Whole numbers are the numbers we use for counting, starting from 0 and going on forever like 0, 1, 2, 3, 4, and so on. These numbers do not include fractions, decimals, or negative numbers. You can find all whole numbers on a number line, starting at 0 and moving to the right. Whole numbers are a part of the real number system, which means they are used in real-life counting and measuring. However, not all real numbers are whole numbers—because real numbers also include decimals and fractions. In this topic, we learn what whole numbers are, their definition, the symbol used for them, and the properties they follow. These include the closure, commutative, associative, and distributive properties. We also look at addition, multiplication, and how whole numbers relate to natural numbers. You will also see a list of whole numbers from 1 to 100, the smallest whole number, and examples with solutions.

What are Whole Numbers?

Whole numbers are a basic part of the number system and are used often in everyday math and algebra. They include zero and all positive counting numbers like 1, 2, 3, 4, 5, and so on. Whole numbers do not include negative numbers, fractions, or decimals. For example, -2, 1.5, or ¾ are not whole numbers.

In short, whole numbers are made up of zero and all positive integers. These numbers go on forever and are used in many areas of math like addition, subtraction, multiplication, and more. Understanding whole numbers helps in learning more advanced math topics later. You can see whole numbers clearly placed on a number line, starting from 0 and moving to the right.

So, whole numbers are a simple but important group of numbers that form the base for learning many other math concepts.

Definition of Whole Numbers

Any positive number without a fractional or decimal part is referred to as a whole number. This indicates that all whole numbers, such as 0, 1, 2, 3, 4, 5, 6, and 7, are whole numbers. Numbers like -3, 2.7, and \(3{1\over{2}}\) aren’t even close to being whole numbers.

• A whole number, not a negative number (sometimes known as a minus number), must be positive. This implies that it must have a value of 0 or greater. For example, entire numbers are 0, 1, 2, and 3, but -1, -2, and -3 are not.
• Any fractional element cannot be included in a whole integer. That is, figures like 1112, 314, and 756 are not whole numbers, whereas 1, 3, and 7 are. There can’t be any decimal elements in a whole integer. As a result, figures like 3.4, 7.9, and 11.234 are not whole numbers, although 3, 7, and 11 are.
• Whole numbers are those numbers that do not have a fractional or decimal part. Whole numbers begin with the number 0 and include all of the positive integers from 0 to infinity.
• All whole numbers are integers but not all integers are considered to be whole numbers.
• The reason not all integers are whole numbers is that the parameters for an integer include numbers that are negative. For example, integers can include numbers such as -2, -100, and -590. These numbers are not whole numbers because they are negative.

Set of Whole Numbers

The set of natural numbers is usually denoted by the symbol \(\mathbb{N}\).

W = {0, 1,2,3,4,5,6,… }

The set of natural numbers, denoted \(\mathbb{W}\), is a subset of the set of integers, \(\mathbb{W}\).

The set W is a denumerable set. Denumerability refers to the fact that, even though there might be an infinite number of elements in a set, those elements can be denoted by a list that implies the identity of every element in the set. Hence, the set of natural numbers is infinite. It is a superset to a set of even numbers, a set of odd numbers, a set of prime numbers and a set of composite numbers.

Whole Number Symbol

Whole numbers are numbers that have no fractions and are made up of positive integers and zero. The symbol for it is “W,” and the numbers are 0 through 1, 2, 3, 4, 5, 6, 7, 8, 9,…………

Properties of Whole Numbers

There are some properties of whole numbers like closure property, commutative property and associative property. Let us explore these properties on the four binary operations of addition, subtraction, multiplication and division in mathematics. The general properties of operations of whole numbers are as follows:

• Closure Property: The closure property means that a set is closed for some mathematical operation. That is, a set is closed with respect to that operation if the operation can always be completed with elements in the set. Thus, a set either has or lacks closure with respect to a given operation.
  Example: 8 + 10 = 18 is a natural number
• Commutative Property: The commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer. The property holds for Addition and Multiplication, but not for subtraction and division.
  Example: 5 + 4 = 9 & 4 + 5 = 9
  Hence, 2 + 4 = 4 + 2
• Associative Property: The associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.
  Example: 2 + (4 + 1) = 2 + (5) = 7
  (2 + 4) + 1 = (6) + 1 = 7
  Hence, 2 + (4 + 1) = (2 + 4) + 1.
• Multiplicative Identity: Multiplicative identity states that if a number is multiplied to 1 the resultant will be the number itself. 1 is the multiplicative identity of a number.
  Example: -8 x 1 = 1 x (-8) = -8
• Distributive Property: The distributive property of binary operations generalizes the distributive law, which asserts that equality is always true in elementary algebra.
  Example: 3 x (4 + 5) = (3 x 4) + (3 x 5) = 12 + 15 = 27
  2 x (3 + 4) = 2x (7) = 14
  Hence, 2 x (3 + 4) = (2 x 3) + (2 x 4).
  Therefore, Multiplication is distributive over addition

There are 5 properties of natural numbers: Closure Property, Commutative Property, Associative Property, Identity Property and Distributive Property.

