Operations on Large Numbers Class 5 Notes Maths

Large Numbers

Have you ever wondered how we count things that are really big? Think about:

• How many people live in your city?
• How many stars are in the sky? 
  For such huge things, we need large numbers.
  In this Chapter, you’ll learn to read, write, compare, and play with large numbers so you can measure the big things all around us.

Indian Place Value Chart

Following is the table for the Indian Place Value System to let students understand how numbers are arranged and read.

Example:  Write the numeral by separating periods of the following
1. 603274087 
Ans: 60,32,74,087

Sol: 
Step 1: Identify the periods(refer to the table given above):
Crore: 60
Lakh: 32
Thousand: 74
Ones: 087
Step 2: Put commas 
After putting commas according to the periods we get = 60,32,74,087

2. 37007690
Ans: 3,70,07,690

Sol: 
Step 1: Identify the periods(refer to the table given above):
Crore: 3
Lakh: 70
Thousand: 07
Ones: 690
Step 2: Put commas 
After putting commas according to the periods we get = 3,70,07,690

Number Names & Expanded Form

1. Number Names

To write number name for any given number, we must follow the following steps

Step 1: Break the Number into Periods
Identify the crore, lakh, thousand, and ones periods.
Step 2: Arrange & Combine the periods
Start from the highest place value (left) to the lowest (right),
and combine all the words you will get your number name.

We will refer to some solved examples for better clarity of the concept

Example 1: Write number name for the following
1. 76,23,45,809
Ans: Seventy-six crore twenty-three lakh forty-five thousand eight hundred nine.

Sol: 
Step 1: Break it down
Crore: 76
Lakh: 23
Thousand: 45
Ones: 809
Step 2: Arrange and combine
Seventy-six crore twenty-three lakh forty-five thousand eight hundred nine.

2. 1,05,08,406
Ans: One crore five lakh eight thousand four hundred six.

Sol:
Step 1: Break it down
Crore: 1
Lakh: 05
Thousand: 08
Ones: 406

Step 2: Arrange and combine
One crore five lakh eight thousand four hundred six.

2. Expanded Form

Refer to the following example to understand the concept of expanded form

Example 2: Write 45,76,90,213 in expanded form

Sol: 
Step 1: The given number is 45 crore 76 lakh 90 thousand 2 hundred thirteen.
The given number in the place value chart is as shown below.

Step 2: Write the table into number name format
Thus, 45,76,90,213 = 4 ten crores + 5 crores + 7 ten lakhs + 6 lakhs+ 9 ten thousands + 0 thousands + 2 hundreds+ 1 ten + 3 ones

Step  3: Mathematically we can write it as

= 4 x 100000000 + 5 x 10000000 + 7 x 1000000 + 6 x 100000 + 9 x 10000 + 0 x 1000 + 2 x 100 + 1 x 10 + 3

= 400000000 + 50000000 + 7000000 + 600000 + 90000 + 0 + 200 + 10 + 3

Comparing & Ordering Large Numbers

In this section, we will  learn about the following three things

• Comparing Large Numbers
• Ordering Large Numbers
• Formation of Greatest and Smallest Number

1. Comparing Large Numbers

For comparing large numbers , we have two cases:

Case 1: Unequal number of digits  
The number having more digits is greater than the number having fewer digits.

Example 1: Which is greater: 8,93,10,009 or 11,00,73,640? 

Sol: On counting both of the digits we found out that,
11,00,73,640 has 9 digits whereas 8,93,10,009 has 8 digits. 
and 9 is greater than 8.
therefore, 11,00,73,640 > 8,93,10,009.

Case 2: Equal number of digits
Start from the left and compare the digits until they differ. 

Example 2: Compare 8,31,46,205 and 8,31,57,610?

Sol: As we can see both of these numbers have same number of digits,
therefore we start comparing from the left until both of them differ.

The digits differ at the ten thousands place.
 5 > 4,
So, 8,31,57,610 > 8,31,46,205.

2. Ordering Large Numbers:

In this , we will arrange large numbers into ascending or descending order according to their values.
Example 3: Arrange the following numbers in descending order.

