Session 1: International Number System and Indian Number System – Definition, Chart, Interactives and Examples

During the Vedic period (6th and 7th centuries BC), motivated by astronomy, a numerical system and basic mathematical operations developed in northern India. Hindu cosmology required the mastery of very large numbers, so used a set of 10 symbols. This system was introduced to Middle Eastern mathematicians, who later introduced the system to Europe.

Different numeral systems were used across different countries and periods. The system adopted in India became known as the Indian numeral system.

Indian Number System (Indian Numeral System)

The Indian numeral system can be distinguished from the other numeral systems based on the nomenclature followed for different groups or periods. In this system, starting from the right, the groups are ones, thousands, lakhs and crores.

Place Value Chart – Indian Number System

The following place value chart depicts the different periods and places according to the number of digits in a number.

Reading and Writing Numbers in the Indian Number System

In the Indian system of numeration, numbers are written using a set of 10 symbols—1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Each digit of a number gets its value in this system depending on its place in the place value chart. While writing the number names of the numerals in the Indian numeral system, separators (commas) are marked to distinguish different periods. The first three digits from the right of a number make ones period. The following two digits make thousands period, the next two as lakhs period and the next two as crores period. The digits in the same period are read together (except ones), and the name of the period is read along with them. The following figure shows how to read and write the number name for 85419723, with the help of the Indian place value chart. 

Indian Number System: Expanded Form

We can stretch a number to see how much value each digit is worth. This is called expanded form. In the Indian numeral system, the place value of a digit decreases 10 times from left to the right and increases 10 times from right to the left. 

Let us write 4,19,723 in expanded form. 

4,19,723 is written in expanded form as 4,00,000+10,000+9,000+700+20+3.

It means there are four lakhs, one ten thousand, nine thousands, seven hundreds, two tens, and three ones in this number. We can easily know the place value of each digit in a number through its expanded form.

1) Express 91,827 in expanded form. 

In the number 91,827, 

The place value of 9 is 9 ten thousands=9×10,000

The place value of 1 is 1 thousands=1×1,000

The place value of 8 is 8 hundreds=8×100

The place value of 2 is 2 tens=2×10

The place value of 7 is 7 ones=7×1

In the expanded form, the number 91827 can be written as

91,827=90,000+1,000+800+20+7

International Number System (International Numeral System)

The International numeral system is the most commonly used in the world. The International numeral system uses separators (commas) differently from the Indian norm. Instead of grouping the rightmost three digits (until the hundreds place), and after that group by sets of two digits as in the Indian system, the International system groups digits by threes.

Place Value Chart – International Number System

Look at the place value chart below to understand the period system in the International number system.

Reading and Writing Numbers in the International Number System

The International system corresponds to the Indian system till the first five digits. For the places higher than ten thousand, the names no longer correspond. In the International system, after the ten thousands place, we have hundred thousands, millions, ten millions, hundred millions, billions, and so on; there are new words for after third place. The following figure shows how to read and write the number name for 85419723, with the help of the International place value chart. 

Formation of Numbers with Given Digits

In the formation of numbers with the given digits, we may say that a number is an arranged group of digits. Numbers may be formed with or without the repetition of digits. 

Case 1: When the digits are not repeated

Let’s understand the formation of numbers without the repetition of digits using an example.

2) Write all three-digit numbers using the digits 3, 6, 9, taking each digit only once.

We are required to write three-digit numbers using the digits 3, 6, 9, and the repetition of digits is not allowed.

Keeping 3 at ones place, three-digit numbers are 693 and 963.

Keeping 6 at ones place, three-digit numbers are 396 and 936.

Keeping 9 at ones place, three-digit numbers are 639 and 369. 

Hence, all possible three-digit numbers are 693, 963, 396, 936, 639 and 369.

3) Find the number of three-digit numbers that can be formed by the digits 4, 7 and 0 using each digit only once.

The given digits are 4, 7 and 0 and repetition of digits in any number is not allowed.

Keeping 0 at ones place, three-digit numbers are 740 and 470.

Keeping 0 at tens place, three-digit numbers are 704 and 407.

The digit 0 cannot be put at hundreds place because that would make it a two-digit number.

Therefore, the three-digit numbers are 740, 470, 704 and 407. 

Hence the total number of three-digit required numbers are 4.

Case 2: When the digits are repeated

Let’s understand the formation of numbers with the repetition of digits using the following example.

4) Write all possible two-digit numbers that can be formed by using the digits 2, 7 and 8; repetition of digits is allowed.

We are required to write two-digit numbers. The given digits are 2, 7 and 8 and repetition is allowed.

From the given digits, the possible ways of choosing two digits are 2, 7; 2, 8; 7, 8.

Using the digits 2 and 7, the numbers are 27, 72, 22 and 77.

Using the digits 2 and 8, the numbers are 28, 82, 22, and 88.

But the number 22 has already been taken, so the new numbers are 28, 82 and 88.

Using the digits 7, 8, the numbers are 78, 87, 77 and 88. 

But 77 and 88 have already been taken, so the new numbers are 78 and 87.

Hence, the required numbers are 27, 72, 22, 77, 28, 82, 88, 78 and 87.

