MANISH KUMAR
MATHEMATICS NUMBER SYSTEM
INTRODUCTION: - In our day to life we deal wit different types of numbers which can be broady classified as follows CLASSIFICATIN OF NUMBERS
NUMBERS
REAL NUMBERS
RATIONAL NUMBERS
FRACTIONS
EVEN NUMBERS
IMAGINARY NUMBERS
IRRATIONAL NUMBERS
INTEGERS
ODD NUMBERS
NATURAL NUMBERS WHOLE NUMBERS
PRIME NUMBERS (i)
Natural numbers (N) : Set of all non-fractional numbers from 1 to N = {1,2,3,4..... }
(ii)
Whole numbers(W) : Set of all non-fractional numbers from zero to , W = {0,1,2,3,4..... }
(iii)
Integers (I or Z) : Set of all non-fractional numbers from {- ..... to ..... + } I or Z = { - .... -3, -2, -1, 0, +1, +2, +3.... + } Positive integers : {0, 1, 2 ,3....}, Negative integers : {..... -4, -3, -2, -1}
(iv)
Prime numbers : All natural numbers that have one & itself as their factor are prime numbers. Ex : 2, 3, 5, 7
(v)
Composite numbers : All natural numbers which are not prime numbers are composite numbers. Ex : 4, 6, 8, 9 (1 is neither prime nor composite number.)
A.
RATIONAL NUMBERS :Rational numbers : - These are real numbers which can be expressed in the form of
p . Where p and q are q
integers and q 0. Note: (i) whole numbers and integers are rational numbers. (ii) Types of rational :- (a) Terminating decimal numbers and (b) Non-terminating repeating (recurring) decimal numbers are rational numbers.
MANISH KUMAR Ex.
2 37 17 , , , 3 15 19
MATHEMATICS -3 0, 10, 4.33, 7.123123123.....
MANISH KUMAR
MATHEMATICS
A-1.
FINDING RATIONAL NUMBERS BETWEEN TWO NUMBES :-
Ex -1
Find 4 rational numbers between 4 and5.
Method (i)
a = 4, b= 5, n = 4
x (n 1) 4 ( 4 1) 4 5 20 n 1 4 1 5 5
b (n 1) 5 ( 4 1) 5 5 25 n 1 4 1 5 5 20 21 22 23 24 25 , , , , , 5 5 5 5 5 5
Ex-2
Find 3 rational number between
Method (ii)
a
6 7 , 5 5
6 7 ,b ,n 3 5 5
7 6 n a 5 5 5 1 d= n 1 3 1 4 20 7
a+d=
6
6 1 24 2 25 5 20 20 20
a + 2d
6 1 24 1 25 2 5 20 20 20
a + 3d
6 3 24 3 27 5 20 20 20
24 25 26 27 28 , , , , 20 20 20 20 20 A-2.
RATIINAL NUMBER IN DECIMAL REPRESENTATION :-
(i)
Terminating decimal : In this a finits number of digit occours after decimal.
Ex.
1 1 0.25, 0.5 4 2
MANISH KUMAR
MATHEMATICS
(ii)
Non terminating & Repeating (Recurring decimal) :- The remainder never become zero.
Ex.
2 0.181818 0.18 11
Ex.
2 0.181818...... 0.18 11
8 2.666666 2.6 3
8 2.6666..... 2.6 3
A-3.
REPRESENTATION OF RATIIONAL NUMBERS ON A NUMBER LINE:-
(i)
Positive fraction :
3 Divide a unit into 6 equal parts. 6
MANISH KUMAR (ii)
Negative fraction:
MATHEMATICS 4 divide a unit into 9 equal parts 9
MANISH KUMAR
MATHEMATICS
(iii)
Terminating decimals :
(iv)
Non terminating & repeating decimals : Visualize 4.26 on the number line, up to 4 decimals places.
4.26 = 4.262626....
a c e and , are three rational numbers then. b d f
A-4.
PROPERTIES OF RATIONAL NUMBERS :- If
(i)
a c c a [commutative law of additional] b d d b
(ii)
a c e a c e [associative law of additional] b d f b d f
(iii)
a a a a a a 0, is called the additive inverse of b b b b b b
(iv)
a c c a [commutative law of multiplication] b d d b
(v)
a c e a c e [associative law of multiplication] b d f b d f
(iv)
a c e a c a e [distributive law] b d f b d b f
A.5.
