Whole Numbers: As kids, we all have studied numbers and integers. We use numbers to count things. There are various kinds of numbers in a number system, such as; natural numbers, whole numbers, real numbers, odd and even numbers, etc. A collection of numbers that includes all positive integers and 0 is known as a whole number.
Whole numbers are real numbers that do not contain fractions, decimals, or negative values. Examples of Whole numbers are, 0,1,2,3,4,5,6……so on. We make use of numbers in our everyday life for telling the cost of items, telling time, counting objects, representing or exchanging money, measuring the temperature, etc. Let us study more about whole numbers definition and examples, properties of whole numbers and more.

Natural numbers refer to a set of positive integers and on the other hand, natural numbers and zero \(\left( 0 \right)\) form a set, referred to as whole numbers. In other words, every student must know that all natural numbers are whole numbers.

Whole Numbers are the set of natural numbers and \(0.\) The set of whole numbers is written as \(\left\{{0,1,2,3,…} \right\}.\)

Properties of Whole Numbers

The basic operations on whole numbers are addition, subtraction, multiplication, and division, which further leads to four main properties of whole numbers that are listed below:

1. Closure Property
2. Associative Property
3. Commutative Property
4. Distributive Property

1. Closure Property

When two whole numbers \(a\) and \(b\) are added or multiplied, the result \(\left({a + b} \right)\) or ab will always be a whole number.
Example: \(6 + 3 = 9\) (whole number), \(9 \times 2 = 18\) (whole number)
The closure property is not true in the subtraction of whole numbers.
Example: \(6 – 3 = 3\) (whole number), \(3 – 6 = – 3\) (not a whole number)

2. Commutative Property

The sum or the product of two whole numbers remain the same even after interchanging the order of the numbers.
Let \(a\) and \(b\) be two whole numbers, according to the commutative property \(a + b = b + a\) or \(ab = ba.\)
Example: Sum, \(11 + 7 = 18\) and \(7 + 11 = 18.\)
Product, \(7 \times 6 = 42\) and \(6 \times 7 = 42.\)
This property does not hold good for subtraction or division.
Example: Difference, \(7 – 2 = 5\) and \(2 – 7 = – 5,\) which is not a whole number.
Division, \(\frac{4}{2} = 2\) and \(\frac{2}{4} = \frac{1}{2},\) which is not a whole number.

3. Associative Property

The sum or product of three whole numbers remains unchanged by grouping the numbers in any order.
Let \(a,b\) and \(c\) be three whole numbers, according to the associative property \(\left({a + b} \right) + c = a + \left({b + c} \right)\) or \(a \times \left({b \times c} \right) = \left({a \times b} \right) \times c.\)
Example: When we add three numbers, say, \(9,11,7,\) we get the same sum:
\(9 + \left({11 + 7} \right) = 27\)
\(\left({9 + 11} \right) + 7 = 27\)
\(\left({9 + 7} \right) + 11 = 27\)
Similarly, when we multiply any three numbers \(7,9,2,\) we get the same product no matter how the numbers are grouped:
\(7 \times \left({9 \times 2} \right) = 126\)
\(\left({7 \times 9} \right) \times 2 = 126\)
\(\left({2 \times 7} \right) \times 9 = 126\)
The associative property is not true for subtraction and division.

The associative property is not true for subtraction and division.

4. Distributive Property

This property states that the multiplication of a whole number is distributed over the sum of the whole numbers.

