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NUMBER SYSTEM
Real Number
Rational or irrational numbers are called real number. Since all
rational and irrational numbers can be represented on number
line, that’s why number line is called real number line. The set
of real numbers is denoted by R.
Rational Numbers
Rational numbers are those numbers, which can be expressed
p
, where p and q are integers and q ≠ 0
in the form
q
(Division by 0 is not defined). Every integer p is a rational
p
number because it can be written as
.
1
If rational number is converted into decimal form, then
(i) Either they are terminating decimal.
(ii) Or they are non-terminating, but repeating decimals.
Example
13
(i)
4
7
(ii)
3
=
=
Example
(i) Convert
12
8
1.5
=
(ii) Convert
17
4
2.
4.25
=
Terminating Decimal To Rational Number
2.33333……
Example
Rational number is said to be in its simplest form if its
numerator and denominator have no common factor other
p
is a rational number, then it is in simplest
that one. So, if
q
form if p and q have no common divisor other than 1.
(i) Convert 6.4 into the form
6.4
=
64
in its simplest form.
80
4
16
=
5
20
8.25
(i) Convert
Number
64
10
p
.
q
32
5
=
(ii) Convert 8.25 into the form
Example
Rational
p
.
q
Steps
1. Remove decimal point from numerator.
2. Write 1 in denominator and put as many Zeros (0) as
decimal places in the number.
3. Reduce the fraction to its simplest form by dividing the
numerator and denominator by their common divisor.
3.25
Conversion Between
Decimal Number
17
into decimal form.
4
We can convert a number in decimal form to the form
Simplest Form Of A Rational Number
64
=
80
12
into decimal form.
8
And
Rational number can be converted to decimal form. For this,
we divide the numerator of rational number with its
denominator till the quotient is up to desired decimal places.
The conversion of rational number to decimal form can be
categorized in the following ways: 1. Rational Number To Finite/Terminating Decimals
The rational numbers for which the division terminates after a
finite number of steps or with finite decimal part are known as
finite/terminating decimal.
=
825
100
=
p
.
q
33
4
3. Rational Number To Repeating/Recurring Decimal
Some times on dividing the numerator of a rational number
with its denominator, the quotient starts repeating itself. Such
decimals are called Repeating/Recurring decimal.
Steps
1. Take the numerator of rational number as dividend and
denominator as divisor.
2. Divide the numerator by denominator.
3. Divide the numerator by denominator till quotient starts
repeating itself.
Example
(i) Convert
8
into decimal form.
3
8
= =
2.6666......
2.6
Steps
3
1. Take the numerator of rational number as dividend and
denominator as divisor.
16
2. Divide the numerator by denominator till the remainder is
(ii) Convert
into decimal form.
45
0 (Zero).
16
= =
0.3555.....
0 .3 5
45
4. Repeating Decimal To Rational Number
To convert a terminating decimal to a rational number, the
following steps are used.
p
and name the equation (i).
q
1.
Put the number equal to
2.
Multiply the equation with that power of 10, so only
repeating decimal remain on the left side of decimal.
That means, if only 1 digit is repeating then multiply the
equation by 10, if 2 digits are repeating then multiply the
equation by 100, if three 3 digits are repeating then
multiply the equation by 1000 and so on.
After multiplying, name the equation (ii).
Subtract equation (i) from equation (ii).
Write the decimal in the simplest form.
Example
(i) Convert 0.35 into the form
Let
p
.
q
……………(i)
Multiplying both sides by 100
p
= 35.353535……
100
q
……………(ii)
Subtracting (i) from (ii)
p
p
–
= 35.353535…… – 0.353535……
100
q
q
⇒
Example
(ii) Convert 36.5424242…… into the form
Multiplying both sides by 10
p
= 365.424242……
10
q
Multiplying (i) by 100
p
= 36542.424242……
1000
q
⇒ 990
p
35
=
q
99
⇒
Let
p
= 23.434343……
q
Multiplying both sides by 100
p
= 2343.434343……
100
q
p
.
q
⇒
……………(i)
……………(ii)
Subtracting (i) from (ii)
p
p
–
= 2343.434343…… – 23.434343……
100
q
q
⇒ 99
⇒
p
= 36.5424242……
q
p
.
q
……………(i)
……………(ii)
……………(ii)
Subtracting (ii) from (iii)
p
p
– 10
= 36542.424242…… – 365.424242……
1000
q
q
p
= 35
q
(ii) Convert 23.43 into the form
Determine the number of digits after the decimal point
which do not have bar on them. Let there be n digits
