Class 6 Maths Chapter 2 Whole Numbers

Class 6 Maths Chapter 2 Whole Numbers
Syllabus
1.Knowing Our Numbers
2.Whole Numbers
3.Playing with Numbers
4.Basic Geometrical Ideas
5.Understanding Elementary Shapes
6.Integers
7.Fractions
8. Decimals
9.Data Handling
10.Mensuration
11.Algebra
12. Ratio and Proportion
13.Symmetry
14.Practical Geometry
Revision Notes on Whole Numbers
Natural Numbers
All the positive counting numbers starting from one are called Natural Numbers.
Predecessor and Successor
If we add 1 to any natural number, we get the next number, which is called the Successor of that number.

12 + 1 = 13

So 13 is the successor of 12.

If we subtract 1 from any natural number, we get the predecessor of that number.

12 – 1 = 11

So 11 is the predecessor of 12.

Remark: There is no predecessor of 1 in natural numbers.

Whole Numbers
Whole numbers are the collection of natural numbers including zero. So, the zero is the predecessor of 1 in the whole numbers.

Whole Numbers

Number Line
To draw a number line, follow these steps-

Draw a line and mark a point 0 on it.

Now mark the second point to the right of zero and label it as 1.

The distance between the 0 and 1 is called the unit distance.

Now you can mark other points as 2, 3, 4 and so on with the unit distance.

Number Line

This is the number line for the whole numbers.

1. The distance between two points

The distance between 3 and 5 is 2 units. Likewise, the distance between 1 and 6 is 5 units.

2. The greater number on the number line

The number on the right is always greater than the number on the left.

As number 5 is on the right of the number 2, Hence 5>2.

3. A smaller number on the number line

The number on the left of any number is always smaller than that number.

As number 3 is on the left of 7, so 3 < 7.

Addition on the Number Line
If we have to add 2 and 5, then start with 2 and make 5 jumps to the right. As our 5th jump is at 7 so the answer is 7.

Addition on the Number Line

The sum of 2 and 5 is 2 + 5 = 7

Subtraction on the Number Line
If we have to subtract 6 from 10, then we have to start from 10 and make 6 jumps to the left. As our 6th jump is at 4, so the answer is 4.

Subtraction on the Number Line

The subtraction of 6 from 10 is 10 – 6 = 4.

Multiplication on the Number Line
If we have to multiply 4 and 3, then Start from 0, make 4 jumps using 3 units at a time to the right, as you reach to 12. So, we say, 3 × 4 = 12.

Multiplication on the Number Line

Properties of Whole Numbers
1. Closure Property
Two whole numbers are said to be closed if their operation is also the whole number.

Operation Meaning Example Closed or not
Addition

Whole numbers are closed under addition as their sum is also a whole number.

2 + 5 = 7

Yes

Subtraction

Whole numbers are not closed under subtraction as their difference is not always a whole number.

9 – 2 = 7

2 – 9 = (-7) which is not a whole number.

No

Multiplication

Whole numbers are closed under multiplication as their product is also a whole number.

9 × 5 = 45

Yes

Division

Whole numbers are not closed under division as their result is not always a whole number.

5 ÷ 1 = 5

5 ÷ 2 =, not a whole number.

No

Comparing Numbers

1. Compare 4978 and 5643…….

5643 is greater as the digit at the thousands place in 5643 is greater than that in 4978.

2. Compare 9364,8695,8402 and 7924

9364 is the greatest as it has the greatest digit at the thousands place in all the numbers.

Whereas 7924 is the smallest as it has the smallest digit at the thousands place in all the numbers.

3. Special case

Compare 56321 and 56843

Here, we will start by checking the thousands place.As the digit 5 at ten thousand place is same so we will move forward and see the thousands place. The digit 6 is also same so we will still move on further to check the hundreds place.

The digit at the hundreds place in 56843 is greater than that in 56321

Thus 56843 is greater than 56321

Proper Order

  • If we arrange the numbers from the smallest to the greatest then it is said to be an Ascending order.

  • If we arrange the numbers from the greatest to the smallest then it is said to be Descending order.

Example

Arrange the following heights in ascending and descending order.

Heights in ascending and descending order.

Ascending order – 90 < 160 < 170 < 185 < 230

Descending order – 230 >185 >170 > 160 > 90

Number Formations

Form the largest and the smallest possible numbers using 3,8,1,5 without repetition

Largest number will be formed by arranging the given numbers in descending order – 8531

The smallest number will be formed by arranging the given numbers in ascending order – 1358

Introducing 10,000

99 is the greatest 2-digit number.