Learn about Set Builder Notation

Four fundamental operations on whole number

The Four fundamental operations on whole numbers are Addition, Subtraction, Multiplication and Division, which are as follows:

Addition of Whole Numbers

The addition of two whole numbers results in the total amount or sum of those values combined.

• Closure Property of Addition: The sum of any two whole numbers is always a whole number. This is called the ‘Closure property of addition’ of whole numbers. Thus, N is closed under addition. If a and b are any two whole numbers, then (a + b) is also a whole number.
  Example: 6 + 11 = 17 is a whole numbers
• Commutative Property of Addition: The addition of two whole numbers is commutative. A binary operation is commutative if changing the order of the operands does not change the result. If a and b are any two whole numbers, then i.e. a + b = b + a
  Example: 6 + 7 = 13 & 7 + 6 = 13
  Hence, 2 + 4 = 4 + 2
• Associative Property of Addition: Addition of whole numbers is associative. If a, b and c are any three whole numbers, then a + (b + c) = (a + b) + c
  Example:7 + (4 + 1) = 7+ (5) = 12
  (7 + 4) + 1 = (11) + 1 = 7
  Hence, 7 + (4 + 1) = (7 + 4) + 1.

Learn about Arithmetic Mean and Complex Numbers

Subtraction of whole numbers

After the addition of whole numbers, the next operation is the subtraction of whole numbers. The word subtraction means to take out a number from another number. We know that sometimes subtraction can result in a negative number. However, in the case of whole numbers, this case isn’t possible. The reason is that negative numbers are not whole numbers hence even if we get a question that is resulting in a negative number, we will simply say it is not the case of whole numbers because you are studying subtraction in whole numbers.

• Closure Property of Subtraction: The difference between any two whole numbers need not be a natural number. Hence N is not closed under subtraction.
  Example: 3 – 8 = -5 is a not natural number.
• Commutative Property of Subtraction : The subtraction of two whole numbers is not commutative. If a and b are any two whole numbers, then (a – b) ≠ (b – a)
  Example: 4 – 1 = 3
  1 – 4 = -3
  Hence, 4 – 1 ≠ 1 – 4. Therefore, the commutative property is not true for subtraction.
• Associative Property of Subtraction : Subtraction of whole numbers is not associative. If a, b, c and d are any three whole numbers, then a – (b – d) ≠ (a – b) – d
  Example: 3 – (8 – 2) = 3 – 6 = -3
  (3 – 8) – 2 = -5 – 2 = -7
  Hence, 3 – (8 – 2) ≠ (3 – 8) – 2. Therefore, the associative property is not true for subtraction.

Learn about Greatest Integer Function

Multiplication of whole numbers

The multiplication of a natural number and a sum is equal to the sum of the multiplication of the natural number for each of the addends.

• Closure Property of multiplication : The product of two whole numbers is always a natural number. Hence N is closed under multiplication. If a and b are any two whole numbers, then a x b = ab is also a natural number
  Example: 5 x 7 = 35 is a natural number
• Commutative Property of multiplication : Multiplication of whole numbers is commutative. If a and b are any two whole numbers, then a x b = b x a
  Example: 2 x 8 = 16
  8 x 2 = 16
  Hence, 2 x 8 = 8 x 2. Therefore, commutative property is true for multiplication.
• Associative Property of multiplication : Multiplication of whole numbers is associative. If a, b and c are any three whole numbers, then a x (b x c) = (a x b) x c
  Example:3 x (5 x 6) = 3 x 30 = 90
  (3 x 5) x 6 = 15 x 6 = 90
  Hence, 3 x (5 x 6) = (3 x 5) x 6.Therefore, the associative property is true for multiplication.
• Multiplicative Identity of multiplication : The product of any natural number and 1 is the whole number itself. ‘One’ is the multiplicative identity for whole numbers. If a is any natural number, then a x 1 = 1 x a = a
  Example: 8 x 1 = 1 x 8 = 8
• Distributive Property of Multiplication over Addition : Multiplication of whole numbers is distributive over addition. If a, b and c are any three whole numbers, then a x (b + c) = ab + ac
  Example:3 x (4 + 5) = (3 x 4) + (3 x 5) = 12 + 15 = 27
  2 x (3 + 4) = 2x (7) = 14
  Hence, 2 x (3 + 4) = (2 x 3) + (2 x 4).Therefore, Multiplication is distributive over addition.
• Distributive Property of Multiplication over Subtraction : Multiplication of whole numbers is distributive over subtraction. If a, b and c are any three whole numbers, then a x (b – c) = ab – ac
  Example:2 x (5 – 1) = (2 x 5) – (2 x 1) = 10 – 2 = 8
  2 x (5 – 1) = 2 x (4) = 8
  Hence, 2 x (5 – 1) = (2 x 5) – (2 x 1). Therefore, multiplication is distributive over subtraction.

Division of whole numbers

The division of two numbers has the following form: “dividend : divisor = quotient”. The first number is called the dividend, the second is the divisor and the result is called the quotient.

