- 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.
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)
Set-theoretic definition on top of R.
Author: Wolfgang Schreiner
Last Modification: November 16, 1999