999 is the greatest 3-digit number

9999 is the greatest 4-digit number

Observation

  • If we add 1 to the greatest single digit number then we get the smallest 2-digit number

(9 + 1 = 10)

  • If we add 1 to the greatest 2- digit number then we get the smallest 3-digit number

(99 + 1 = 100)

  • If we add 1 to the greatest 3- digit number then we get the smallest 4-digit number

(999 + 1 = 1000)

Moving forward, all the above situations are same as adding 1 to the greatest 4-digit number is the same as the smallest 5-digit number. (9999 + 1 = 10,000), and it is known as ten thousand.
 

Place Value

It refers to the positional notation which defines a digits position.

Example

6931

Here, 1 is at one's place, 3 is at tens place, 9 is at hundreds place and 6 is at thousands place
 

Expanded form

It refers to expand the number to see the value of each digit.

Example

6821 = 6000 + 800 + 20 + 1

= 6 × 1000 + 8 × 100 + 2 × 10 + 1×1

Introducing 1,00,000

As above pattern if we add 1 to the greatest 5-digit number then we will get the smallest 6-digit number

 (99,999 + 1 = 1,00,000)

This number is called one lakh.

Larger Numbers

To get the larger numbers also, we will follow the same pattern.

We will get the smallest 7-digit number if we add one more to the greatest 6-digit number, which is called Ten Lakh.

Going forward if we add 1 to the greatest 7-digit number then we will get the smallest 8-digit number which is called One Crore.

Remark

1 hundred = 10 tens

1 thousand = 10 hundreds

= 100 tens

1 lakh  = 100 thousands

= 1000 hundreds

1 crore = 100 lakhs

= 10,000 thousands

Pattern

9 + 1 = 10

 99 + 1 = 100

 999 + 1 = 1000

 9,999 + 1 = 10,000

99,999 + 1 =1,00,000

9,99,999 + 1 = 10,00,000

99,99,999 + 1 = 1,00,00,000
 

Reading and Writing Large Numbers

We can identify the digits in ones place, tens place and hundreds place in a number by writing them under the tables O, T and H.

AS:

CroresLakhsThousandsOnes

Ten Crores (TC) 

Crores (C)

Ten Lakhs (TL)

Lakhs (L)

Ten Thousands (TTh)

Thousands (Th)

Hendreds (H)

Tens (T)

Ones (O)

(10, 00, 00, 000)

(1,00,00,000)

(10, 00, 000)

(1,00,000)

(10,000)

(1000)

(100)

(10)

(1)

Example

Represent the number 5, 21, 05, 747

Reading and Writing Large Numbers

Use of Commas

We use commas in large numbers to ease reading and writing. In our Indian System of Numeration, we use ones, tens, hundreds, thousands and then lakhs and crores.

 We use the first comma after hundreds place which is three digits from the right. The second comma comes after two digits i.e. five digits from the right. The third comma comes after another two digits which is seven digits from the right.

Example

5,44,12,940

Remark: We do not use commas while writing number names

International System of Numeration

MillonsThousandsOnes
Hundred MillionTen MillionMillionHundred ThousandsTen ThousandsThousandsHundred Tens Ones
100,000,00010,000,0001,000,000100,00010,0001,000100101

Example

341,697,832

Expanded form: 3 x 100,000,000 + 4 x 10,000,000 + 1 x 1,000,000 + 6 x 100,000 + 9 x 10,000 + 7 x 1,000 + 8 x 100 + 3 x 10 + 2 x 1

Remark: If we have to express the numbers larger than a million then we use a billion in the International System of Numeration:

1 billion = 1000 million

Large Numbers in Practice

10 millimeters = 1 centimeter

1 meter = 100 centimeters

= 1000 millimeters

1 kilometer = 1000 meters

1 kilogram = 1000 grams.

1 gram = 1000 milligrams

. 1 litre = 1000 millilitres

1 litre = 1000 millilitres
 

Let’s Solve Some Problems

Example: 1

To fill an order, the factory dyed 336 yards of silk in yellow and 37 yards in pink. How many yards of silk did it dye for that order?

Solution:

Yards of yellow silk = 336

Yards of pink silk = 37

Total yards of silk dyed = 336 + 37

= 371

Thus, 371 yards of silk was dyed for the order.
 

Example: 2

There are 8797 children in a town, 6989 go to school. How many children do not go to school?

Solution:

Total children = 8797

Children who go to school = 6989

Children who do not go to school = 8797 – 6989

= 1808

Thus 1808 children of the town do not go to school.
 

Example: 3

There are 24 folders and each has 56 sheets of paper inside them. How many sheets of paper are there altogether?

Solution:

Thus there are 1344 sheets of paper altogether.
 

Example: 4

$5,876 is distributed equally among 26 men. How much money will each person get?

Solution:
Money received by 26 men = 5876

Money received by one man = 5876 ÷ 26

Thus each man got $226.

Estimation

It is a rough calculation of value. We use estimations when we have to deal with large numbers and to do the quick calculations.

