Whole Numbers- Definition and Properties

What Are Whole Numbers?

Whole numbers are the numbers you use every day to count things. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and so on — that's your whole number set. No fractions. No decimals. No negatives. Just clean, whole values sitting on the number line.

The definition is straightforward: whole numbers include zero and all positive integers. They start at zero and go infinitely upward. There's no upper limit.

That's it. That's the definition. Now let's clear up the confusion that trips most people up.

Whole Numbers vs Natural Numbers

This is where students get tangled. Math textbooks don't agree on whether zero counts as a natural number.

Natural numbers = sometimes start at 1, sometimes at 0. Depends on who you ask.

Whole numbers = always include 0, then 1, 2, 3, and beyond. Always.

Here's the practical breakdown:

• If your textbook says natural numbers start at 1 → whole numbers = natural numbers + zero
• If your textbook says natural numbers start at 0 → whole numbers = natural numbers

Check your course materials. Your teacher decides which definition applies. Don't guess.

Properties of Whole Numbers

Whole numbers follow specific rules. These properties determine what happens when you add, subtract, multiply, or divide them. 📐

Closure Property

Whole numbers are closed under addition and multiplication. This means:

• Add any two whole numbers → you get a whole number
• Multiply any two whole numbers → you get a whole number

Example: 7 + 4 = 11 ✓ (whole number)

Example: 7 × 4 = 28 ✓ (whole number)

But here's the catch: whole numbers are not closed under subtraction or division. You can subtract 3 from 5 and get 2 (whole). But subtract 5 from 3 and get -2 (NOT whole). Division has the same problem — 7 ÷ 2 = 3.5 (NOT whole).

Commutative Property

Order doesn't matter for addition and multiplication.

• 4 + 7 = 7 + 4 = 11 ✓
• 4 × 7 = 7 × 4 = 28 ✓

Subtraction and division? Not commutative. 7 - 4 ≠ 4 - 7. Simple as that.

Associative Property

How you group numbers doesn't change the result for addition and multiplication.

• (4 + 7) + 2 = 4 + (7 + 2) = 13 ✓
• (4 × 7) × 2 = 4 × (7 × 2) = 56 ✓

Again, subtraction and division don't follow this rule. Grouping matters there.

Distributive Property

Multiplication distributes over addition. This one matters for factoring and expanding expressions.

Formula: a × (b + c) = (a × b) + (a × c)

Example: 3 × (4 + 5) = 3 × 9 = 27

And: (3 × 4) + (3 × 5) = 12 + 15 = 27 ✓

This property works both ways. Use it to simplify calculations or break down complex problems.

Identity Properties

Additive Identity: Zero is the additive identity. Any whole number plus 0 equals that number.

23 + 0 = 23 ✓

Multiplicative Identity: One is the multiplicative identity. Any whole number times 1 equals that number.

23 × 1 = 23 ✓

Memorize these. They're the foundation for everything that comes later in algebra.

Whole Numbers on a Number Line

Whole numbers occupy a specific spot on the number line. They start at zero and extend infinitely to the right.

Each whole number is exactly 1 unit apart from its neighbors. The spacing is uniform.

Negative numbers sit to the left. Fractions and decimals sit between the whole numbers. Whole numbers don't overlap with either group.

How to Work with Whole Numbers

Here's the practical part. How do you actually use this knowledge?

Step 1: Identify Whole Numbers in a Problem

Look at the numbers given. Check for:

• Positive values ✓
• Zero ✓
• No negative signs
• No fractions or decimal parts

If all conditions check out, you're working with whole numbers.

Step 2: Choose the Right Property

Before solving, ask yourself what operation you're performing.

• Adding or multiplying? Order and grouping don't matter — rearrange for convenience.
• Subtracting or dividing? Order matters. Be careful.

Step 3: Check Your Answer

After solving, verify your result is still a whole number (unless the problem involves division, where fractions are expected).

Quick Example

Solve: 47 × (12 + 8)

Using distributive property:

47 × 12 + 47 × 8 = 564 + 376 = 940

Or just compute inside the parentheses first:

47 × 20 = 940

Same answer. Both methods work. Pick whichever is faster for you.

Common Mistakes to Avoid

• Assuming whole numbers are closed under division. They're not. 12 ÷ 5 = 2.4 — not a whole number.
• Confusing whole numbers with integers. Integers include negatives. Whole numbers do not.
• Forgetting the zero property. Any number × 0 = 0. This one trips people up constantly.
• Applying commutative property to subtraction. 10 - 3 ≠ 3 - 10.

Quick Reference Table

Summary

Whole numbers are 0, 1, 2, 3, and everything after. They include zero. They exclude negatives, fractions, and decimals.

They follow specific properties under addition and multiplication: closure, commutativity, associativity, and distributivity. They don't follow these properties under subtraction and division.

Know the difference between whole numbers, natural numbers, and integers. Know which properties apply where. That's the whole game. 🎯