Class 6 – NumberSystem
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Study Material
Number: A number is a concept which exists only in mind. The earliest concept of number was a thought that allowed people to mentally picturize the size of some collection of objects. To write down a number we use a symbol called “Numeral”. Numeral: A group of digits denoting a number is called ‘Numeral’. Digits: The numerals 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are called digits. The counting numbers were used by people in the Stone Age for counting things like property, cattle, enemies, etc. The very first stage of learning mathematics begins with counting of objects from ‘1’ onwards. Natural numbers: The counting numbers starting from 1 are called natural numbers. These are denoted by the letter ‘N’. N = {1, 2, 3, 4, 5, 6 …} (i) Representing a number in figures is called Notation. (ii) The act of reading numbers in words when expressed by means of numerals is called Numeration. 
(iii) In our number system we use 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 (10 digits) to represent the numerals. So it is called positional number system with base ‘10’ (base10 system). There are two types of number systems. i) Hindu Arabic (or) Indian system. ii) International (or) English system. Hindu Arabic system: In the HinduArabic numeration system, ten ones are replaced by one ten, ten tens are replaced by one hundred, ten hundreds are replaced by one thousand, 10 one thousands are replaced by 10 thousands, and so on. The periods in HinduArabic numeration starting from extreme right are Units, Thousands, Lakhs, Crores, etc. i) In units period, there are 3 places which are ones, tens and hundreds. ii) In thousands period, there are 2 places which are thousands and ten thousands. 

iii) In lakhs period, there are 2 places which are lakhs and ten lakhs.
iv) Crores period consists of any number of places.
So, in Hindu Arabic system commas are used after 3, 2, 2,…… from right to left with respect to periods.
CRORES PERIOD 
LAKHS PERIOD 
THOUSANDS PERIOD 
ONES PERIOD 

Ten Crores 
Crores 
Ten lakhs 
Lakh 
Ten Thousands 
Thousands 
Hundreds 
Ten 
Units 
Example: The number 345678909 is written in HinduArabic system by using commas as 34, 56, 78, 909.
And read as Thirty four crores, fifty six lakhs, seventy eight thousands, nine hundred and nine.
International System: In this system of numeration, periods are made with commas after every three digits from the right. The periods in this system are made as follows:
BILLIONS PERIOD 
MILLIONS PERIOD 
THOUSANDS PERIOD 
ONES PERIOD 

Hundred billions 
Ten billions 
Billions 
Hundred millions 
Ten millions 
Millions 
Hundred Thousands 
Ten Thousands 
Thousands 
Hundreds 
Ten 
Units 
Conversion from HinduArabic system to English System:
Hindu Arabic system 
English system 
One lakh 
Hundred thousands 
Ten lakhs 
One million 
One crore 
Ten millions 
Ten crores 
Hundred millions 
Example: The date of birth of the great Indian mathematician Srinivasa Ramanujan is 22nd December 1887, written as 22121887.
The number written in HinduArabic and international system as:
i) In International system is twenty two millions, one hundred twenty one thousands, eight hundred and eighty seven.
ii) In Hindu Arabic system is two crores, twenty one lakhs, twenty one thousands, eight hundred and eighty seven.
Facevalue of a digit in a number: The facevalue of a digit in a numeral is the digit itself at whatever place it may be.
Example: In the number 869456
The facevalue of 9 is 9
The facevalue of 6 is 6
Placevalue of a digit in a numeral: The placevalue of a digit in a number depends on the place it occupies in the place value chart.
Example: In the number 32568
1. The placevalue of 5 = 5 × 100 = 500 ( 5 in hundreds place)
2. The placevalue of 3 = 3 × 10,000 = 30,000 ( 3 in ten thousands place)
Place value and face value: Every digit of a number has two valuesthe face value and the place value. The face value of a digit is the value of the digit itself and does not change, while the place value of the digit changes according to its position in the number. This can be shown as:
Number 
Digit 
Face value 
Place value 
67,923 
6 
6 
60,000 
7 
7 
7,000 

