Number System - Integers and their Properties (Closure, Commutative, Associative, Distributive)

Integers consist of the set of whole numbers and their negative counterparts, represented as $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$. On a horizontal number line, $0$ acts as the origin; positive integers extend to the right (increasing in value), and negative integers extend to the left (decreasing in value).

The Closure Property states that for any two integers $a$ and $b$, the results of addition ($a + b$), subtraction ($a - b$), and multiplication ($a \times b$) are always integers. However, division does not satisfy the closure property because an integer divided by another (like $5 \div 2$) may result in a fraction.

The Commutative Property implies that the order of integers does not affect the outcome of addition or multiplication. Mathematically, $a + b = b + a$ and $a \times b = b \times a$. Imagine a balance scale where swapping the positions of two weights does not change the equilibrium. Note: This property does not hold for subtraction or division.

The Associative Property refers to the grouping of three or more integers. For addition, $(a + b) + c = a + (b + c)$, and for multiplication, $(a \times b) \times c = a \times (b \times c)$. Visually, this means shifting parentheses within an expression does not change the final sum or product.

The Distributive Property of Multiplication over Addition/Subtraction states that $a \times (b + c) = (a \times b) + (a \times c)$. You can visualize this as finding the area of a large rectangle by splitting it into two smaller rectangles and adding their areas together.

The Additive Identity of an integer is $0$, because adding $0$ to any integer $a$ results in the integer itself ($a + 0 = a$). The Multiplicative Identity is $1$, because multiplying any integer $a$ by $1$ keeps the value unchanged ($a \times 1 = a$).

Sign Rules for Multiplication and Division dictate that the product or quotient of two integers with like signs is always positive ($+ \times + = +$ or $- \times - = +$), whereas the product or quotient of two integers with unlike signs is always negative ($+ \times - = -$ or $- \times + = -$).