Download NUMBER SYS LEC -1

J E E
(Mathematics) Class IX
Number System
7
3
and  on real number line.
3
7
–
For
3
:
7
For 
7
3
Q
P
O
–7 –6 –5 –4 –3 –2 –1
0

Divide segment OA into 7 equal parts

count 3 parts from 0
3
: 
7
3
7
A
B
C
1
2
3
Divide equal segment OP, PQ into 3 equal parts

Count 7 parts from 0 to left

For mixed fraction convert to proper fraction first.
EXAMPLE 3 :
Represent 2
in
IIT I N EET I B OARD S I FO UNDATIO N I OTH ERS
EXAMPLE 2 :
Represent
N
o n lin e p a d h o ...
3
on a number line
8
2
Q
A
B
0
1
2
3
 3
2 8  2  8 


To represent 2
3
3
: After 2 represent
8
8

Divide BC into 8 parts

Count 3 parts B to C
OR
Represent 2
3 19

on number line like example 2.
8
8
3
8
C
3
JE E
N
in
o n lin e p a d h o ...
(Mathematics) Class IX
Number System
IIT I NEET I BOARDS I FOUNDATIONI OTHERS
Irrational Numbers :
If a number cannot be written in the form of
p
where p and q are integers and q  0, then the number is
q
called as irrational number.
2, 3 5 and 2  5 etc.
Example :
Real Numbers :
Numbers which can represent actual physical quantities in a meaningful way are known as real numbers.
Real numbers includes all rational and irrational numbers.
Prime Numbers :
All natural numbers which have 1 and itself only as their factors are called prime numbers.
Example :
2, 3, 5, 7, 11, 13, 17, 19, ..... etc.
Composite Numbers : All natural numbers which are not prime are composite numbers.
Co-prime Numbers : If HCF of given numbers is 1, then they are co-primes.
Example :

4 and 9
Any two consecutive numbers will always be co-primes.
Imaginary Numbers :
All the numbers that have their square is negative are called imaginary numbers.
Example :
3i, –2i,
1 etc.
i  1
Rational Numbers :

A rational number
a
a
is positive if both a and b have same sign and
is negative if both have
b
b
opposite sign

Every rational number can be expressed as either a terminating decimal or a recurring decimal.

Every integer can be expressed as
Example :
3
p
, where q = 1.
q
3
5
9
, 5  , 9  etc.
1
1
1
EXAMPLE 1 :
State whether following statements are true/false. Give reason for your answer.
(i)
Every rational number is a whole number
(F)
(ii) Every whole number is an integer
(T)
(iii) Every rational number is an integer
(F)
Representing Rational Numbers On Number Line :
In a rational number, the numeral below bar is denominator, represents the number of equal parts into
which upper part i.e., numerator has been divided. Draw a line and mark a point 0 on it to represent
number ‘0’ (zero). The positive rational numbers will be represented by points on right side of 0 and
negative will be on left side of 0.
(Mathematics) Class IX
Number System
J E E
N
o n lin e p a d h o ...
in
IIT I N EET I B OARD S I FO UNDATIO N I OTH ERS
LECTURE # 01
INTRODUCTION :
Natural Numbers :
Counting numbers are called as natural numbers
Natural numbers (N) = {1, 2, 3, 4, .....}
Whole Numbers :
If we include zero (0) to collection of natural numbers, then all together from whole numbers
Whole numbers W = {0} + N
W = {0, 1, 2, 3, 4, .....}
Integers :
If we include negative of natural numbers to collection of whole numbers then all together from collection
of integers.
I or Z = {....., –2, –1, 0, 1, 2, 3, .....}

0 is non-negative and non-positive integer (neither negative nor positive).
Non-negative integers : 0, 1, 2, .....
Non-positive integers : ....., –3, –2, –1, 0
Rational Numbers :
The numbers which can be expressed in the form of
p
, q  0 and q and p are integers. The set of rational
q
numbers is denoted by Q.
1 2 1

Q   , , , ..... etc.
3 9 4

Note : All natural numbers, whole numbers and integers are rational.
Q=N+Q+Z

1 2 10
 
and so on all are equivalent rational numbers.
2 4 20

Two rational numbers are said to be equivalent if we divide each decimal value will be the same.
Difference Between Fraction And Rational Number :
A fraction is a number that expresses part of a whole as a quotient of integers (denominator  0) or as a
repeating or terminating decimal. So every fraction is a rational number but every rational number is not
fraction.
Example :

4
is rational but not a fraction
1
The fractions are a subset of the rational numbers
The rationals contain the integers, and fraction don’t.