without bar just after the decimal point.
Multiply both sides of x by such power of 10, so that only
repeating decimals are remaining on the right side of the
decimal point.
Use the method of converting pure recurring decimal to
the form p/q and obtain the value of x.
Note
In case of repeating non-terminating with whole part zero and
decimal part only with repeating digits only, simply remove the
decimal and in denominator put as many 9 (nine) as there are
repeating decimal digit.
Let
p
= 0.353535……
q
⇒ 99
3.
4.
Steps
3.
4.
5.
2.
p
= 2320
q
p
2320
=
q
99
p
= 36177
q
p
36177
=
q
990
p
12059
=
q
330
Irrational Number
The number, which can not be written in the form
p
, where
q
p and q are integers and q ≠ 0 , is called irrational number.
If irrational number is converted into decimal form, then it is
neither terminating and nor repeating.
Example
(i)
2
(ii)
3 = 1.732050807........
= 1.414215........
p
q
While converting a recurring decimal that has one or more
digit before the repeating digits, it is necessary to isolate the
repeating digits. In order to convert a mixed recurring decimal
to the form p/q, we follow the following steps:
Surd
Steps
1. Obtain the mixed recurring decimal and write it equal to
any variable, say x .
is a rational number and a1 / n is an irrational number. Here
a is called radicand and n is called order of the surd.
Conversion Of Mixed Recurring Decimal To From
Note
Because irrational numbers are non-terminating and nonrepeating in the decimal form, that is why some times we take
approximate value of irrational number to solve the problem.
If a is a rational number and n is a positive, so n a is surd, if a
Every surd is a radical, but every radical is not a surd. In other
word, if a is a real number and n is a positive integer then
a is not a surd, if a is irrational or a1 / n is rational.
Every surd is made up of two parts, i.e. rational part and
irrational part. The part outside the radical sign is called
rational part and the part under the radical sign is called
irrational part.
A surd which has rational factor other than one (unit) is called
mixed surd.
n
Example
(i)
6 3 5 is a surd with of order 3. Its rational part is 6 and its
irrational part is
3
5
5
(ii) 20 12 is a surd with of order 5. Its rational part is 20
5
12
.
Note
1. If the rational part of surd is 1, then it need not be
mentioned.
Example
=
1 2
2.
2
If the order of surd is 2, then it need not be mentioned.
Example
2
=
3
3
Types Of Surd
Surds are divided into various categories depending upon their
characteristics.
1. Like Surds
Surds with same irrational factor and of same order are called
like surds.
Example
(i)
2
(ii)
7
3 , 82 3
7
10 , 5 10
2. Unlike Surds
Surds with different irrational factor or of different order are
called unlike surds.
Example
(i)
62 2 , 62 5
(ii) 72 3 , 73 3
3. Pure Surd
A surd which has one (unit) as its rational factor is called pure
surd.
Example
(i)
(ii)
(iii)
(i)
2 3
(ii) 53 11
(iii) 207 3
5. Depending Upon Number Of Terms
.
and its irrational part is
Example
Depending upon the terms, the surds are divided into various
categories.
(i) Monomial Surd
A surd consisting of only one term is called monomial surd.
Example
(i)
2
3
(ii)
5
7
(iii)
2 +
5−2 6
=
2 +
3+2−2 6
=
2 +
=
2 +
=
2+ 3− 2
=
3
( 3) − ( 2)
( 3 − 2)
2
2
− 2( 3 )( 2 )
2
(ii) Binomial Surd
An expression consisting of two terms, where both terms are
monomial surds or one term is monomial surd and the other is
a rational number is called monomial.
In other words, either it is sum/difference of two monomial
surds or sums/difference of a surd and rational number.
Example
(i)
5 3 − 2 7
(ii) 2 + 3
(iii) 4 3 − 6 5
(iii) Trinomial Surd
An expression consisting of three terms, where all the tree
terms are monomial surds or one term is monomial and other
term is binomial surd is called trinomial surd.
Example
(i)
2+ 3+ 5
2
3
(ii)
3
4
7
4. Mixed Surd
7+ 2 + 3
(iv) Conjugate Surd
Two binomial surds which differ only in the sign (+ or − )
between the terms are called conjugate surd.
Example
(i)
3 − 2 and
3+ 2
(ii) 4 2 + 3 5
If surds are unlike surds, then first convert them to like surds.
and 4 2 − 3 5
Laws Of Surd
Example
There are few law which are used to simplify the mathematical
operations, where surds are involved. These laws are as
follows : 1.
n n
a
= a

