. – user26486 Feb 19 '15 at 16:30
  • We do need other fields than $\mathbb{C}$, e.g., the fraction field of a polynomial ring. If you speak of "numbers", then usually this means an element in $\mathbb{R}$ or $\mathbb{C}$. – Dietrich Burde Feb 19 '15 at 16:32
  • Your example is not about number systems. It is about the definition of the absolute value. –  Feb 19 '15 at 16:38
  • @user314, thanks for pointing out the question. – chatur Feb 19 '15 at 16:45
  • @avz2611 "imaginary numbers were not even imagined" Sounds like a form of poetry :) +1 – imranfat Feb 19 '15 at 16:48
  • In split-complex numbers the modulus (Minkowski norm) of the split-complex unity is imaginary. But there are no numbers with negative modulus. – Anixx Mar 20 '21 at 20:25

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Yes, there are numbers other than the complex numbers. One immediate example that is apparently different than the complex numbers can be seen here. Tersely, let $\varepsilon$ be such that $\varepsilon \neq 0$ and $\varepsilon^{2}=0$. What use does $\{a+b\varepsilon|a,b \in \mathbb{R}\}$ have? Do we lose any nice properties that we are accustomed to when we operate in this set? Do we ever need this set?

Sloan
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