Note: When the numbers of digits are increased, more and more numbers are formed.

Formation of Greatest and Smallest Numbers

Case 1: When the digits are not repeated

When none of the given digits is zero, we arrange the digits in ascending order to form the smallest number and in descending order to form the greatest number.

For example, the smallest 4-digit number that can be formed by the digits 9, 7, 3 and 8 is 3789, and the greatest 4-digit number that the digits can form is 9, 7, 3 and 8 is 9873.

When one of the digits is zero, we place 0 at the second place from the left and arrange the remaining digits in ascending order to form the smallest number. 

For example, the smallest 4-digit number formed using the digits 9, 0, 4 and 2 is 2049. 

We arrange the digits, including 0, in descending order to form the greatest number. 

For example, the greatest 4-digit number that can be formed by using the digits 9, 0, 4 and 2 is 9420.

Case 2: When digits are repeated

When none of the given digits is zero, first form the smallest or greatest number using the given digits only once without repetition. Then, in the number formed, repeat the digit by the number of times allowed. 

For example, the smallest 3 digit number formed using the digits 3 and 6 is 336, while the greatest digit 3 digit number formed using the same digits is 663.

When one of the digits is zero, first form the smallest or greatest number using the given digits only once without repetition. If the digit present at the leftmost place is to be repeated twice, then the repeated digits lie on either side of the zero in the second place. 

For example, the smallest 4-digit number formed by 3, 5 and 0 repeating 3 twice is 3035, and the greatest 4-digit number that can be formed by 3, 5 and 0 repeating 3 twice is 5330.

International Number System and Indian Number System – Examples

Example 1

Using commas, rewrite the number 56473829 in the Indian and the International numeral system. How will you read this number according to both systems?

In the Indian numeral system, we group the rightmost three digits (until the hundreds place), and after that, group by sets of two digits and insert commas. We divide the number into periods and read the period separately.

Indian numeral form: 5,64,73,829

Read as: Five crore, sixty four lakh, seventy three thousand, eight hundred and twenty nine.

In the International numeral system, we group the digits by threes and insert commas after every three digits. The same given number, 56473829, can be written in the International numeral system in the following way:

International numeral form: 56,473,829

Read as: Fifty six million, four hundred seventy three thousand, eight hundred and twenty nine.

Example 2

Find the difference of the place values of two 6’s in 868126030. 

The given number can be written as 86,81,26,030.

The place value of 6 at crores place=6×10000000=60000000

The place value of 6 at thousands place=6×1000=6000

∴ the required difference=60000000−6000=5,99,94,000

Example 3

In a five-digit number, the digit in the thousands place is 7, and the digit in the hundreds place is 3 less than 7. The number has no tens. The ones place digit is 40 divided by 8. The ten thousands place digit is the sum of the hundreds and ones digit. Find the number.

We have, 

Digit in the thousands place =7

Digits in the tens place =0       [ ∵ the number has no tens ]

Digit in the hundreds place =7−3=4

Digit in the ones place=40÷8=5.

Digit in the ten thousands place = Digit in the hundreds place + Digit in the ones place =4+5=9   

Hence, the required number=(9×10000)+(7×1000)+(4×100)+(0×10)+(5×1)                                                                                                   =90000+7000+400+0+5 =97405

Example 4

Write all possible two-digit numbers that can be formed using the digits 4, 7 and 8 when repetition of digits is not allowed.

Out of the given digits, the possible ways of choosing two digits are 4, 7; 4, 8; 7,8.

Using the digits 4 and 7, the numbers are 47 and 74.

Using the digits 4 and 8, the number are 48 and 84.

Using the digits 7 and 8, the numbers are 78 and 87. 

Hence, all possible two-digit numbers are 47, 74, 48, 84, 78, 87.

Example 5

Write the smallest and the greatest 4-digit number that can be formed by the digits 7, 5, 8, 2.

We arrange the given digits in ascending order to form the smallest number.

Hence, the smallest number is 2,578.

We arrange the given digits in descending order to form the greatest number.

Hence, the greatest number is 8,752.

Example 6

Write the smallest 6-digit number having three different digits.

We know that the smallest 6-digit number is 100000. To generate the smallest 6-digit number with three different digits, we must first choose the three smallest digits. 

Clearly, 0, 1 and 2 are the smallest digits.

Now, we will put them in ascending order at the place having the highest place value. Clearly, the left-most place is the place having the highest place value. 

The digit 0 cannot be put at this place because that would make it a 2-digit number.

So, we put the next smallest digit, i.e., 1, in the leftmost place of the number and the largest digit, i.e. 2, in the ones place. All other places are filled with zeros. 

Hence, the required number is 100002.

Example 7

Write the greatest 5-digit number having three different digits.

We will have to use the three largest digits to write the greatest 5-digit number having three different digits. 

Clearly, 9, 8 and 7 are the three greatest digits. 

To write the greatest 5-digit number using 7, 8 and 9, we put the greatest digit 9 at the place having the highest place value. The smallest digit, 7 is out at the rightmost place, i.e., ones place and the digit 8 is put at the tens place. All the other places are filled by 9.

Hence, the required number = 99987