CONVERSION OF NON TERMINATING & REPEATING DECIMALS NUMBERS TO THE FORM P q :Non-terminating repeating decimals, are basically two types.
(i)
Pure recurring decimals :- A decimal in which all the digits after the decimal point are repeated these types of decimals are known as pure recurring decimals.
Ex.
0.6 .0.16 0.123 are pure recurring decimals.
MANISH KUMAR (ii)
MATHEMATICS
Mixed recurring decimals : - A decimal in which at least one of the digits after the decimal point is not repeated and then some digit or digits are repeated. These type of decimals are known as mixed recurring decimals.
Ex.
2.16. 0.35. 0.785 are mixed recurring decimals. (a) In order to convert a pure recurring decimal to the form
Step
p , we follow the following steps :q
(i)
Obtain the repeating decimal and put it equal to x (say).
(ii)
Write the number in decimal from by removing bar from the top of repeating digit and listing repeating digits at least twice.
Ex.
x = 0.8 Has x = 0.888
(iii)
Determine the number of digits having bar on their heads.
(iv)
If the repeating decimal has 1 place repetition, multiply by 10; a two place repetition. multiply by 100; a three place repetition, multiply by 1000 and so on.
(v)
Subtract the number in step (ii) from the number obtained in step (iv).
(vi)
Divide both sides of the equation by the coefficient of x.
(vii)
Write the rational number in its simplest form.
Ex. 1
Express of the 0.3 in the form
Sol.
Let x = 0.3 Then, x 0.33333....
p : q
...... (i)
Here we have only one repeating digit after decimal so we multiply both sides by 10.
10x = 3.33333...
.......(ii)
On subtracting (i) from (ii), we get 10x - x = (3.33333.....) - (0.33333....) 9x = 3
3 9
x
0.3
9 3 i.e., 0.33333..... 9 9
Ex.2
Show that 1.272727 = 1.27 can be expressed in the form
Sol.
Let x = 1.27 Then, x = 1.27272727....
p where p and q are integers and q 0. q
..... (i)
Here we have only two repeating digit after decimal so we multiply both sides by 100.
100 × = 127.272727.... On subtracting (i) from (ii), we get 99x = (127.272727...) - (1.272727....)
MANISH KUMAR
MATHEMATICS
99x = 126
x
126 14 99 11
Hence, 1.27
14 11
Ex. 3
Convert the 23.43 number in the form
Sol.
Let x = 23.43 Then1
p . q
x = 23.434343..... ....(i) Here we have only two repeating digit after decimal so we multiply both sides by100. Multiplying both sides of (i) by 100, we get 100 x = 2343.4343.... ....(ii) Subtracting (i) from (ii), we get 100x - x = (2343.4343...) - (23.4343....) 99x = 2320 2320 x 99 Aliter method : We have,
23.43 23 0.43 43 23.43 23 99
23.43
Ex.4
If
Sol.
[Using the above rule, we have 0.43
43 ] 99
23 99 43 99
2277 43 2320 99 99
1 2 3 4 5 without actually doing the long division. 0.142857 , write the decimal expression of . . and 7 7 7 7 7 Thus, we have 2 1 2 0.285714 7 7 3 1 3 0.428571 7 7 4 1 4 0.571428 7 7 5 1 5 0.714285 7 7 p (b) In order to convert a mixed recurring decimal to the form , we follow the following steps: q (i) Obtain the mixed recurring decimal and write it equal to x (say). (ii) 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. n (iii) Multiply both sides of x by 10 so that only the repeating decimal is on the right side of the decimal point.
MANISH KUMAR
MATHEMATICS
(vi) Use the method of converting pure recurring decimal to the form Ex.
Show that 0.2353535... = 0.235 can be expressed in the form
Sol.
Let x = 0.235 . Then.
p and obtain the value of x. q
p , where p and q are integers and q 0 . q
10x = 0.235
10x = 2 + 0.35
10 x 2
10 x
2 99 35 99
10 x
198 35 233 233 10x x 99 99 990
A.6.