Consider \(b\) and \(c\) are multiplied with the same number \(a\) and the products are added, then \(a\) can be multiplied with the sum of \(b\) and \(c\) to get the same answer. This situation can be represented as: \(a \times \left({b + c} \right) = \left({a \times b} \right) + \left({a \times c} \right).\)
Example: \(a = 8,b = 12\) and \(c = 2 \Rightarrow 8 \times \left({12 + 2} \right) = 112\) and \(\left({8 \times 12} \right) + \left({8 \times 2}\right) = 112.\)
The distributive property is true for subtraction as well.
\( \Rightarrow a \times \left({b – c} \right) = \left({a \times b} \right) – \left({a \times c} \right).\)
Example: \(a = 8,b = 12\) and \(c = 2 \Rightarrow 8 \times \left({12 – 2} \right) = 80\) and \(\left({8 \times 12} \right) – (8 \times 2) = 80.\)

Special Properties of Whole Numbers

There are some special properties of whole numbers other than the main properties discussed in the above section,

1. Additive identity
2. Multiplicative identity
3. Multiplication by zero
4. Division by zero

1. Additive Identity

When a whole number is added to \(0,\) its value remains unchanged, i.e., if \(x\) is a whole number, then \(x + 0 = 0 + x = x.\)
Example: \(4 + 0 = 4 = 0 + 4.\)
The number \(0\) is called the additive identity for the whole number.

2. Multiplicative identity

Any whole number multiplied by \(1\) gives the same value of the whole number, i.e., if \(x\) is a whole number, then \(x \times 1 = x = 1 \times x.\)
Example: \(6 \times 1 = 6 = 1 \times 6.\)
The number \(1\) is called the multiplicative identity for the whole number.

3. Multiplication by Zero

Any whole number multiplied by \(0\) always gives \(0,\) i.e., \(x \times 0 = 0 = 0 \times x.\)
Example: \(5 \times 0 = 0 = 0 \times 5.\)

4. Division by Zero

Division of a whole number by \(0\) is not defined, i.e., if \(x\) is a whole number, then \(\frac{x}{0}\) is not defined.

LEARN ABOUT INDIAN NUMBER SYSTEM

What are Natural Numbers?

The natural numbers are \(1,2,3,4…..\) So, the collection of numbers starting from 1 to infinity is called natural numbers. Natural numbers are also known as counting numbers.

Suppose a group of students are going on a trip, and the teacher wants to count the number of students. She starts from \(1,2,3,\) and so on. Thus, natural numbers are counting numbers.

Also check,

Odd Numbers	Even Numbers	Rational Numbers
Composite Numbers	Real Numbers	Natural Numbers

Whole Numbers Symbol

Whole numbers are represented by the alphabet \(”W”\) in capital letters.
\(W = \left\{{0,1,2,3,4,5,6,7,8,9,10, \ldots } \right\}\)

Smallest Whole Number

Whole numbers start from zero (from the definition of whole numbers). Thus, \(0\) is the smallest whole number. The concept of zero was first introduced by a Hindu astronomer and mathematician, Brahmagupta, in \(628\,A.D.\) Although zero has no value, it is used as a placeholder. So, zero is neither a positive nor a negative number.

Predecessor and Successor

Every number has a predecessor and a successor. We get predecessor by subtracting \(1\) from a number and successor by adding \(1\) to the number.
Example: The predecessor of \(7\) is \(7 – 1 = 6\)
The successor of \(7\) is \(7 + 1 = 8.\)

Whole Numbers vs Natural Numbers

From the following figure, we can understand that natural numbers are a subset of whole numbers.

Difference Between Whole Numbers and Natural Numbers:

The following are the differences between whole numbers and natural numbers:

Whole number	Natural Number
The set of whole numbers is, \(W = \left\{ {0,1,2,3,…} \right\}\)	The set of natural numbers is, \(N = \left\{{1,2,3,…} \right\}\)
The smallest whole number is \(0.\)	The smallest natural number is \(1.\)
Every natural number is a whole number.	Every whole number is a natural number, except \(0.\)

Whole Numbers on Number Line

All the integers starting from \(1\) represent the natural numbers, whereas all the positive integers and zero represent the whole numbers. Both whole numbers and natural numbers can be represented on the number line. The following figure shows the set of natural numbers and whole numbers on the number line.

Solved Examples

Q.1. Are 99,225,647,4381 whole numbers?
Ans: Yes. \(99,225,647,4381\) are all whole numbers.

