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Rationals can't solve $x^2=2$, and reals can't solve $x^2=-1$. Is there any problem that cannot be solved by complex numbers but can be solved by non-standard numbers?

Every polynomial with coefficients in $C$ can be solved by numbers in $C$, can every equation* be solved by numbers in $C$?

*(that can not be simplified to $1=0$)

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    Your first Q's A depends on how you define a number since you already know C. Your 2nd Q: yes, because C is a field and if manipulations are also defined in this field.2011-11-19
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    Certainly not every equation has a solution in $\mathbb{C}$. As a trivial example, $1=2$, or to make it look more like an equation in a variable, $z+1-z=2$. As a less trivial example, consider $z \bar{z} = -1$.2011-11-19
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    The equation $e^z=0$ cannot be "simplified" to $1=0$, but it doesn't have any solutions in $\mathbb{C}$.2011-11-19
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    Why would you use the tag [nonstandard-analysis]?2011-11-19
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    @ZevChonoles The equation $e^z=0$ can be "simplified" by multiplication with $e^{-z}$. Guess what you get...2011-11-19
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    @Thomas: That's true, but as long as we're playing that game, the statement that an equation $f(z)=0$ can't be solved is then *equivalent* to the statement that $1=0$, since $f$ has no zeros and is therefore invertible. Wouldn't you agree that there is at least content in the statement that there is no solution to $e^z=0$?2011-11-19
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    @ZevChonoles Good point, you are right.2011-11-19
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    @ZevChonoles: I think that statement is exactly what Holowitz was looking for.2012-12-03

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