Chapter Notes: Whole Numbers

 Table of contents Introduction Whole numbers on a number line Operations on a number line Predecessor and Successor Properties of addition Properties of subtraction Properties of multiplication Properties of division

## Introduction

• Numbers play an important role in our life.
• We use numbers in our day to day life to count things.
• While counting we use numbers to represent any quantity, to measure any distance or length.

3 apples and 3 pepper                                  Discount                          City distance                                             Length of the table

• The counting numbers starting from 1, 2, 3, 4, 5, ……… are termed as  natural numbers.
• The set of counting numbers and zero are known as whole numbers.
•  Whole numbers are 0, 1, 2, 3, 4, 5, 6, 7,…….. and so on.

Example: Write the next three natural numbers after 10999.
Solution: 10999 + 1 = 11000
11000 + 1= 11001
11001 + 2 = 11002.
Thus, 11000, 11001, 11002 are the next three natural numbers after 10999.

## Whole numbers on a number line

### Number Line

A pictorial representations of numbers evenly marked on a straight line is known as a number line.

• To mark whole numbers on a number line draw a horizontal line and mark a point on it as 0.
•  Extend this line towards right direction.
• Starting from 0, mark points 1, 2, 3, 4, 5, 6, 7, 8, 9….on a line at equal distance towards right side.
• There is no whole number on the left of zero. Therefore zero is the smallest whole number.

### Comparison using Number Line

The number line also helps us to compare two whole numbers. i.e., to decide which of the two given whole numbers is greater or smaller.

• A whole number is greater than all the whole numbers which lie to the left of it on the number line.
• A whole number is less than all the whole numbers which lie to the right of it on the number line
Example:  We can say that 5 is less than 9 and write 5 < 9.
We can also say that 5 is greater than 4 and write 5 > 4.

Example: Identify which whole number comes first on the number line in each pair and write the appropriate sign (> or <) between them:
(a) 507 and 503
(b) 338 and 342

Sol:  (a) 503 is on the left side of 507 on the number line. So, 507 > 503.

(b) 338 is on the left side of 342 on the number line. So, 342 > 338.

## Operations on a number line

For eg: Add 2 and 5. i.e 2 + 5.

• Start from 0, 2 jumps towards right. You reach at 2.
• Start from 2, 5 jumps towards right. You reach at 7.
• Therefore, 2 + 5 = 7.

### Subtraction

For eg: Subtract 4 from 7.  i.e 7 - 4.

• Start from 0, jump directly to 7.
• Start from 7,  4 jumps towards left. You reach at 3.

### Multiplication

For eg:  Multiply 3 and 2 i.e. 3 x 2

• Multiplication means repeated addition. Thereofore 3 x 2 means 3 added twice.
• Start from 0. Make 2 jumps (each jump of 3 units).You reach at 6.

### Division

For eg: 6 ÷ 3 = 2.

• Start from 6 and subtract 3 for a number of times till 0 is reached.
• The number of times 3 is subtracted gives the quotient.

Ques: Using the number line, determine the following products:
(a) 3 x 3
(b) 4 x 2

Sol: (a)

• Start from 0. Make 1 jump of 3 steps towards right. You reach at 3.
• Start from 3.  Make 1 jump of 3 steps towards right. You reach at 6.
• Start from 6. Make 1 jump of 3 steps towards right. You reach at 9.

Therefore, 3 x 3 = 9

(b)

• Start from 0.  Make 1 jump of 4 steps towards right. You reach at 4.
• Start from 4.  Make 1 jump of 4 steps towards right. You reach at 8.

Therefore, 4 x 2 = 8.

## Predecessor and Successor

### Predecessor

• The number which comes right before a given number is called its predecessor.
• The predecessor of a whole number found by subtracting 1 from it.

Number - 1 = Predecessor

### Successor

• The number which comes right after a given number is called its successor.
• The successor of a whole number found by adding 1 to it.

Number + 1 = Successor.

### Closure property

• For any two whole numbers a and b, their sum  a + b is always a whole number.
• For eg:  12 + 45 = 57. Here, 12, 45 and 57 all are whole numbers.

