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Whole Numbers

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08 September 2020

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Hey there!  Today, We’re back with something new to learn, something you will enjoy! This time, you will learn about whole numbers and natural numbers, their properties, and how to work with them. You will also see some examples and solve some practice questions based on the topic.
So let's get started!

Table of Contents

1. Introduction and Understanding of Whole Numbers
2. Definition
3. Importance of Zero
4. Properties of Whole Numbers
5. Solved Examples
6. Computation of Whole Numbers
7. Summary
8. Practice Questions
9. Related Topics
10. FAQS


Introduction and Understanding of Whole numbers

Whole numbers are a set of numbers(0,1,2,3,4....), which include natural numbers or counting numbers and zero. In counting numbers, zero is not included as we do not start counting from zero.
Therefore, we can say that all whole numbers are real numbers, but not all the real numbers are whole numbers.

Whole numbers


Definition

Whole numbers are the set of numbers which are basically natural numbers in addition with zero. But they don’t include fractional or decimal value. Zero as a whole represents nothing or a null value. The symbol to represent whole numbers is the alphabet ‘W’ in capital letters. Therefore, we can represent the set of whole numbers as follows:
W = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…
Here are some facts about whole numbers, which will help you understand them better:

  • All natural numbers are whole numbers
  • All counting numbers are whole numbers
  • All positive integers including zero are whole numbers
  • All whole numbers are real numbers.

Importance of Zero

The concept of zero was first defined by a Hindu astronomer and mathematician Brahmagupta in 628. 
In simple language, zero is a number that lies between the positive and negative numbers on a number line. As such zero carries no value, though it is used as a placeholder. So zero is neither a positive number nor a negative number, but it is an even number.


Properties of whole numbers

There are four basic properties of whole numbers, namely:

  • The closure property
  • The commutative property
  • The associative property
  • The distributive property

Closure Property

The closure property states that when two whole numbers are added, their sum is always a whole number. Therefore, if a and b are two whole numbers and a + b = c, then c is also a whole number. For example: 3 + 4 = 7 (whole number).

Commutative Property

This property states that change in the order of addition does not change the value of the sum. Let a and b be two whole numbers, commutative property states that a + b = b + a. 
For example, a = 10 and b = 19 ⇒ 10 + 19 = 29 = 19 + 10.
It means that the whole numbers are closed under addition. This property also holds true for multiplication, but not for subtraction or division.
For e.g: 7 x 9 = 63 or 9 x 7 = 63

Associative Property

The associative property states that when we add three or more whole numbers, whatever may be the way in which the numbers are grouped, the value of the sum remains the same. Therefore, if a, b, and c are three whole numbers, then a + (b + c) = (a + b) + c = (a + c) + b. 
For example, 10 + (7 + 12) = (10 + 7) + 12 = (10 + 12) + 7 = 29.

Distributive Property

This property states that the multiplication of a whole number is distributed over the sum of the whole numbers. It means that when two numbers, take for example a and b are multiplied with the same number c and are then added, then the sum of a and b can be multiplied by c to get the same answer. This situation can be represented as:
a × (b + c) = (a × b) + (a × c). 
Let a = 10, b = 20 and c = 7 ⇒ 10 × (20 + 7) = 270 and (10 × 20) + (10 × 7) = 200 + 70 = 270.

The same is true for subtraction as well. For e.g we have a × (b − c) = (a × b) − (a × c). 
Let a = 10, b = 20 and c = 7 ⇒ 10 × (20 − 7) = 130 and (10 × 20) − (10 × 7) = 200 − 70 = 130.



So you have learnt all the four properties of whole numbers. Now let’s learn how to apply them. Here are some solved examples for you:

Solved Examples

Example 1



 

Multiply 26 × 15 by using a property.

Solution:

26 × 15 = 26 × (10 + 5) = (26 × 10) + (26 × 5) = 260 + 130 = 390.

 
Example 2


 

Solve (121 × 16) − (121 × 6) by the distributive property.

Solution:

(121 × 16) − (121 × 6) = 121 × (16 − 6) = 121 × 10 = 1210.

 
Example 3

 

Use associative property for the following mathematical equation:
2 + 3 + 7 = 7 + 3 + 2 = 2 + 7 + 3

Solution:

 (2 + 3) + 7 = 2 + (3 + 7).

 
Example 4

 

Use commutative property for the mathematical equation:   7*4=28

Solution:

  7*4=28  and 4*7=28


Computation of Whole Numbers

Under this topic, you will learn calculations with whole numbers through different operations like addition, subtraction, multiplication, and division.

Addition on the number line 

The addition of whole numbers can be shown on the number line as follows:
Let us take an example: (3 + 4)

Addition on number line

Start from 3. Since 4 is being added to 3, make 4 jumps to the right; from 3 to 4, 4 to 5, 5 to 6, and 6 to 7 as shown above. The tip of the last arrow in the fourth jump is at 7. Therefore the sum of 3 and 4 is 7, i.e. 3 + 4 = 7.

Subtraction on the number line 

The subtraction of two whole numbers can also be shown on the number line in the following way: Let us find (7 – 3).