DIRECT METHOD :-
35 99
35 0.35 99
p (complete number)-(number formed by Non - repeating digit) form = q No. of 9 as no. of repeating digits after that write no. of 0 as no. of non repeating digits. Ex.
35 0 35 99 99
(i)
0.35
(ii)
0.435
435 4 431 990 990
A-7.
DETERMINING THE NATURE OF THE DECIMAL EXPANSIONS OF RATIONAL NUMBERS:-
(a)
Let x be a rational number whose decimal expansion terminates. Then we can expressed x in the form m
p , q
n
where p and q are co-primes, and the prime factorization of q is of the form 2 × 5 . where m, n are nonnegative integers. (b)
Let x =
p m n be a rational number, such that the prime factorisation of q is of the form 2 × 5 where m, n are q
non-negative integers. Then, a has a decimal expansion which terminates. (c)
Let x
p m n be rational number, such that the prime factorisation of q is not of the form 2 × 5 . where m, n q
are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating. Ex.
(i)
189 189 23 189 8 189 1512 3 125 53 2 53 ( 2 5)3 103
(ii)
17 2.83333.... 6 (we observe that the prime factorisation of the dominators of these rational numbers are not of the form 2m × 5n. where m,n are non-negative integers.)
MANISH KUMAR (iii)
MATHEMATICS
17 17 3 8 2 50 (So, the denominator 8 of Hence
(iv)
17 m n is of the form 2 × 5 . where m,n are non-negative integers.) 8
17 has terminating decimal expansion. 8
64 64 455 5 7 13 m
n
(Clearly, 455 is not the 2 × 5 . So, the decimal expansion
64 is non-terminating repeating.) 455
MANISH KUMAR
MATHEMATICS
TEST OF DIVISIVILITY : A positive integer N is divisible by (i) 2 if and only if the last digit (nit’s digit) is even. (ii) 3 if and only if the sum of all the digits is divisibly by 3. (iii) 4 if and only if the number formed by last two digits is divisibly by 4. (iv) 5 if and only if the last digit is either 0 or 5. (v) 8 if and only if the number formed by last three digits is divisibly by 8. (vi) 9 if and only if the sum of all the digits is divisibly by 9. (vii) 11 if and only if the difference between the sum of digits in the odd places (starting from right) and sum of the digits in the even places (starting from right) is a multiple of 11. (viii) 25 if and only if the number formed by the last two digits is divisibly by 25.
B.
IRRATIONAL NUMBER :Irrational numbers:- A number is called irrational number. If it can not be written in the form
p , where & q q
are integers and q o Ex. (i)
2 , 3, 3 2 , 2 3, 2 3 .....etc. Non-terminating & non-repeating decimal numbers are Irrational numbers.
Ex.1 Sol.
Prove that 2 is not a rational number. Let us find the square root of 2 by long division methods as shown below. 1.414215 1
2.000000000000
+1
1
24
100
4
96
281
400
+1
281
2824
11900
+4
11296
28282
60400
+2
56564
282841
383600
+1
282841
2828423 3
10075900 8485269
28284265
159063100
+5
141421325
28284270
11641775 2 = 1.414215
Clearly, the decimal representation of (a)
2 is neither terminating nor repeating.
We shall prove this by the method of contradiction If possible, let us assume that a Then- 2 where a, b are integers having no common factor other then 1. b
2
2
2
a (squaring both sides) b
2 is a rational number.
MANISH KUMAR
MATHEMATICS
a2
2
b2 a = 2b2 2
2 divides a2
2 divides a
Therefore let a = 2c for some integer c.
a2 = 4c2
2b2 = 4c2
b2 = 2c2
2 divides b2
2 divides b
Thus, 2 is common factor of a and b. But, it contradicts out assumption that a and be have no common factor other than 1. So, our assumption that Hence,
2 is irrational.
Ex.2
Prove that
So.
Let
3
2 is a rational, is wrong.
3
3 is irrational.
3 be rational
p where p and q z and p. q have no common factor except 1 also q > 1. q
p 3 3 q
Cubing both side
p3 q3
3
Multiply both sides by q2
p3 3q2 Hence L.H.S. is rational since p. q have no common factor. q
p3 , q also have no common factor while R.H.S. is an integer. L.H.S. R.H.S. which contradicts our assumption that Ex.3
Prove that 2 +
Sol.