n×
 n an = (an )1/n = a


2.
3.
4.
n
a ×nb
n
a
n
b
mn
=
n

= a


ab
a
b
mn
a =
n
=
1
n
 mn
a =

a
m 1/n
a

= (a1/n )1/m = a1/n × 1/m = a1/mn = mn a 

7
Operation Between Surds
While performing mathematical operations on surds, we have
to follow various rules.
1. Surd Addition
1
3
3
1

3  6 − 5 − 
3

(i) 6 3 − 5 3 −
=
 18 − 15 − 1 
3 

3


=
=
2
3
3
(ii) 5 2 − 4 64 − 6 8
=
=
=
=
=
=
5.21/2 − 26 ×1/4 − 23/6
5.21/2 − 23/2 − 21/2
5.21/2 − 23×1/2 − 21/2
5 2 − 23 − 2
5 2 −2 2 − 2
2 2
3. Multiplication
Only same order surds can be multiplied. In such case, we
multiply rational part with rational part and irrational part with
irrational part.
Only like surds can be added. In such case, we add rational
part with rational part and write the irrational part as it is.
If surds are of different order, first convert them to same
order surds.
If surds are unlike surds, then first convert them to like surds
Example
Example
(i)
=
1
6 3 +5 3 +
3
3
1

3  6 + 5 + 
3

(i)
23 7 × 33 5
=
63 35
(ii) 32 2 × 53 3 × 6 4 4
12 6
=
3×5×6
34 3
3
=
90 12 26 × 26 × 34
=
9012 212 × 34
(ii)
5 2 + 4 64 + 6 8
=
90 × 2
=
5 2 + 26 + 23
=
1
5.2 2
+
=
1
5.2 2
3
2
+2
=
 18 + 15 + 1 
3 