Q.2. Solve \(9 \times \left({5 + 10} \right)\) using the distributive property.
Ans: Distributive property of multiplication over the addition of whole numbers is:
\(x \times \left({y + z} \right) = \left({x \times y}\right) + \left({x \times z} \right)\)
\( \Rightarrow 9 \times \left({5 + 10}\right) = \left({9 \times 5} \right) + \left({9 \times 10} \right)\)
\( = 45 + 90\)
\( = 135\)
Therefore, \(9 \times \left({5 + 10} \right) = 135.\)

Q.3. Add 25, 36, 15 in three ways. Indicate the property used.
Ans: To add \(25,36,15\)
Way I: \(25 + \left({36 + 15} \right) = 25 + 51 = 76\)
Way II: \(\left({25 + 36} \right) + 15 = 61 + 15 = 76\)
Way III: \(\left({25 + 15} \right) + 36 = 40 + 36 = 76\)
Here, we have used associative property.

Q.4. Anok buys 8 containers of milk from one shop and 10 containers of the same kind of container from another shop. If the capacity of each container is the same and the cost of each container is \(₹120,\) find the total money spend by Anok.
Ans: Anok buys \(10\) containers of the same capacity from another shop, cost of \(1\) container \(=₹120\)
Total money spent by Anok \( ₹= \left[{8 \times 120 + 10 \times 120} \right]\)
Using the distributive property of whole numbers, \(a \times \left({b + c} \right) = \left({a \times b} \right) + \left({a \times c} \right),\) where \(a,b,c\) are whole numbers
\( =₹ 120 \times \left({8 + 10} \right)\)
\(₹ = 120 \times 18\)
\( ₹= 2160\)
Thus, the total money spent by Anok is \(₹2160.\)

Q.5. Multiply \(25 \times 15 \times 4\) by using a property.
Ans: Using the associative property of whole number, \(a \times \left({b \times c} \right) = \left({a \times b} \right) \times c\)
\(\Rightarrow 25 \times 15 \times 4 = \left({25 \times 4} \right) \times 15\)
\( = 100 \times 15\)
\( = 1500.\)
Thus, \(25 \times 15 \times 4 = 1500.\)

Q.6. 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?
Ans: We observe that the scores of Suresh in the first inning is equal to the score of Ramesh in the second innings, and the score of Suresh in the second innings is equal to the score of Ramesh in the first innings.
By the commutative property of whole numbers, \(a + b = b + a\)
\( \Rightarrow 52 \times 78 = 78 + 52 = 130\)
Thus, both Sachin and Ajay had equal scores.

Summary

This article taught us the definition of natural numbers and whole numbers.We studied the difference between whole numbers and natural numbers. Then we learnt the number line representation of the whole numbers and natural numbers.

Then, we briefly understood the different properties related to whole numbers with examples. At last, we have solved some topic-related examples to make the student understand the concept correctly.

FAQs on Whole Numbers

Following are some common questions which candidates may have in their mind regarding Whole Numbers:

Q1. Define whole numbers, or what are whole numbers?
Ans: Whole numbers in Math is the set of positive integers and \(0.\) In other words, it is a set of natural numbers, including \(0.\) Decimals, fractions, negative integers are not part of whole numbers.

Q2. What are the four properties of whole numbers?
Ans: The four properties of whole numbers are:
1. Closure property.
2. Associative property.
3. Commutative property.
4. Distributive property

Q3. What is the use of whole numbers?
Ans: These are numbers that we are the most used to working with, including zero. We see whole numbers on nutrition labels or signs on the highway telling us how many miles are to the next city. 

Q4. What is the smallest whole number?
Ans: Zero is the smallest whole number.

Q5. Which numbers are not whole numbers?
Ans: A negative integer, fractions, part of rational numbers and decimals do not belong to whole numbers.

Q6. Which is the largest whole number?
Ans: There is no largest whole number. Every whole number has an immediate successor or a number that comes after. So the whole numbers are infinite to count, and thus, there is nothing such largest whole number.

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