 8,36,04,295 ; 40,34,51,276 ; 3,04,63,500 ; 89,72,340 ; 99,00,32,941

Sol: 
Step 1: Line up the numbers by their place values

Step 2: 
There are two 9-digit numbers, two 8-digit numbers and one 7-digit number.
Comparing 9-digit numbers.
9 TC > 4 TC, 
So, 99,00,32,941 > 40,34,51,276
Comparing 8-digit numbers.
8 crore > 3 crore, ∴ 8,36,04,295 > 3,04,63,500 
7-digit number is the smallest.
Hence, the numbers written in descending order are:
99,00,32,941 > 40,34,51,276 > 8,36,04,295 > 3,04,63,500 > 89,72,340.

3. Formation of  Smallest & Greatest Numbers

Sol: 
1. Smallest Digit: 
For writing the smallest number, we place the smallest digit at the leftmost place and write the other digits in ascending order.
Here, 0 is the smallest digit but we cannot begin a number with 0.
So, we write the next higher digit which is 2 in this case.
So, the smallest 7-digit number is 2045689
2. Greatest Digit:
For writing the Greatest Number, We place the greatest digit at the leftmost place and the other digits in descending order.
Here, 9 is the greatest digit 
So, the greatest 7-digit number is 9865420

Case 2:  Repetition of Digits is Allowed
Example 5: Write the smallest and the greatest 7-digit number by using the digits 0, 1, 4, 6, 5, 8 and 9, where repetition is allowed.

Sol: Greatest Number: 9999999
Smallest number : 1000000

Arithmetic Operations on Large Numbers

•  Addition of Large numbers
• Subtraction of Large Numbers

1. Addition

Example: The statistics on the Internet suggest that the estimated population of Russia and Mexico is 14,39,64,709 and 13,07,59,074, respectively. If the population of the USA is 32,67,66,748, find out whether the population of Russia and Mexico taken together is more than that of the USA or not.

Sol: 

• The population of Russia and Mexico is 27,47,23,783
• Population of USA = 32,67,66,748
• Since, 3 ten crores > 2 ten crores, the population of the USA is more than the population of Russia and Mexico taken together.

2. Subtraction

Example : The population of Greater Mumbai and Chennai as per the 2011 census was 1,63,68,084 and 64,24,624, respectively. How much more is Greater Mumbai’s population as compared to Chennai?

Sol: 

• The population of Greater Mumbai = 1,63,68,084
• Population of Chennai = 64,24,624
• Population difference  = 1,63,68,084 – 64,24,624 =  99,43,460
• 
• Thus, the population of Greater Mumbai is 99,43,460 more than that of Chennai.
• You can check subtraction by adding the subtrahend to the difference. If the sum is equal to the minuend, then the subtraction has been carried out correctly.
• 

International System of Writing Numbers

In International system, the number 10 lakh is called a million and furthermore every bigger calculations is in millions.

The chart can be extended to its left to include more than 9 places. The period just to the left of the millions period is called the billions period. Then, we have the trillions period and so on. 
On comparing the two systems of numerations, we can observe the following relationships.

• 10 lakh = 1 million
• 1 crore = 100 lakh = 10 million
• 10 crore = 100 million
• 100 crore = 1,000 million = 1 billion
• 1000 crore = 10,000 million = 10 billion

Did You Know

• The Indian system, starting from the right after the first period of 3 digits all other periods are of 2 digits.
• In the International system, each period consists of three digits. 
• We place comma or leave space after every 3 digits.

Example 1: Write in numerals:
(a) 8 million 
(b) Eighty-five million three hundred five thousand nine hundred

Sol: 
(a) 8 million = 8,000,000 
(b) Step by step breakdown of the numbers

1. Eighty-five million = 85,000,000

2. Three hundred five thousand = 305,000

3. Nine hundred = 900
On adding we get, 85,305,900

Example 2: Number names in international system:

1. 23,450,789

Sol: Number Name for the same is given below
Number Name: Twenty-three million four hundred fifty thousand seven hundred eighty-nine

2. 105,067,092

Sol: Number Name for the same is given below
Number Name: One hundred five million sixty-seven thousand ninety-two

Example 3: Write ‘sixty-nine crore four lakh forty-one thousand six hundred twenty-seven’ in the International system.

Sol: The numeral in Indian system is 69 04 41 627. 
On writing it in International system, we get,
Numeral : 690,441,627
Number Name : Six hundred ninety million four hundred forty-one thousand six hundred twenty-seven.