3


6
4
1
×6
24
1
=
=
=
=
5.2 2 + 2
+ 2
3×
1
6
1
+ 22
3×
1
2
1
+ 22
5 2 + 23 + 2
5 2 +2 2 + 2
=
=
=
12
43
( )
26 × 3 4 × 22
3
12 4
180 × 3
3
4×
1
12
1
180 × 3 3
180 3 3
Note
Like surds and same order surds are different things.
1. In case of like surd, both the radicand and order of the
surds must be same.
Example
8 2
37 10
2. Surd Subtraction
Only like surds can be subtracted. In such case, we subtract
the rational part with rational part and write the irrational part
as it is.
× 5
12
34 × 6
3
=
2
12
=
2.
and
87 10 .
In case of same order surd, only the order of the surd
must be same, but radicand may be different.
Example
37 4
and
(i) Rationalising factor of
87 6
4. Division
Only same order surds can be divided. In such case, we divide
rational part with rational part and irrational part with irrational
part.
1
53
×
=
×
If surds are of different order, first convert them to same
order surds.
=
1
53
× 53
=
5(1/3 + 2/3)
Example
=
=
53/3
5
62 8 ÷ 22 4
=
32 2
5 is
3
52 , i.e
3
25 .
1
1−
5 3
1
53
(i)
3
3−1
5 3
2
2. Rationalising Factor Of Binomial Surd
If binomial surd is the form a ± b
(ii) 106 162 ÷ 23 3
factor is a m b .
6
=
106 162 ÷ 2 32
=
56 18
then its rationalizing
(i) If binomial surd is in the form a +
rationalising factor is a −
b , then its
b.
(ii) If binomial surd is in the form a −
Rationalization
rationalising factor is a +
b , then its
b.
Process of converting an irrational numerator/denominator
into rational numerator/denominator is called rationalization.
Surd in denominator of a fraction makes the fraction
complicated. With rationalization, the surd in denominator is
changed to a rational number which makes the term simpler.
Note
Simplest rationalising factor of binomial surd is its conjugate
surd.
Rationalizing Factor
Example
If two surds on multiplying with each other result into a
rational number, then each one of them is called rationalizing
factor of the other.
Example
(i)
2 and
2 ×
=
=
8 are rationalizing factor of each other.
( 2 )2 − ( 3 )2
2 − 3
−1
7 − 4 is
7 +
4.
( 7 − 4) × ( 7 + 4)
(ii) 3 5 and
5 are ratioanalizing factor of each other.
3 5 × 5
= 3× 5
= 15
3 ×3 9
=
3
=
=
=
( 7 )2 − ( 4 )2
7 − 4
3
3. Rationalising Factor Of Trinomial Surd
3
3 and
3
=
=
=
=
(ii) Rationalising factor of
4
3
( 2 + 3) × ( 2 − 3)
8
16
(iii)
(i) Rationalising factor of ( 2 + 3 ) is ( 2 − 3 ) .
9 are rationalizing factor of each other.
3
Treat the trinomial surd as binomial surds. For this, group any
two terms and treat the grouped terms as one term and the
remaining surd as second term.
3 × 32
If the trinomial surd is a ± b ± c , then after grouping the
1/3
terms ( a ± b ) ±
3
2×1/3
×3
1/3
=
3
=
1 2
+
33 3
=
1+ 2
3 3
=
3
33
c.
2/3
×3
So, rationalise factor of
( a m b) m
1. Rationalising Factor Of Monomial Surd
Rationalising factor of monomial surd
Example
n
1−
a is a
1
n
.
c
a± b± c
and m
ab .
Note
In case of trinomial surd, even after rationalization once, a
surd still exists. To remove that surd, we need to rationalize
the expression again. So, in case of trinomial surd, we have to
do the rationalization two times.
Example
(i) Rationalising factor of
(4 + 2 6 ) .
