Whole Numbers - Introduction to Whole Numbers

Natural Numbers and Whole Numbers: The counting numbers $1, 2, 3, \dots$ are called natural numbers. When we include $0$ to the set of natural numbers, we get the collection of Whole Numbers $0, 1, 2, 3, \dots$. Visually, this can be represented as a set where natural numbers are a subset inside the larger circle of whole numbers.

Predecessor and Successor: Given any whole number $n$, the number that comes immediately after it is its successor, calculated as $n + 1$. The number that comes immediately before it (except for $0$) is its predecessor, calculated as $n - 1$. For example, the successor of $19$ is $20$, and the predecessor of $19$ is $18$. Note that the whole number $0$ has no predecessor.

The Number Line: A number line for whole numbers is a horizontal line starting from a point marked $0$ and extending infinitely to the right. Points are marked at equal intervals and labeled $1, 2, 3, \dots$. The distance between any two consecutive points is called a unit distance. Numbers increase as we move to the right and decrease as we move to the left.

Operations on the Number Line: Addition is represented by moving to the right; for $4 + 3$, start at $4$ and move $3$ units right to land on $7$. Subtraction is represented by moving to the left; for $7 - 2$, start at $7$ and move $2$ units left to reach $5$. Multiplication is shown as equal jumps from $0$; for $2 \times 3$, make $3$ jumps of $2$ units each starting from $0$.

Closure Property: Whole numbers are 'closed' under addition and multiplication. This means if $a$ and $b$ are whole numbers, then $(a + b)$ and $(a \times b)$ will always be whole numbers. However, they are not closed under subtraction or division, as $3 - 5$ or $5 \div 2$ do not result in whole numbers.

Commutative and Associative Properties: Addition and multiplication are commutative ($a + b = b + a$) and associative $( (a + b) + c = a + (b + c) )$. This means the order or grouping of numbers does not change the result. Visually, a grid of $3 \times 4$ dots contains the same number of dots as a grid of $4 \times 3$ dots.

Identity Elements: The number $0$ is the additive identity because $a + 0 = a$ for any whole number $a$. The number $1$ is the multiplicative identity because $a \times 1 = a$. Adding zero or multiplying by one leaves the original value unchanged.

Distributive Property: Multiplication distributes over addition. This is written as $a \times (b + c) = (a \times b) + (a \times c)$. Visually, this is like calculating the total area of two adjacent rectangles with the same height $a$ but different widths $b$ and $c$.