Revision Notes on Number System

Number

The digits and group of digits are called number.

1. Number name

2. Types of number

3. Divisibility rule

4. Comparing Numbers

5. Number Formations

6. Place Value

7. Expanded form

8. Estimation

9. Roman Numerals

1. Comparing Numbers

* Using by sign

* Smallest and Greatest number

* 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.

Types of questions

1. Comparing natural numbers

2. Comparing decimals numbers

3. Comparing fractional numbers

4. Comparing roman numbers

2. 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

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

Ans- Largest even no = 5318, Smallest even no- 1358

* Form the largest and the smallest odd numbers using 3,8,0,1 without repetition

Ans- Largest odd no = 5381, Smallest even no- 3581

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.
3. Place Value and Face Value स्थानीय मान और अंकित मान

Place value 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
Que-1. Find the place value of underline digit in number 47820.

Ans- 8 hundreds = 800

Que-2. Find the sum of place value and face value of 8 in number 47820.

Ans- 8 hundreds + 8 = 800 + 8 = 808

Que-3. Find the difference of place value of both 8 in number 87820.

Ans- 80000 - 800 = 79200

Que-1. Find the product of place value and face value of 8 in number 47820.

          Ans- = 800 × 8 = 6400

4. Expanded form

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

Example

6821 = 6 thousands + 8 hundreds + 2 tens + 1 once

= 6 × 1000 + 8 × 100 + 2 × 10 + 1×1

= 6000 + 800 + 20 + 1

= 6 × 10³ + 8 × 10² + 2 × 10¹ + 1×10⁰

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:

 

 

Crores		Lakhs		Thousands		Ones
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

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

Millons			Thousands			Ones
Hundred Million	Ten Million	Million	Hundred Thousands	Ten Thousands	Thousands	Hundred	Tens	Ones
100,000,000	10,000,000	1,000,000	100,000	10,000	1,000	100	10	1

 

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
5. 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

6. Roman Numerals

The system in which numbers are represented by combinations of letters from the Latin alphabets is called roman Numerals.

• In this system 7 letters are used to represent the number.

 

• When we write bar bracket upon the roman number, it’s value increase one thousand time.

Ex-  IV  - 4×1000 = 4000, V  - 5×1000 = 5000, VII  - 7×1000 = 7000, X  - 10×1000 = 10000

•  

 

 

 

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.