3 − 2 − 1 are ( 3 − 2 ) + 1 and
After grouping the terms ( 3 − 2 ) − 1
Now,
[(
] [
]
3 − 2) −1 × ( 3 − 2) + 1
2
( 3 − 2 ) − (1)
=
=
=
(i) Remove irrational number from the denominator
4
2
2+ 3
3 + 2 − 2 6 −1
=
4−2 6
Rationalising again,
(4 − 2 6 ) × (4 + 2 6 )
=
(4)2 − (2 6 )2
=
=
16 − 24
−8
(ii) Rationalizing factor of 7 + 6 + 13 are ( 7 + 6 ) - 13
and
] × [(
7 + 6 ) − 13
=
( 7 + 6 )2 − ( 13 )2
=
=
7 + 6 + 2 42 − 13
42
=
−4 2 − 3
(i) Remove the irrational number from the denominator
3
3
=
3( 7 + 4 )
( 7 )2 − ( 4 )2
7+ 4
1
1
3
.
=
=
6
.
6
2 6
6
6
3
2. Binomial Surd As Denominator
( 3 − 2) + 1
( 3 − 2 )2 − (1)2
3 − 2 +1
3 + 2 − 2 6 −1
3 − 2 +1
4-2 6
3 − 2 +1
(
)
22- 6
Rationalising again,
=
6
( 3 − 2) + 1
( 3 − 2) + 1
3 − 2 +1
2
×
( 3 − 2) − 1
2(2 − 6 )
(ii) Simplify
=
7+ 4
3( 7 + 4)
3
3
3
×
.
7+ 4
×
=
=
6
7− 4
3( 7 + 4)
7−4
1. Monomial Surd As Denominator
2
)
3
=
=
=
(
(ii) Simplify
Note
Same non-zero number can be multiplied by numerator and
denominator of a term. This property is used in rationalization.
×
)
(i) Remove irrational number from denominator
Irrational number as the denominator of a term/expression
makes the term/expression very complicated. Using
rationalization, we can convert the irrational number in the
denominator to a rational number. Rationalization is used to
simplify the term/expression, thus simplify whole calculation.
3
(
3. Trinomial Surd As Denominator
Advantage Of Rationalization
1
)
4 2− 3
−1
=
84
)
=
=
2 × 42
=
=
(
3
2 42
Rationalising again,
2 42 ×
4 2− 3
( 2 )2 − ( 3 )2
7− 4
]
2− 3
4 2− 3
2−3
42 .
7 + 6 ) + 13
(
.
2− 3
=
After grouping the terms ( 7 + 6 ) + 13
Now,
[(
×
4
2+ 3
×
(2 + 6 )
(2 + 6 )
2 3 +3 2 −2 2 −2 3 +2+ 6
[
2 (2)2 − ( 6 )2
=
2+ 2 + 6
2(4 − 6)
=
2+ 2 + 6
−4
=
−2− 2 − 6
4
(ii) Simplify
1
7 + 6 + 13
.
]
1
3 − 2 −1
.
1
( 7 + 6 ) + 13
=
=
×
( 7 + 6 ) − 13
( 7 + 6 ) − 13
( 7 + 6 ) − 13
( 7 + 6 )2 − ( 13 )2
7 + 6 − 13
2 42
Rationalizing again,
2 42
=
(
×
7 + 6 − 13
( 2 ) ( 42 )
=
7 6 + 6 7 − 546
84
Here are few laws of surds. These laws are used to simplify
the mathematical operations, where real numbers are
involved.
7 + 6 + 2 42 − 13
7 + 6 − 13
7 6 + 6 7 − 546
84
Laws Of Real Numbers
7 + 6 − 13
=
=
42
If a is a real number such that a > 0 and p, q are rational
numbers, then
1. ap .aq = ap + q
42
2.
ap ÷ aq =
ap − q
3.
(a )
4.
ap .bp = ( ab )
)(
42
)
p
q
= apq
p
Operations Between Rational And Irrational Numbers
The result of a mathematical operation may be rational or irrational depending on the type of values used in the operation.
Operand
Operand
Rational
Rational
Irrational
Irrational
Rational
Irrational
+
-
Operation
×
Rational
7 × 4 = 28
2 × 11 = 22
Rational/Irrational
÷
Rational
7 + 5 = 12
20 + 14 = 34
Rational/Irrational
Rational
5-3=2
9-3=6
Rational/Irrational
(2 + 3 ) + (2 − 3 ) = 4
(6 + 3 ) − (2 + 3 ) = 4
2 × 8 =4
8 × 2 =2
(2 + 3 ) + ( −2 + 3 ) = 2 3
(9 + 3 ) − (9 − 3 ) = 2 3
2 × 3= 6
10 × 2 = 5
Irrational
Irrational
Irrational
Rational
8÷2=4
42 ÷ 14 = 3
Rational/Irrational
Irrational
4 + 6 =4 + 6
6 − 2 =6 − 2
4 × 3 =4 3
10 ÷ 3 = 10 ÷ 3
5 + 2 =5 + 2
10 − 4 3 = 10 − 4 3
5 × 2 7 = 10 7
Note Rational ≠ 0
8 ÷ 4= 8 ÷ 4
Note Rational ≠ 0