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- Not every element has a square root in
**Q**:- ~
**exists**`x`:`x`*`x`= 2. - sqrt(
`x`) :=**such**`z`:`x`=`z`*`z`. - sqrt(2) is undefined.

- ~
*Proof:*see lecture notes.- Introduce a set
**R**of*real numbers*such that**Q**can be "embedded" into**R**, and- for every non-negative
real number
`a`there is a real number`x`with`a`=`x`*`x`(and consequently sqrt(`a`) is defined).

*Axiomatic characterization*.

Author: Wolfgang Schreiner

Last Modification: November 16, 1999