Category : 6th Class
Number System and Its Operations
Numbers are the symbolic representation of counted objects. There are infinite counting numbers from 1. Some arcdivisible by another whereas some are not divisible. Numbers are differentiated according to their divisibility and factors. A numeral system is a writing system for expressing numbers. The most commonly used system of numerals is HinduArabic numeral system. In this chapter, we will learn about various numeral systems, types of numbers and operation on numbers.
Indian or HinduArabic Number System
This number system was introduced by Indians, and is therefore, called Indian Number System. In this number system 10 is considered as the base.
10 ones = 10, 10 tens = 1 hundred, 10 hundreds = 1 thousand
Hindu  Arabic number system is based on the place value of digits in number
Indian Place Value Chart
crores 
Ten Lakes 
Lakes 
Ten Thousand 
Thousands 
Hundred 
Tens 
Ones 


2 
9 
8 
7 
3 
5 
The number two lakh ninetyeight thousand seven hundred and thirtyfive is written by placing 2 at the place of "lakhs', 9 at the place of "Ten thousands', 8 at "Thousands', 7 at "Hundreds', 3 at "Tens' and 5 at "Ones',
Place Value
If a number contains more than one digit then the place occupies by each digit is its place value. In the number 732 the number 7 occupies the place of hundreds, therefore, the place value of 7 is seven hundred.
Face Value
The face value of a number does not change regardless of the place it occupies. Therefore, the face value of a number is the number itself.
International Place Value Chart
Billions 
Millions 
Thousand 




Ten billion 
Billion 
Hundred million 
Ten million 
Million 
Hundred thousand 
Ten 
Thousand 
Hundred 
tens 
Ones 

5 
6 
8 
4 
3 
2 
5 
4 
3 
1 
The above chart is the international place value chart. The number 5, 684, 325, 431 is read as five billion, six hundred eightyfour million, three hundred twentyfive thousand, four hundred thirtyone.
Comparison between Indian and International Number System
International 
Hundred billion 
Ten billion 
Billion 
Hundred million 
Ten million 
Million 
Hundred thousand 
Ten Thousand 
Thousand 
Hundred 
Ten 
Ones 
Indian 
Kharab 
Ten arab 
Arab 
Ten crore 
Crore 
Ten lakes 
Lakes 
Ten thousand 
Thousand 
Hundred 
Ten 
Ones 
According to the place value of International number system, which one is six million five hundred fifteen thousand two hundred twentyone?
(a) 60515221 (b) 6515221
(c) 65150221 (d) 600515221
(e) None of these
Answer (b)
What is the reduced form of the following expanded from:
\[7\text{ }\times \text{ }10000\text{ }+\text{ }5\text{ }\times \text{ }100\text{ }+\text{ }4\text{ }\times \text{ }10\text{ }+\text{ }6\].
(a) 70546 (b) 7546
(c) 75460 (d) 7546
(e) None of these
Answer (a)
Explanation: \[7\text{ }\times \text{ }10000\text{ }+\text{ }5\text{ }\times \text{ }100\text{ }+\text{ }4\text{ }\times \text{ }10\text{ }\text{ }6\text{ }=\text{ }70000\text{ }+\text{ }500\text{ }+\text{ }40\text{ }+\text{ }6\text{ }=7546\]
Roman Numerals
This numeric system is represented by Roman numerals which are the combinations of letters from the latin alphabet. Roman numerals are based on seven symbols as. given below:
Symbol 
I 
V 
X 
L 
C 
D 
M 
Value 
1 
5 
10 
50 
100 
500 
1000 
The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 are expressed in Roman numerals as I, II, III, IV, V, VI VII, VIII, IX, X
The rules for this system are:
Note: Only I, X, C and M can be repeated. But the symbols V, L and D are never repeated.
Note:
(a) The symbols V, L and D are never written to the left of a symbol of greater value ie. V, L and D are never subtracted.
(b) The symbol I can be subtracted from V and X only.
(c) The symbol x can be subtracted from L, M and C only.

