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- Not every element has a square root in
**R**:- ~
**exists**`x`:`x`*`x`= -1. - sqrt(
`x`) :=**such**`z`:`x`=`z`*`z`. - sqrt(-1) is undefined.

- ~
*Proof:*We prove**forall**x in**R**:`x`*`x`!= -1. Take arbitrary`x`in**R**. If`x`>= 0, then`x`*`x`>= 0. If`x`< 0, then also`x`*`x`>= 0.- Introduce a set
**C**of*complex numbers*such that**R**can be "embedded" into**C**, and- for every complex number
`a`there is a complex number`x`with`a`=`x`*`x`(and consequently sqrt(`a`) is defined).

*Set-theoretic definition on top of R*.

Author: Wolfgang Schreiner

Last Modification: November 16, 1999