Set of Numbers

Sets of Numbers

• Positive integers
  $\mathbb{Z}^{+} = \set{1,2,3,4,...}$
• Natural integers
  $\mathbb{N} = \set{0,1,2,3,4,...}$
• Negative integers
  $\mathbb{Z}^{-} = \set{-1,-2,-3,-4,...}$
• Integers
  $\mathbb{Z} = \mathbb{Z}^{-} \cup \set{0} \cup \mathbb{Z}^{+}$
• Rational numbers
  $\mathbb{Q} = \set{\frac{p}{q} | q \not = 0 \land p, q \in \mathbb{Z}}$
• Irrational numbers
  Limits of sequences of rational numbers (which are not rational numbers)
• Real numbers
  $\mathbb{R} = \mathbb{Q}^{c} \cup \mathbb{Q}$

Complex numbers are taught in a separate set of lectures, and not included under real analysis lectures.

Axiomatic Definiton of Real Numbers

Set of real numbers is a set satisfying all these axioms:

• Field axioms
• Order axioms
• Completeness axiom

Archimedean Property