5000 
10000 
50000 
100000 
500000 
1000000 
Value 
\[\overline{V}\] 
\[\overline{X}\] 
\[\overline{L}\] 
\[\overline{C}\] 
\[\overline{D}\] 
\[\overline{M}\] 
Arrange the following Roman numerals in descending order.
MD, MDCCC, DCCCXC, CM
(a) MDCCC > MD> DCCCXC >CM
(b) MDCCC > DCCCXC > MD >CM
(c) MDCCC > CM> DCCCXC > MD
(d) MDCCC > MD> CM>DCCCXC
(e) None of these
Answer (d)
Explanation:
Clearly, MDCCC = 1800, MD = 1500, CM = 900 and DCCCXC = 890
\[\therefore \] Option (d) is correct.
TYPES OF NUMBERS
Natural Numbers
Every counting number is called a natural number. 1, 2, 3, 4, 5, etc. are natural numbers. Zero is excluded from the natural numbers.
Natural numbers are represented by N (First capital letter of its name).
Whole Numbers
When 0 is included with counting numbers, it becomes whole number. Whole number is represented by W (First letter of its name). Whole numbers (W) = {0, 1, 2, 3, 4 .....}.
Prime Numbers
The numbers which have only two factors, 1 and the number itself are called prime numbers. The numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, etc are prime numbers.
Composite Numbers
The numbers which have more than two factors are called composite numbers. actors of 4 and 6 are: 1, 2, 4 and 1, 2, 3, 6 respectively. Hence, these are composite numbers.
Coprime
If H.C.R (Highest Common Factor) of two numbers is 1 then the numbers are called coprime. The numbers 21 & 22 are coprime as both have no common factors other than 1.
Perfect Numbers
The natural numbers whose sum of positive divisors (excluding the number itself) is equal to the number itself are called perfect numbers. The first perfect number Is 6.
Note:
Euclid proved that \[{{2}^{p1}}\,({{2}^{p}}1)\] is an even perfect number whenever \[{{2}^{p}}1\] is prime and o is a natural number
Successor
Successor of every number comes just after the number Successor of 25 = 25 + 1 = 26. Successor of 4573 = 4573 + 1 = 4574.
Predecessor
Predecessor of every number comes just before the number Predecessor of 25 is obtained by subtracting 1 from the number Predecessor of 23 = 23  1 = 22,
Choose the composite numbers from the following numbers:
87, 67, 45, 34, 23, 27, 33.
(a) 45, 87, 34, 27, 33 (b) 45, 87, 67, 33
(c) 33, 27, 23, 34 (d) Ali the above
(e) None of these
Answer (a)
Explanation: In the given numbers 23 and 67 are prime numbers and rest are composite numbers.
How many prime numbers are there between 10 to 50?
(a) 10 (b) 11
(c) 13 (d) 18
(e) None of these
Answer (b)
Explanation: Prime numbers between 10 and 50 are:
11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47.
PROPERTIES OF ADDITION
Closure Property
The sum of two whole numbers is always a whole number. If a and b are two whole numbers, then their addition (a + b) is also a whole number.
Commutative Property
The sum of two and more whole numbers remains same even if the order of the numbers are changed.
Associative Property
The addition of a set of numbers is same regardless of how the numbers are grouped. The associative property involves 3 or more numbers.
Additive Identity
Zero (0) is called the additive identity of every whole number. When 0 is added to the whole number its identity does not change or number remains unchanged.
Additive Inverse
Additive inverse of a is  a and additive inverse of  a is a, therefore, the sum of number with its additive inverse is always zero.
PROPERTIES OF SUBTRACTION
PROPERTIES OF MULTIPLICATION
Closure Property
If a and b are whole numbers, then their product \[a\times b\] is also a whole number. Let us consider two whole numbers 3 and 4, their product is 12, which is also a whole number.
Commutative Property
The product of the whole numbers remains same even if the order of the multiplication s changed. In other words if a and b are whole numbers/ their product \[a\text{ }\times \text{ }b\text{ }=\text{ }b\text{ }\times \text{ }a\]
For example: \[5\text{ }\times \text{ }6\text{ }=\text{ }30\], on changing their order, \[6\text{ }\times \text{ }5\text{ }=\text{ }30\],. Thus\[5\,\,\times \,\,6\,\,\times \,\,=\,\,6\,\,\times \,\,5\].