### Commutative property

• For any two whole numbers a and b, a +b = b + a.
• We can add any two whole numbers in any order.
For eg:  12 + 45 = 45 + 12

### Associative property

• For any three whole numbers a, b and c, (a + b) + c = a + (b + c).
• This means the sum is regardless of how grouping is done.
For eg:  31 + (24 + 38) = (31 + 24) + 38

• For every whole number a, a + 0 = a.
• Therefore ‘0’ is called the Additive identity.
E.g. 19 + 0 = 19

Example: Find the sum of 435, 216 and 165
Sol: 435 + 216 + 165
Now, 5 + 5 = 10. So, we add 435 + 165 first.
= (435 + 165) + 216
= 600 + 216 = 816

Example: Find the sum by suitable arrangement:
(a) 837 + 208 + 363
(b) 1962 + 453 + 1538 + 647

Sol: (a) 837 + 208 + 363
Now, 7 + 3 = 10.
So, we add 837 + 363 first.
= (837 + 363) + 208
= 1200 + 208 = 1408

(b) 1962 + 453 + 1538 + 647
Now, 2 + 8 = 10 .So, we make one group of (1962 + 1538)
3 + 7 = 10.  Next we make another group of (453 + 647)
= (1962 + 1538) + (453 + 647)
=3500 + 1100 = 4600

## Properties of subtraction

### Closure property

• For any two whole numbers, a and b, if a  > b then a – b is a whole number.
• If a < b then a – b is never a whole number.
•  Closure property is not always applicable to subtraction.
E.g. 150 – 100 = 50, is a whole number but 100 – 150 = -50 is not a whole number.

### Commutative property

• For any two whole numbers a and b, a – b ≠  b – a .
• Hence subtraction of whole number is not commutative.
E.g  16 – 7 = 9 but  7 – 16 ≠ 9

### Associative property

• For any three whole numbers a, b and c, (a – b) – c ≠ a – (b – c).
• Hence  subtraction of whole numbers is not associative.
E.g.  25 – (10 – 4) = 25 – 6 = 19
Also,  (25 – 10) – 4 = 15 – 4 = 11. This means that 25 – (10 – 4) ≠ (25 – 10) – 4

(iv) If 𝒂 is any whole number other than zero, then 𝒂 – 𝟎 = 𝒂 but 𝟎 − 𝒂 is not defined.

For eg:
(i) 18 – 5 = 13 but 5 – 18 is not defined in whole numbers.
(ii) 30 – 12 = 18 but 12 – 30 is not defined in whole numbers

(v) If 𝒂, 𝒃 and 𝒄 are whole numbers such that 𝒂 – 𝒃 = 𝒄, then 𝒃 + 𝒄 = 𝒂

For eg:
(i) If 25 – 16 = 9 then 25 = 9 + 16,
(ii) If 46 – 8 = 38 then 46 = 38 + 8

Example: Solve the following:

(i) 367 – 99
= 367 + (– 100 + 1)
= 367 – 100 + 1
= (367 + 1) – 100
= 368 – 100
= 268

(ii) 5689 – 99
= 5689 + (- 100 +1)
= 5689 – 100 + 1
= (5689 + 1) – 100
= 5690 – 100
= 5590

### Closure property

• For any two whole numbers a and b,their product  a x b is always a whole number.
• E.g. 12 x 7 =  84. Here,  12, 7 and 84 all are whole numbers.

### Commutative property

• For any two whole numbers a and b, a x b = b x a.
• Order of multiplication is not important.
E.g  11 x 6 =  66 and   6 x 11 = 66 Therefore, 11 x 6 = 6 x 11.

### Associative property

• For any three whole numbers a, b and c, (a x b) x c = a x (b x c).
• This means the product is regardless of how grouping is done.
E.g   8 x (4 x 5) = 8 x 20 = 160;   (8 x 4) x 5 = 32 x 5 = 160 Therefore,  8 x (4 x 5) = (8 x 4) x 5
• We can explain the associative property with the help of following example
• Count the number of dots in figure (a) and figure (b)
•  In figure (a), there are 2 rows and 2 columns which means 2 x 2 dots in each box. So the total number of dots are (2 x 2) x 3 = 12
• In figure (b), there are 3 rows and 2 columns which means 3 x 2 dots in each box. So the total number of dots are 2 x (3 x 2) = 12
• This explain the associative property of multiplication.

