Whole Numbers: Definition, Properties, Formulas and Examples
Whole Numbers are the non-negative integers starting from 0 and continuing without an end: 0, 1, 2, 3, and so on. This page covers their definition, number-line representation, predecessor and successor, arithmetic properties, formulas, whole numbers examples, and common whole numbers problems.
What Are Whole Numbers?
The set of whole numbers is written as W = {0, 1, 2, 3, ...}. It is an infinite set because there is no greatest whole number. Every natural number is a whole number, but 0 is a whole number that is usually not included in the set of natural numbers.
Whole Numbers Video Lessons
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Types of Numbers Classification
Understand the classification of numbers, including natural, whole, integer, rational, irrational, real and complex numbers, and see how these categories are related in the classification tree.
Whole Numbers Formula & Tricks
Important Formulas
The successor is the whole number immediately after n.
The predecessor is the whole number immediately before n. Zero has no predecessor in the set of whole numbers.
This formula gives the sum of all whole numbers from 0 to n.
The endpoints are included in the count. For example, from 4 to 9, the count is 9 - 4 + 1 = 6.
Quick Tricks
Multiplying any whole number by 0 gives 0. Division by 0 is undefined, so 0 cannot be used as a divisor.
For consecutive whole numbers, subtract the lower endpoint from the upper endpoint and add 1.
Addition and multiplication of whole numbers always produce whole numbers. Subtraction and division may produce a negative number, fraction, or undefined result.
Whole Numbers Concepts
Representation on the Number Line
Each number to the right is 1 greater than the previous number. Therefore, 7 is greater than 4 because 7 lies farther to the right. The distance between two whole numbers a and b, where b ≥ a, is b - a units.
Successor and Predecessor
For a whole number n, successor = n + 1. For n > 0, predecessor = n - 1. Zero has a successor, 1, but it has no predecessor within the whole-number set because -1 is not a whole number.
Properties of Addition and Multiplication
Closure means the result remains a whole number. For whole numbers a, b, and c: a + b and a × b are whole numbers; a + b = b + a; a × b = b × a; (a + b) + c = a + (b + c); and (a × b) × c = a × (b × c). The distributive property is a × (b + c) = a × b + a × c.
Closure Under Subtraction and Division
Subtraction may give a negative number, and division may give a fraction or may be undefined when the divisor is zero. A subtraction result is a whole number only when the minuend is at least as large as the subtrahend. A division result is a whole number only when the divisor is non-zero and divides the dividend exactly.
Even and Odd Whole Numbers
Even whole numbers have the form 2k, and odd whole numbers have the form 2k + 1, where k is a whole number. The last digit of an even number is 0, 2, 4, 6, or 8. The last digit of an odd number is 1, 3, 5, 7, or 9.
Whole Numbers Data Table
Use the structured data below while following the explanation and examples.
Whole Numbers and Related Number Sets
Comparison of whole numbers with natural numbers and integers
| Number set | Meaning | Examples |
|---|---|---|
| Whole numbers | Non-negative integers including 0 | 0, 1, 2, 3 |
| Natural numbers | Counting numbers; commonly starts from 1 | 1, 2, 3, 4 |
| Integers | Positive numbers, negative numbers and 0 | -3, -2, -1, 0, 1 |
Note: Some mathematical conventions include 0 in the natural numbers, but whole numbers always include 0.
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Whole Numbers Revision Points
Key rules and formulas for quick revision
- Whole numbers are W = {0, 1, 2, 3, ...}.
- Zero is the smallest whole number, and there is no greatest whole number.
- Successor of n is n + 1.
- Predecessor of n is n - 1 for n > 0; zero has no whole-number predecessor.
- Whole numbers are closed under addition and multiplication.
- Whole numbers are not closed under subtraction and division.
- Division by zero is undefined.
- The number of whole numbers from a to b, inclusive, is b - a + 1.
Whole Numbers FAQs
Is 0 a whole number?
Yes. Zero is the smallest whole number, and the whole-number set is W = {0, 1, 2, 3, ...}.
What is the difference between whole numbers and natural numbers?
Whole numbers include 0 and all positive integers. Natural numbers commonly begin with 1, although some conventions also include 0.
What is the sum of the whole numbers from 0 to 20?
Use n(n + 1) / 2: 20 × 21 / 2 = 210.
How many whole numbers are there from 15 to 32 inclusive?
The count is 32 - 15 + 1 = 18.
Are whole numbers closed under subtraction?
No. For example, 3 - 8 = -5, and -5 is not a whole number.
Are whole numbers closed under division?
No. For example, 5 ÷ 2 = 2.5, which is not a whole number. Division by zero is also undefined.