Number Sets
Table of Contents

We will look at some common number sets that will arise frequently in the modern algebra section:
Symbol for Number Set  Name of Number Set  Examples 

$\mathbb{R}$  Real Numbers  1922.33 2, 0, 1.331, $\pi$, 3/2, … 
$\mathbb{Q}$  Rational Numbers  All real numbers q that can be written in the form $q = \frac{a}{b}$ where b ≠ 0. For example, 2/5, 9/2, 3/1, 27/6, … 
$\bar{\mathbb{Q}}$ or $\mathbb{R}  \mathbb{Q}$  Irrational Numbers  All real numbers q that cannot be written in the form $q = \frac{a}{b}$. For example, $\pi, \tau, e$, … (Note that notation for irrational numbers is not standard). 
$\mathbb{N}$  Natural Numbers  All whole numbers starting from 1 upwards, for example 1, 2, 3, … (Note that sometimes the natural numbers are defined to include 0, while other times they are not). 
$\mathbb{Z}$  Integers  All positive and negative whole numbers including zero, for example …, 2, 1, 0, 1, 2, … 
$\mathbb{Z}^+$  Positive Integers  All positive whole numbers, for example 1, 2, 3, … 
$\mathbb{I}$  Imaginary Numbers  Any number in the form $\sqrt{a}$ where a is negative, for example $\sqrt{3}$ 
$\mathbb{C}$  Complex Numbers  Any number in the form $z = a + bi$ where "a" is considered the real part and "bi" is considered the imaginary part ($i = \sqrt{1}$). For example, 3 + 2i, 4 + 0i, 0 + 2i, etc… 