Estimating to the nearest tens by rounding off

If the digit in the ones places is:

5 or higher, round tens place up

4 or lower, leave tens place as is

Firstly to estimate we need to see where does the number lies.

Here 38 lie between 30 and 40

Secondly, we will see if it is 5 or higher.

Yes it is higher than 5 i.e. 38

Thus the number 738 is rounded off to 740.
 

Estimating to the nearest hundreds by rounding off

Round off the number 867 nearest to the hundreds.

It lies between 800 and 900

Now we have to check for tens place. If it is greater than 50 then we will round it off to the upper side and if it is less than 50 then we will round it off on the lower side.

It is 67, which is greater than 50 and is closer to 900.

Thus 867 is rounded off to 900
 

Estimating to the nearest thousands by rounding off

The Numbers from 1 to 499 are rounded off to 0 as they are nearer to 0, and the numbers from 501 to 999 are rounded off to 1000 as they are nearer to 1000

And 500 is always rounded off to 1000.

Example

Round off the number 7690 nearest to thousands.

It lies between 7000 and 8000

And is closer to 8000

Thus 7690 is rounded off to 8000.

To estimate sum or difference

Estimate: 3,210 + 12,884

Solution

3,210 will be rounded off to 3000.

12,884 will be rounded off to 13000.

 3000+ 13000

Estimated solution = 16000

Actual solution = 3,210 + 12 884

= 16,094

To estimate products

Estimate: 73 × 18

Solution

73 will be rounded off to 70

18 will be rounded off to 20

70 × 20

Estimated solution = 1400

Actual solution = 73 × 18

= 1314

Using Brackets

We use brackets to indicate that the numbers inside should be treated as a different number thus the bracket should be solved first.

Example

8 + 2 = 8 + 10

= 18

Whereas if we use brackets

(8 + 2) × 5 = 10 

= 50

Expanding brackets

Brackets help in the systematic calculation.

Example

3 × 109 = 3 × (100 + 9)

= 3 × 100 + 3 × 9

= 300 + 27

= 327

Roman Numerals

Numbers in this system are represented by combinations of letters from the Latin alphabets.

Roman Numerals

Rules:

a. If we repeat a symbol, its value will be added as many times as it occurs:

Example

II is equal 2

XX is 20

XXX is 30.

b. We cannot repeat a symbol more than three times and some symbols like V, L and D can never be repeated.

c. If we write a symbol of lesser value to the right of a symbol of larger value then its value will be added to the value of the greater symbol.

VI = 5 + 1 = 6,   XII = 10 + 2 = 12 and LXV = 50 + 10 + 5 = 65.

d. If we write a symbol of lesser value to the left of a symbol of larger value then its value will be subtracted from the value of the greater symbol.

IV = 5 – 1 = 4, IX = 10 – 1 = 9 XL= 50 – 10 = 40, XC = 100 – 10 = 90.

e. The symbols V, L and D can never be subtracted so they are never written to the left of a symbol of greater value. We can subtract the symbol “I” from V and X only and the symbol X from L, M and C only.2. Commutative Property

Two whole numbers are said to be commutative if their result remains the same even if we swap the positions of the numbers.


Operation Meaning Example Commutative or not

Addition


The addition is commutative for whole numbers as their sum remains the same even if we interchange the position of the numbers.


2 + 5 = 7


5 + 2 = 7


Yes


Subtraction


Subtraction is not commutative for whole numbers as their difference may be different if we interchange the position of the numbers.


9 – 2 = 7


2 – 9 = (-7) which is not a whole number.


No


Multiplication


Multiplication is commutative for whole numbers as their product remains the same even if we interchange the position of the numbers.


9 × 5 = 45


5 × 9 = 45


Yes


Division


The division is not commutative for whole numbers as their result may be different if we interchange the position of the numbers.


5 ÷ 1 = 5


1 ÷ 5 =, not a whole number.

3. Associative Property

The two whole numbers are said to be associative if the result remains the same even if we change the grouping of the numbers.


Operation Meaning Example Associative or not

Addition


The addition is associative for whole numbers as their sum remains the same even if we change the grouping of the numbers.


3 + (2 + 5) = (3 + 2) + 5


3 + 7 = 5 + 5


10 = 10


Yes


subtraction


Subtraction is not associative for whole numbers as their difference may change if we change the grouping of the numbers.


8 – (10 – 2) ≠ (8 – 10) – 2


8 - (8) ≠ (-2) – 2


0 ≠ (-4)


No


Multiplication


Multiplication is associative for whole numbers as their product remains the same even if we change the grouping of the numbers.


3 × (5 × 2) = (3 × 5) × 2


3 × (10) = (15) × 2


30 = 30


Yes


Division


The division is not associative for whole numbers as their result may change if we change the grouping of the numbers.


24 ÷ 3 ≠ 4 ÷ 2


8 ≠ 2





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