Number classes

SectionA.1Number classes

Although mathematicians ultimately regard the fundamental undefined object as the set, in the context of this course we will ultimately work with several mathematical

DefinitionA.1.1

We will take several familiar number classes as objects which we assume we already understand. These are:

natural numbers, $\mathbb{N}$: the set of positive integers (also known as whole numbers or counting numbers), which we enumerate as $1, 2, 3,$ … We will not define these at any point during the course.
integers, $\mathbb{Z}$: the set of integers, including those positive, negative and zero, which we enumerate as … $-2, -1, 0, 1, 2,$ … We will not formally define these at any point in the course, although we will discuss how they could be defined in terms of the set $\mathbb{N}\text{.}$
rational numbers, $\mathbb{Q}$: the set of rational numbers, i.e., those numbers which may be written as a fraction of two integers of which the denominator is nonzero. After covering equivalence relations, we will formally define $\mathbb{Q}$ in terms of $\mathbb{Z}\text{.}$
algebraic numbers, $\overline{\mathbb{Q}}$: the set of algebraic numbers, i.e., those numbers which are roots of a polynomial with integer coefficients. For example, $\sqrt{2}$ is algebraic because it is a root of the polynomial $x^2 - 2\text{.}$ Similarly, the imaginary unit $i$ is algebraic because it is a root of $x^2 + 1\text{.}$ Although it is not easy to prove, $\pi$ is not an algebraic number. While you may not be familiar with algebraic numbers, they are in many ways a more natural class to work with than the real numbers.
real numbers, $\mathbb{R}$: the set of real numbers, i.e., those numbers which correspond to all possible places on a number line, or alternatively, all numbers represented by infinite decimal expansions. These include, for example, $\sqrt{2}, \pi, e$ and others, but not any numbers involving the imaginary unit $i\text{.}$ If time allows and there is sufficient interest, we will cover the construction of the real numbers at the end of the semester.
complex numbers, $\mathbb{C}$: the set of complex numbers, i.e., all those numbers of the form $a + bi$ where $a,b \in \mathbb{R}$ and $i$ is the imaginary unit satisfying the relation $i^2 = -1\text{.}$ These can also be defined as those numbers which are roots of a polynomial with real coefficients (in fact, we can even restrict the polynomial to have degree two or less). This set includes all previous classes mentioned.

DefinitionA.1.2Even integers

An integer $n \in \mathbb{Z}$ is said to be even if it can be written as $n = 2k$ for some integer $k \in \mathbb{Z}\text{.}$

DefinitionA.1.3Odd integers

An integer $n \in \mathbb{Z}$ is said to be odd if it can be written as $n = 2k + 1$ for some integer $k \in \mathbb{Z}\text{.}$

ExerciseA.1.4Parity

Prove that every integer is either even or odd, but never both.

TheoremA.1.5Properties of number classes

Let $\mathbb{F}$ denote any of the number classes mentioned in Definition 1. Then

$\mathbb{F}$ is closed under addition and multiplication. That is, if $x,y \in \mathbb{F}$ then $x+y, xy \in \mathbb{F}\text{.}$
Addition and multiplication are commutatitive in $\mathbb{F}\text{.}$ That is, if $x,y \in \mathbb{F}$ then $\begin{equation*} x+y = y+x, \qquad xy = yx. \end{equation*}$
Addition and multiplication are associative in $\mathbb{F}\text{.}$ That is, if $x,y \in \mathbb{F}$ then $\begin{equation*} (x+y)+z = x+(y+z), \qquad (xy)z = x(yz). \end{equation*}$
Multiplication distributes over addition in $\mathbb{F}\text{.}$ That is, if $x,y \in \mathbb{F}$ then $\begin{equation*} (x+y)z = xz+yz, \qquad z(x+y) = zx+zy. \end{equation*}$
For any class $\mathbb{F}$ except for $\mathbb{N}\text{,}$ additive inverses exist. That is, for any $x \in \mathbb{F}\text{,}$ there exists $y \in \mathbb{F}$ (namely, $y = -x$ ) so that $\begin{equation*} x+y = 0. \end{equation*}$
For any class $\mathbb{F}$ except for $\mathbb{N}, \mathbb{Z}\text{,}$ every nonzero element has a multiplicative inverse. That is, for any $x \in \mathbb{F}\text{,}$ if $x \not= 0\text{,}$ then there exists $y \in \mathbb{F}$ (namely, $y = \frac{1}{x}$ ) so that $\begin{equation*} xy = 1. \end{equation*}$

DefinitionA.1.6Divisibility

For integers $a,b \in \mathbb{Z}\text{,}$ we say $a$ divides $b$ if there is some integer $k \in \mathbb{Z}$ for which $b = ak\text{.}$ In this case we say $a$ is a divisor or multiple of $b\text{.}$

DefinitionA.1.7Prime

A natural number $p \in \mathbb{N}$ is said to be prime if $p \gt 1$ and the only divisors of $p$ are $1$ and itself. A natural number greater than $1$ which is not prime is said to be composite.