Natural Numbers and Integers
1. What Are Natural Numbers?
The natural numbers (also called counting numbers) are the most fundamental set of numbers in mathematics. They are the numbers we use to count objects.
Formal Definition: The set of natural numbers is denoted by the symbol $\mathbb{N}$ and is defined as: $$\mathbb{N} = \{1, 2, 3, 4, 5, 6, 7, 8, \ldots\}$$
Note: Some mathematicians include $0$ in the natural numbers, writing $\mathbb{N}_0 = \{0, 1, 2, 3, \ldots\}$. In this course, we will consider natural numbers to start from $1$ unless stated otherwise.
Key Properties of Natural Numbers:
| Property | Description | Example |
|---|---|---|
| Closure under Addition | Adding two natural numbers always gives a natural number | $3 + 5 = 8 \in \mathbb{N}$ |
| Closure under Multiplication | Multiplying two natural numbers always gives a natural number | $4 \times 7 = 28 \in \mathbb{N}$ |
| NOT Closed under Subtraction | Subtracting natural numbers may NOT give a natural number | $3 - 5 = -2 \notin \mathbb{N}$ |
| Well-Ordered | Every non-empty subset has a smallest element | Smallest natural number is $1$ |
2. What Are Integers?
The integers extend the natural numbers by including zero and negative whole numbers. This extension was necessary to solve equations like $x + 5 = 3$.
Formal Definition: The set of integers is denoted by the symbol $\mathbb{Z}$ (from the German word Zahlen, meaning "numbers") and is defined as: $$\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$$
The Three Subsets of Integers:
Negative Integers
$\mathbb{Z}^- = \{\ldots, -3, -2, -1\}$
Numbers less than zero
Zero
$\{0\}$
Neither positive nor negative
Positive Integers
$\mathbb{Z}^+ = \{1, 2, 3, \ldots\}$
Same as natural numbers
Properties of Integers:
| Property | Addition | Multiplication |
|---|---|---|
| Closure | $a + b \in \mathbb{Z}$ | $a \times b \in \mathbb{Z}$ |
| Commutative | $a + b = b + a$ | $a \times b = b \times a$ |
| Associative | $(a+b)+c = a+(b+c)$ | $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ |
| Identity | $a + 0 = a$ | $a \times 1 = a$ |
| Inverse | $a + (-a) = 0$ | Not always in $\mathbb{Z}$ |
| Distributive | $a \times (b + c) = a \times b + a \times c$ | |