Let 2 +
3
3 is Irrational.
3 is irrational.
3 be a rational number equal to r
2 3 r
3 r 2 Here L.H.S. is an irrational number while R.H.S. r - 2 is rational L.H.S. R.H.S. Hence it contradicts our assumption that 2 +
2+ B-1
3 is rational.
3 is irrational.
PROPERTIES FO IRRATIIONAL NUMBERS : (i) Negative of an irrational number is an irrational number. E.g.:-
3, 4 5 are irrational.
MANISH KUMAR
MATHEMATICS
(ii) Sum and difference of a rational and an irrational number is an irrational number. Two number’s are 2 and Sum = 2 +
3 is a irrational number.
Difference = 2 Also Ex.
3
3 , is an irrational number.
3 2 is an irrational number. 3
Two number’s are 4 and Sum = 4 +
3
3
3 is an irrational number.
Difference = 4 -
3
3 is an irrational number.
(iii)
Sum and difference of two irrational number is not necessarily an irrational number.
Ex.
Two irrational numbers are
3. 2 3
3 2 3 3 3. is an irrational.
Sum =
Difference = 2 3 3 3 is an irrational. Ex.
Two irrational number are
5
3 5 . is an irrational number.
Sum =
5 3 is an irrational number.
Difference = Ex.
3.
Two irrational numbers are
3, 3
3 3 0 , which is rational.
Sum =
Difference 3 3 2 3. which is irrational Ex.
Two irrational numbers are 2 +
3 and 2 -
3
Sum = 2 3 2 3 4 . a rational number. Two irrational number are
3 3, 3 3
Difference 3 3 3 3 6 . a rational number. Ex.
Two irrational numbers are Sum =
3 2. 3 2
3 2 3 2 2 3 an irrational.
(iv)
Product of a rational number with an irrational number is not always irrational.
Ex.
2 is a rational number and
3 is an irrational
2 3 2 3 , an irrational. Ex.
0 a rational and
3 an irrational
0 3 0 a rational.
MANISH KUMAR (v) Ex.
MATHEMATICS
Product of a non-zero rational number with an irrational number is always irrational.
4 4 4 3 3 is an irrational 3 3 3
MANISH KUMAR
MATHEMATICS
(vi)
Product of an irrational with an irrational is not always irrational.
Ex.
(i)
3 3 3 3 9 3 a rational number.
(ii)
2 3 3 3 2 3 3 2 6 6 an irrational number.
(iii)
3
(iv) (v)
3 32 3 32 33 3 a rational number 3
3
3
2 3 2 3 (2) 3 = 4 - 3 = 1 a rational number. 2 3 2 3 2 3 (2) 3 2(2) 3 4 3 4 2
2
2
2
2
3 7 4 3 an irrational number.
IMPORTANT NOTE :(a)
2 2 it is not a irrational number.
(b)
2 5
2 5 10
2, 5, 5 2 are called imaginary numbers. 2 i 2 where i(iota)
B-2.
1
REPRESENTATION OF IRRATIONAL NUMBERS ON A NUMBER LINE :=
2, 3, 5 on a number line
Ex.1 Sol.
OB =
2 = OF
OC =
3 = OG
OD =
4 = OH
OE =
5 =OI
Insert a rational and an irrational number between 2 and 3. If a and b are two positive rational numbers such that ab is not perfect square of a rational number, then
ab is an irrational number lying between a and b. Also, if a, b are rational numbers, then
ab is a rational number between them. 2
A rational number between 2 and 3 is
23 2.5 2
ab 2 3 6
An irrational number between 2 and 3 is Ex.2
Find two irrational numbers lying between
2 and
3.
Sol.
We known that, if a and b are two distinct positive irrational number, than between a and b.
irrational number between
2 and
3 is
2 3
6 61 / 4
ab is an irrational number lying
MANISH KUMAR
MATHEMATICS
irrational number between
2 and 6
1/4
is
2 61 / 6 21 /* 4 61 / 8
Hence, required irrational numbers are 61/4 and 21/4 × 61/8 Ex.3
Identify
45 as rational number or irrational number.
Sol.