• Closure Property of Division : When we divide a natural number by another natural number, the result does not need to be a natural number. Hence, N is not closed under multiplication.
  Example: When we divide the natural number 3 by another natural number 2, we get 1.5 which is not a natural number.
• Commutative Property of Division : The division of whole numbers is not commutative. If a and b are two natural then a ÷ b ≠ b ÷ a
  Example: 4 ÷ 1 = 4
  1 ÷ 4 = 0.25
  Hence, 2 ÷ 1 ≠ 1 ÷ 2. Therefore, the Commutative property is not true for division.
• Associative Property : Division of whole numbers is not associative. If a, b and c are any three whole numbers, then a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c
  Example: 5 ÷ (6 ÷ 3) = 5 ÷ 2 = 2.5
  (5 ÷ 6) ÷ 3 = 0.834 ÷ 3 = 0.275
  Hence, 5 ÷ (6 ÷ 3) ≠ (5 ÷ 6) ÷ 3.Therefore, Associative property is not true for division.

Also, learn about Mean Deviation.

Whole Numbers on a Number Line

Whole numbers can be easily shown on a number line. A number line is a straight line where numbers are placed in order. On this line:

• All the numbers to the right of 0 (like 1, 2, 3, etc.) are natural numbers.
• When we include 0 along with these natural numbers, we get the set of whole numbers.

So, on the number line:

• Natural numbers = 1, 2, 3, 4, …
• Whole numbers = 0, 1, 2, 3, 4, …

In short, whole numbers are natural numbers plus zero, and they are shown starting from 0 and moving right on the number line.

Smallest Whole Number

Whole numbers start at 0 and go to infinity, without including negative values. The smallest whole number is zero because there is no positive number less than 0. Zero is the integer denoted 0 that, when used as a counting number, means that no objects are present. It’s the only integer (and, by extension, the only real number) that’s neither negative nor positive. Nonzero refers to a number that is not zero. A zero of a function is also known as a root of a function. Zero is a whole number and so an integer, it is known as a neutral integer because it is neither negative nor positive. 0 is an integer because it is a whole number that can be expressed without a remainder.

Key Facts About Whole Numbers

• Zero (0) is counted as a whole number, but it is not a natural number.
• The first few whole numbers are: 0, 1, 2, 3, and 4.
• The smallest whole number is 0.
• Whole numbers do not include:
  • Negative numbers (like -1, -2, etc.)
  • Fractions (like ½ or ¾)
  • Decimals (like 2.5 or 3.1).

Whole Numbers vs Integers 

Whole numbers and integers are both sets of numbers used in mathematics, but they differ in their range and elements.

1. Inclusion of Zero: Both whole numbers and integers include the number 0. This makes 0 a part of both sets.
2. Positive Numbers:
   Both sets contain all positive counting numbers like 1, 2, 3, and so on.
3. Negative Numbers:
   This is the key difference between the two.
   • Whole numbers do not include any negative numbers.
   • Integers include all negative numbers such as -1, -2, -3, etc., along with zero and positive numbers.
4. Fractions and Decimals:
   Neither whole numbers nor integers include fractions or decimal values. They are made up of complete values only (no parts or points).
5. Set Representation:

• The set of whole numbers is written as:
• Set of Whole Numbers: {0, 1, 2, 3, …}
• Set of Integers: {…, -3, -2, -1, 0, 1, 2, 3, …}

Whole Numbers 1 to 100

Now let’s see the first 100 Whole numbers starting from 1 in their numerical as well as their alphabetical form.

Difference Between Natural Numbers and Whole Numbers

The difference between a natural number and whole numbers are as follows:

Solved Examples of Whole Numbers

Example 1: Find the product using distributive property: (a) 237 × 103

Solution: 237 × 103

237 × (100 + 3)

Property: a × (b + c) = a × b + a × c

Therefore, 237 × (100 + 3)

= 237 × 100 + 237 × 3

= 23700 + 711

= 24411

Example 2: Suresh scored 48 runs in the first innings and 72 runs in the second innings.
Ramesh scored 72 runs in the first innings and 48 runs in the second innings.
Who had a higher total score?

Solution: 
Suresh's total score = 48 + 72 = 120
Ramesh's total score = 72 + 48 = 120

We observe that the scores are the same, just added in different order.
According to the commutative property of addition,
a + b = b + a, so the total remains the same.

Answer:
Both Suresh and Ramesh had equal total scores of 120 runs.

Example 3: Say whether each of the following statements is true or false for whole numbers.

a.) 5 is a whole number.
b.) Every whole number is positive.
c.) Whole numbers do not include fractions.
d.) -2 is a whole number.
e.) All whole numbers are real numbers.

Solution:

a.) True – 5 is a whole number.
b.) False – 0 is a whole number but it is not positive.
c.) True – Whole numbers do not include fractions or decimals.
d.) False – Negative numbers like -2 are not whole numbers.
e.) True – All whole numbers belong to the set of real numbers.

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