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Motivation

• 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
  1. Q can be "embedded" into R, and
  2. 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.