Number System - Whole Numbers: Properties and Number Line Operations

Whole Numbers: The set of whole numbers, denoted by $W$, includes all natural numbers starting from $1$ along with the number $0$. It is written as $W = \{0, 1, 2, 3, \dots\}$. There is no largest whole number as the sequence is infinite.

The Number Line: Whole numbers are represented on a horizontal line starting from $0$ at the extreme left. Points are marked at equal intervals (units) moving to the right ($0, 1, 2, 3, \dots$). On this line, a number is greater than any number to its left and smaller than any number to its right.

Operations on Number Line: To perform addition ($a + b$), start at $a$ and move $b$ units to the right. To perform subtraction ($a - b$), start at $a$ and move $b$ units to the left. For multiplication ($a \times b$), start from $0$ and take $a$ jumps of $b$ units each (or $b$ jumps of $a$ units) to the right.

Closure Property: Whole numbers are closed under addition and multiplication, meaning the sum or product of any two whole numbers is always a whole number ($a + b \in W$ and $a \times b \in W$). However, they are not closed under subtraction or division as the result may not be a whole number (e.g., $3 - 5 = -2$, which is not a whole number).

Commutative Property: The order of numbers does not change the result for addition ($a + b = b + a$) and multiplication ($a \times b = b \times a$). This property does not apply to subtraction or division.

Associative Property: When adding or multiplying three or more whole numbers, the grouping of numbers does not change the final result: $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$.

Distributive Property: Multiplication distributes over addition and subtraction: $a \times (b + c) = (a \times b) + (a \times c)$ and $a \times (b - c) = (a \times b) - (a \times c)$. This is highly useful for simplifying complex calculations.

Identity Elements: $0$ is the additive identity because adding it to any number does not change the value ($a + 0 = a$). $1$ is the multiplicative identity because multiplying any number by it leaves the number unchanged ($a \times 1 = a$).