Calculus of a Single Variable

Chapter 0: Numbers & Functions

0.1 Some Types of Numbers

The set of natural numbers $\mathbb{N} = \{1,2,3... \}$ ; you can add two natural numbers together to obtain another natural number, but what about subtraction?
The set of integers $\mathbb{Z}=\{..., -3,-2,-1,0, 1, 2, 3,...\}$ ; you can add or subtract two integers to obtain another integer, but what about division?
The set of rational numbers $\mathbb{Q}=\{\frac{p}{q}\,:\,p\in\mathbb{Z},\,q\in\mathbb{N}\}$ ; you can add, subtract, multiply and divide two rational numbers two obtain another rational number. However there is no number $x\in\mathbb{Q}$ such that $x^{2}=2$ . Thus, in view of Pythagoras’ theorem, rational numbers are not adequate for describing the lengths of line segments.

0.2 The Set of Real Numbers The set of real numbers $\mathbb{R}$ consists of all numbers of the form $\pm\, a_{0}\,.\,a_{1}\,a_{2}\,a_{3}\,...$ where $a_{0}$ is a non-negative integer and $a_{1}\,,a_{2}\,,...$ are digits $(0,1,...,9)$ . The set $\mathbb{R}$ includes numbers like $\sqrt{2}$ and $\sqrt{3}$ , which do not belong $\mathbb{Q}$ . Such numbers are called irrationals. We have the familiar operations of arithmetic on $\mathbb{R}$ , and the familiar ordering $<$ . The inequality $a<b$ corresponds to saying that $a$ is to the left of $b$ on the real number line.