### Multiplicative identity

• For any whole number a, a x 1 = a.
• Since any number multiplied by 1 doesn’t change its identity
• Hence 1 is called as multiplicative identity of a whole number.
E.g. 21 x 1 = 21

### Multiplication by zero

• For any whole number a, a x 0 = 0,
•  E.g 25 x 0 = 0

### Distributive property of multiplication over Addition

• This property is used when we have to multiply a number by the sum.
• If 𝒂, 𝒃&𝒄 are any three whole numbers, then 𝒂 × (𝒃 + 𝒄) = 𝒂 × 𝒃 + 𝒂 × 𝒄.

Example: Find the product by suitable rearrangement:
(i) 4 × 1768 × 25
(ii) 2 × 166 × 50
(iii) 285 × 4 × 75
(iv) 625 × 279 × 16

Sol:
(i) 4 × 1768 × 25= (4 × 25) × 1768 (by commutative property)
= 100 × 1768 = 176800
(ii) 2 × 166 × 50= (2 × 50) × 166 (by commutative property)
= 100 × 166 = 16600
(iii) 285 × 4 × 75= 285 × (4 × 75) (by commutative property)
= 285 × 300 = 85500
(iv) 625 × 279 × 16= (625 × 16) × 279 (by commutative property)
= 10000 × 279 = 2790000

### Closure property

• For any two whole numbers a and b, a ÷ b is not always a whole number.
• Hence closure property is not applicable to division.
For eg:  68 and 5 are whole numbers but 68 ÷ 5 is not a whole number.

### Commutative property

• For any two whole numbers a and b, a ÷ b ≠ b ÷ a.
• This means division of whole number is not commutative.
For eg: 16 ÷ 4 ≠ 4 ÷ 16

### Associative property

• For any 3 whole numbers a, b and c,(a ÷ b)  ÷ c ≠ a ÷ (b ÷ c)
•  For eg: consider (80 ÷ 10) ÷ 2 = 8 ÷ 2 = 4 80 ÷ (10 ÷2) = 80 ÷ 5 = 16 (80 ÷ 10) ÷ 2 ≠80 ÷ (10 ÷2)
• Hence division does not follow associative property.

### Division by 1

• For any whole number a, a ÷ 1 = a.
• This means any whole number divided by 1 gives the quotient as the number itself.
For eg: 14 ÷ 1 = 14;
26 ÷ 1 = 26

### Division of 0 by any whole number

• For any whole number, a ≠ 0, 0 ÷ a = 0,
• This shows zero divided by any whole number (other than zero) gives the quotient as zero.
For eg: (i) 0 ÷ 1 = 0;
(ii) 0 ÷ 25 = 0;

### Division by 0

• To divide any number, say 7 by 0, we first have to find  out a whole number which when multiplied by 0 gives us 7. This is not possible.
• Therefore, division by 0 is not defined.

Example: Solve the following
(i) 636 ÷ 1

(ii) 0 ÷ 253
(iii) 246 – (121 ÷ 121)
(iv) (45÷ 5) – (9÷ 3)
Sol: (i) 636 ÷ 1 = 636 (∵ a ÷1 = a )
(ii) 0 ÷ 253 = 0 (∵ 0 ÷ a = 0)
(iii) 246 – (121 ÷ 121)
= 246 – (1)
= 246 – 1
= 245
(iv) (45÷ 5) – (9 ÷ 3)
= 9 – 3 = 6

The document Whole Numbers Chapter Notes | Maths Olympiad Class 6 is a part of the Class 6 Course Maths Olympiad Class 6.
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## FAQs on Whole Numbers Chapter Notes - Maths Olympiad Class 6

 1. How are whole numbers represented on a number line?
Ans. Whole numbers are represented on a number line by marking points at equal intervals starting from zero in the positive direction. Each point represents a whole number.
 2. What are the basic operations that can be performed on a number line with whole numbers?
Ans. The basic operations that can be performed on a number line with whole numbers include addition, subtraction, multiplication, and division.
 3. What is the predecessor of a whole number on a number line?
Ans. The predecessor of a whole number on a number line is the whole number that comes before it when moving to the left on the number line.
 4. How can we find the successor of a whole number on a number line?
Ans. The successor of a whole number on a number line can be found by moving to the right on the number line and identifying the next whole number.
 5. How do whole numbers play a role in everyday life applications?
Ans. Whole numbers are used in everyday life applications such as counting objects, measuring quantities, and representing scores in games and sports.

9 videos|114 docs|49 tests

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