9 
9 
900 

2 
2 
20 

3 
3 
3 
Example: What is the difference between place value and face value of 8 in 478526?
Solution:
The place value of 8 = 8 × 1000 =8000
The face value of 8 = 8
The difference between place value and face value of 8 = 8000 – 8 = 7992
The placevalue or facevalue of ‘0’ is always ‘0’ wherever it may be.
Expanded form of a number: If a number is written as the sum of the place value of all the digits of the number, then the number is said tobe in its expanded form.
Example:4235 = 4000 + 200 + 30 + 5 (expanded form).
Short form of a number: The expanded form of the number can be written as a single number (numeral) then it is called the short form of the number.
Example:70000 + 3000 + 500 + 00 + 7 = 73507 (short form).
Comparison of Numbers: In order to compare two numbers use the following rules:
Rule 1: If the two numbers with different number of digits are given, then the number which has more digits is the greater of the two.
Example:Find the greater among 278654 and 76543671.
Solution: In the given two numbers 278654has 6 digits, while 76543671 has 8 digits.
∴ 76543671 > 278654
Rule 2: If the two numbers to be compared have the same number of digits.
i) First compare the digits in the extreme left place in both the numbers.
ii) If they are of equal value, compare the second digit from the left in both the numbers.
iii) If they are of equal value, compare the third digit from the left in both the numbers.
iv) Continue until you come to unequal digits at the corresponding places.
v) Now the number with such digit is the greater of the two.
Example: Which is greater: 8569734 or 8561867?
Solution: Arrange the two given numbers in the place value chart.
TL 
L 
TTh 
Th 
H 
T 
U 
8 
5 
6 
9 
7 
3 
4 
8 
5 
6 
1 
8 
6 
7 
Both the numbers have 7 digits. At the tenlakhs place, both have the same digit 8.Also at lakhs, and tenthousands both the given numbers have same digits 5 and 6 respectively. But at thousands place the first number has 9 while the second one has 1, Since 9 > 1
∴ 8569734> 8561867.
Ordering of numbers:
i) The arrangement of numbers from the smallest to the greatest is called ascending order.
Example: 560, 567, 662, 665, 786
ii) The arrangement of numbers from the greatest to the smallest is called descending order.
Example: 789, 665, 662, 567, 560 are in descending order.
Rules for formation of the largest and least numbers with required number of digits:
To form the largest number with required number of digits fill the places with the digit 9.
To form the smallest number with required number of digits fill the places with the first digit in the number from left with 1 and rest with zeroes.
Table of forming the greatest and least numbers with required digits:
Number of digits 
Largest Number 
Least Number 
1 
9 
1 
2 
99 
10 
3 
999 
100 
4 
9999 
1000 
5 
99999 
10000 
The smallest number using the given digits, each only once without repetition:
Case I. If none of the given digits is zero.
In this case, arrange the given digits in ascending order.
Examples:
(i) The smallest 3digit number formed by using the digits 7, 3 and 9 is 379.
(ii) The smallest 4digit number formed by using the digits 7, 2, 5 and 9 is 2579.
Case II. If one of the given digits is zero.
In this case, put 0 at second place from the left. Then the remaining places are filled from left to right by the remaining digits in ascending order.
Examples:
(i) The smallest 3digit number formed by using the digits 0, 2 and 5 is 205.
(ii) The smallest 4digit number formed by using the digits 8, 6, 1 and 0 is 1068.
The greatest number using the given digits, each only once without repetition: To form the greatest number, the given digits are arranged in descending order.
Examples:
(i) The greatest 3digit number formed by using the digits 3, 6and 7 is 763.
(ii) The greatest 4digit number formed by using the digits 0, 5,8 and 9 is 9850.
The smallest or greatest number, using the given digits, when repetition of digits is allowed:
In this case, first form the smallest or greatest number using the given digits each only once without repetition. Then, in the number so formed,replace the digit whose repetition is allowed.
Example: Find the greatest and least number of 5 digits using digits 8, 3, 6 with 6 as repetition.
Solution: The least 5 digit number is 36668.
The greatest 5 digit number is 86663.
Exception: If the smallest number to be formed with one of the digits as zero, then in the number, zero has to be in the second place from the left.Thus, if the digit at the left
most places is to be repeated twice then the repeated digits lie on either side of the zero in the second place.
Examples:
(i) The smallest 3digit number formed by using the digits 0 and 7 with ‘7’ as repetition is 707.
(ii) The smallest 4digit number formed by using the digits 0, 5 and 6 with 5 as repetition is 5056.
Example:How many 8digit numbers are there in all?
Solution: The largest 8digit number =99999999
The smallest 8digit number = 10000000.
Number of all 8digit numbers
= (99999999) – (10000000) + 1[1 is for the beginning number]
= (89999999 + 1) = 90000000
= nine crores.
Aliter 1:
The smallest 9 – digit number = 100000000
The smallest 8 – digit number = 10000000
Number of all 8 digit numbers = 100000000 – 10000000
= 90000000 (nine crores)
Aliter 2:
The greatest 8 – digit number = 99999999
The greatest 7 – digit number = 9999999
Number of all 8 digit numbers = 99999999 – 9999999
= 90000000 (nine crores)
Hence, there are in all nine crores of 8digit numbers.