Whole Numbers: Definition, Properties and Examples

Whole numbers are the natural numbers together with zero: 0, 1, 2, 3, 4, and so on. They answer the question "how many?" even when the answer is "none" — for example, zero runs scored in an over, or zero absentees in a class. The set is written as W = {0, 1, 2, 3, …}.

Whole numbers matter because zero changes the rules of arithmetic in important ways. The idea of an additive identity — a number you can add without changing anything — only appears once zero joins the number family.

Key Concepts

• Definition: whole numbers are 0, 1, 2, 3, … — all natural numbers plus zero, with no fractions or negatives.
• Additive identity: 0, because a + 0 = a for every whole number a.
• Multiplicative identity: 1, because a × 1 = a for every whole number a.
• Closure: the sum and product of two whole numbers is always a whole number.
• Commutative and associative: whole numbers can be added or multiplied in any order or grouping.
• Distributive property: a × (b + c) = (a × b) + (a × c), which links multiplication with addition.

Formulas

• Successor of n: n + 1 (for example, the successor of 9 is 10).
• Predecessor of n: n − 1, defined for every whole number except 0.
• Distributive law: a × (b + c) = ab + ac, valid for all whole numbers a, b, c.
• Sum of first n whole numbers (0 to n): n(n + 1) ÷ 2, same as for natural numbers since adding 0 changes nothing.

Worked Example

Verify the distributive property with a = 4, b = 3, c = 5. Left side: 4 × (3 + 5) = 4 × 8 = 32. Right side: (4 × 3) + (4 × 5) = 12 + 20 = 32. Both sides equal 32, so the property holds. As a second check, the predecessor of 1 is 1 − 1 = 0, which is a whole number — but 0 itself has no predecessor in the whole numbers, because −1 is not a whole number.

Common Misconceptions

• "0 divided by a number is undefined." Actually 0 ÷ 5 = 0 is perfectly fine. It is division by zero, like 5 ÷ 0, that is undefined.
• "Whole numbers include negative numbers." They do not. Negatives belong to integers, the next larger set.
• "Every whole number has a predecessor." Every whole number except 0 has one; 0 has no predecessor among whole numbers.

Real-World Applications

• Scores and counts that can be zero: runs in an over, goals in a match, items left in stock.
• Digital systems: binary digits are 0 and 1, both whole numbers.
• Calendars and clocks use whole numbers, including 0 in 24-hour time.

To sum up, whole numbers are the natural numbers plus zero, giving us the additive identity 0 and keeping closure under addition and multiplication. Remember the distributive law a × (b + c) = ab + ac, and that only 0 lacks a predecessor in this set.