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

$$\forall y\in\mathbb{R^+} \; \exists k \in \mathbb{Z}^+ \; \text{s.t.} \; \frac{1}{k} \lt y$$