Subtraction on number line

 

Start from 7. Since 3 is being subtracted, so move towards the left by jumping 1 unit at a time. Make 3 such jumps. We reach point 4. Therefore, we get 7 – 3 = 4. 

Multiplication on the number line 

We will now see the multiplication of whole numbers on the number line. Let us find 4 × 3.

 

Multiplication on number line

Start from 0, move 3 units at a time to the right, make 4 such moves. Where do you reach? You will reach 12. So, we say, 3 × 4 = 12. 

Division on the number line 

The division on the number line can be shown on the number line as follows.
For e.g: 20/4

 

Division on number line

Start from 20, move 4 units at a time to the left. Keep moving in this way until you reach zero. Make the maximum possible jumps. If you do not land on zero in one full jump, then the number you landed on will be the remainder.
In this example, you will reach 0 in 5 full jumps. So we can say that 20/4=5.          


Summary

So here we come to the end of this article. Let’s have a quick revision.
Whole numbers, in short, are nothing but a collection of natural numbers along with zero.

  • Set of Whole Numbers: W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10……}
  • Set of Natural Numbers: N = {1, 2, 3, 4, 5, 6, 7, 8, 9,…}
  • Set of Integers: Z = {….-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,…}
  • Counting Numbers: {1, 2, 3, 4, 5, 6, 7,….}

So did you enjoy learning about whole numbers? Please don’t forget to give your feedback and feel free to write in the comment section below if you have any questions. We have also provided some practice questions for you to solve and get more familiar with the topic. Hope you find it useful :)


Practice Questions

  1. Write the next seven natural numbers after 10999. 
  2. Write six whole numbers occurring just before 10001. 
  3. For each of the following pairs of numbers, state which whole number is on the left of the other number on the number line. Also, complete them with the appropriate sign (>, <) between them.(a) 570, 507 (b) 390, 309 (c) 98365, 56389 (d) 9832415, 10223001
  4. How many whole numbers are there between 37 and 73?
  5. Write the successor of : (a) 2440501 (b) 102199 (c) 1489999 (d) 2342670
  6. Write the predecessor of : (a) 93 (b) 17000 (c) 208390 (d) 76553321 
  7. Which is the smallest whole number?
  8. Find using distributive property : (a) 729 × 101   (b) 5439 × 1001   (c) 823 × 25   (d) 4275 × 125   (e) 505 × 34
  9. Match the following:

                  (i) 427 × 138 = 427 × (8 + 30 +100)         (a) Commutativity under multiplication.
                  (ii) 3 × 49 × 52 = 3 × 52 × 49                    (b) Commutativity under addition. 
                  (iii) 83 + 2007 + 20 = 83 + 20 + 2007       (c) Distributivity of multiplication over addition.

  10. Mark the below statements with true (T) or false (F)
  • Zero is the smallest natural number.
  • 700 is the predecessor of 699.
  • Zero is the smallest whole number
  • 900 is the successor of 899
  • All-natural numbers are whole numbers 
  • The whole number 1 has no predecessor
  • The predecessor of a two-digit number is never a single-digit number 
  • The whole number 14 lies between 15 and 16
  • The successor of a two-digit number is always a two-digit number
  • All whole numbers are natural numbers
  • 1 is the smallest whole number
  • The whole number 0 has no predecessor 
  • The natural number 1 has no predecessor

     


Topics Related to Current Topics

Other topics related to whole numbers are place values, addition, subtraction, multiplication, division, factoring, fractions, decimals, exponents, scientific notations, percents, integers, proportions and word problems etc.


Frequently Asked Questions (FAQs)

1. Define whole numbers.

These are the most basic concepts of mathematics and it is basic counting numbers. The whole number includes numbers such as 0,1,2,...∞.. Furthermore, whole numbers include natural numbers that begin from 1 also they include positive integers along with 0.

2. What are the properties of whole numbers?

The set of whole numbers is closed, associative, and commutative under addition and multiplication. 
You can refer to the section “Properties of Whole Numbers” on this page for the same.

3. What is a natural number?

A natural number is a number that is used for counting.
The set of natural numbers in Mathematics is the set 1, 2, 3, ...

4. Can whole numbers be negative?

The set of whole numbers is the set 0, 1, 2, 3, ...
Hence, whole numbers cannot be negative.

We use Natural numbers in our everyday life, in counting discrete objects, that is, objects which can be counted.

5. With what number do the whole numbers start from?

The whole numbers start with 0.

6. What are some examples of whole numbers?

The set of whole numbers in Mathematics is the set 0, 1, 2, 3, ...
Some examples can be 0, 15, 53, etc.

7. Is fraction also a whole number?

Fractions are not the whole number. Because they do not represent a complete number but if you want a fraction to be a whole number then convert them to the nearest whole number explicitly. For example, if you want 3/5 to be the whole number then it will be 1.

8. What is the symbol for the set of whole numbers?

The set of whole numbers is denoted by the symbol W.

9. What is the difference between the sets of whole numbers and natural numbers?

The only difference between whole numbers and natural numbers is that the set of whole numbers contains an extra number which is 0.

10. Is decimal a whole number?

No, decimals are not the whole numbers, but you can make them a whole number explicitly by rounding them off to the nearest whole number. Such 4.87 can be rounded off to 9 and 5.1 can be rounded off like 5.


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