Similarly, \[10\text{ }\times \text{ }15\text{ }=\text{ }150\]and\[15\text{ }\times \text{ }10\text{ }=\text{ }150\], thus, \[10\text{ }\times \text{ }15\text{ }=\text{ 15 }\times \text{ }10\],
Associative Property
The product of more than two numbers remains same by changing the groups of the numbers. If a, b and c are three numbers then their product\[(a\times b)\times c=a\times (b\times c)\].
For example: \[(4\,\,\times \,\,5)\,\,\times \,\,6=20\,\,\times \,\,\,620\]and\[4\,\,\times \,\,(5\,\,\times \,\,6)\,\,\times \,\,30=120\]. Thus\[(4\,\,\times \,\,5)\,\times 6\,\,=4\times \,\,(5\,\,\times \,\,6)\].
Multiplicative Identity
The product of every whole number with 1 is the number itself.
If a is a whole number then, \[a\,\,\times \,\,1=1\,\,\times \,\,a=a\]
Multiplication of a number by 1 is the number itself, therefore, the identity of the whole number does not change thus 1 is called multiplicative identity of the whole number, i.e.\[5\,\,\,\times \,\,\,1=1\,\,\times \,\,5=5\text{ }and\text{ }10\text{ }\times \text{ }1\text{ }=\text{ 1}\,\,\times \text{ }10\text{ }=\text{ }10\].
Multiplication of Whole Numbers with 0:
When a whole number is multiplied by 0, it becomes equal to zero.
In other words\[O\,\,\times \,\,\,a=a\,\,\times \text{ }0\text{ }=\text{ }0.\text{ i}\text{.}e.\text{ }O\,\,\times \text{ }11\text{ }=\text{ }0\text{ }and\text{ }11\,\,\,\times \,\,\,0\text{ }=\text{ }0\].
Multiplicative Inverse
Multiplicative inverse of a number a is \[\frac{1}{a}\] and multiplication of a and \[\frac{1}{a}\] is 1,
Let us consider a number 4 and its multiplicative inverse \[\frac{1}{4}\]
Hence, its multiplication \[=4\,\,\times \frac{1}{4}=1\]
PROPERTIES OF DIVISION
BODMAS Rule
When a single expression contains many mathematical operations then BODMAS rule is used for the simplification of the expression. The word BODMAS has been arranged according to the priority of the operations.
The letters of BODMAS express the following operations:
B 
Stands for Bracket 
O 
Stands for of 
D 
Stands for Division 
M 
Stands for Multiplication 
A 
Stands for Addition and 
S 
Stands for Subtraction 
Evaluate: \[\mathbf{[}\left\{ \left( \mathbf{563 \times 8} \right)\mathbf{ + 70} \right\}\,\,\mathbf{}\,\,\mathbf{4384] \div 10}\].
(a) 60 (b) 45
(c) 19 (d) 24
(e) None of these
Answer (c)
Explanation: \[[\left\{ \left( 563\text{ }\times \text{ }8 \right)\text{ }+\text{ }70 \right\}\,\,\text{}\,\,\text{4384 }\!\!]\!\!\text{ }\div \text{ }10\].
\[=\text{ }\left( 4504\text{ }+\text{ }70\text{ }\text{ }4384 \right)\text{ }\div \text{ }10\text{ }=\text{ }190\text{ }\div \text{ }10\text{ }=\text{ }19\]
Example:
Product of 23 and 13 is added with 56 then subtracted from 675. Which one of the following is correct for the arrangement of given statement using brackets?
(a) \[675\text{ }\text{ }\left\{ \left( 23\text{ }\times \text{ }13 \right)\text{ }+\text{ }56 \right\}\] (b) \[\left\{ \left( 23\text{ }\times \text{ }13 \right)\text{ }+\text{ }56 \right\}\text{ }\text{ }675\]
(c) \[\left( 675\text{ }+\text{ }56 \right)\text{ }\text{ }\left( 23\text{ }\times \text{ }13 \right)\] (d) All the above
(e) None of these
Answer (a)
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