The World of Numbers - Integers: Expanding the Horizon

Integers are a set of numbers consisting of natural numbers ($1, 2, 3, \dots$), zero ($0$), and the negatives of natural numbers ($-1, -2, -3, \dots$). The set is denoted by the symbol $\mathbb{Z}$.

On a number line, integers to the right of $0$ are positive, and those to the left are negative. As we move right, the value increases; as we move left, the value decreases.

The Absolute Value of an integer $a$, denoted by $|a|$, is its numerical value regardless of its sign. For example, $|-5| = 5$ and $|5| = 5$.

Closure Property: For any two integers $a$ and $b$, the results of $a + b$, $a - b$, and $a \times b$ are always integers. However, division $\frac{a}{b}$ does not always result in an integer.

Commutative Property: Addition and multiplication are commutative ($a + b = b + a$ and $a \times b = b \times a$). Subtraction and division are not commutative.

Associative Property: Addition and multiplication are associative, meaning $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$.

Distributive Property: Multiplication distributes over addition and subtraction: $a \times (b + c) = (a \times b) + (a \times c)$.

Additive Identity: Zero is the additive identity because $a + 0 = a$. Additive Inverse: For every integer $a$, there exists $-a$ such that $a + (-a) = 0$.