We have,
45 9 5 3 5 Since 3 is a rational number and C
5 is an irrational number. Therefore, the product 3 5 45 is an
irrational number. REAL NUMBER :Real numbers :- Rational numbers together with irrational numbers are known as real numbers. Thus a real number is either rational or irrational but can not be simultaneously both. If a real number is not
7 rational, it has be to be irrational and vice versa. 3, , -1.5, 2.3, 5.76245 etc. are some of the rational 5 number. whereas (i)
2. 3.3 7 .5 11. are some examples of irrational numbers.
These are the numbers which can represent actual physical quantities in a meaningful way. These can be represented on the number line. Number line is geometrically straight line with arbitrarily
C-1.
defined zero(origin). GEOMETRICAL REPRESENTATION OF REAL NUMBERS :To represent only real numbers on numbers line we follow the following algorithm. ALGORITHM (i) (ii) (iii)
Obtain the positive real number x (say) Draw a line and mark a point A on it. Mark point B on the line such that AB = X units.
(iv) (v) (vi) (vii)
From point B mark a distance of 1 unit and mark the new point as C. Find the mid-point of AC and mark the points as O. Draw a circle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Length BD is equal to
x
Justification : We have. AB = x units and BC = 1 units. AC = (x+1) units
OA = OC =
OD =
x 1 units. 2
x 1 units. 2
Now, OB = AB - OA = x -
x 1 x 1 2 2
Using Pythagoras theorem in OBD , we have 2
2
2
OD = OB + BD
BD2 = OD2 - OB2
[ OA = OC= OD]
MANISH KUMAR
MATHEMATICS 2
x 1 x 1 BD2 2 2
BD =
( x 2 2 x 1) ( x 2 2 x 1) 4
This shows hat C-2.
2
4 x x4
x exists for all real numbers x > 0.
EXPONENTS OF REAL NUMBERS :(i) Positive exponent :n
For any real number a and a positive integer ‘n’ we define a as n
a = a x a xa x....... xa (n times) n
th
a is called the n power of a. The real number ‘a’ is called the base and ‘n’ is called the exponent of the n
th
power of a. Ex.
33 = 3 × 3 × 3 = 27 For any non - zero real number ‘a’ we define a0 = 1. 0
3 Thus, 40 = 1, 60 = 1, = 1 and so on. 2 (ii) Negative exponent :For any non - zero real number ‘a’ and a positive integer ‘n’ we define a n
1 an
n
n
th
Thus we have defined a for all integral values of n. positive. zero or negative. a is called that n power of a. Ex.
53
1 53
1 1 5 5 5 125
(iii) Rational power [Exponents] :For any positive real number a and a rational number
p p/q p 1/q , where q > 0, we define a = (a ) q
i.e. ap/q is the principal qth root of ap . C-3.
RATIONAL EXPONENTS OF A REAL NUMBER :th
(i) n root of a positive real number th
If a is a positive real number and ‘n’ is a positive integer, then the principal n root of a is the unique positive n
real number x such that x = a. th
The principal n root of a positive real number a is denoted by a
1/n
or
n
a
th
(ii) Principal n root of a negative Real Number th
If a is a negative real number and n is an odd positive integer, then the principal n root of a is defined as - | a|
1/n
th
th
i.e. the principal n root of is minus of the principal n root if | a |. th
Remark : If a is negative real number and n is an even positive integer, then the principal n root if a is not defined, because an even power of a real number is always positive. Therefore (-9) 1/2 is a meaning less quantity, if we confine ourselves to the set of real number, only. C-4.
LAWS OF RATIONAL EXPONENTS :The following laws hold the rational exponents.
MANISH KUMAR
MATHEMATICS
m
n
m+n
(i)
a ×a =a
(ii)
a
(iii)
(a ) = a
(iv)
a =
(v)
a
(vi)
ab m ambm
(vii)
am a m (b 0) b b
(viii) (ix)
a =a m m 0 a a = a = 1
a =a
m
n
m n
mn
1
-n
m/n
m+n
an
= (am)
1/n
1/n m
= (a ) = a
m/n
=
n
am
a n
m
m
Ex.1
bn
b+b+b….n times
Where a, be are positive real numbers and m, n are rational numbers. Simplify each of the following
3 (ii) (32)3 (iii) 4
(i) 52 × 54 Sol.
am an am 1
(i) 52 . 54 = 52+4 = 56 = 15625 (ii) (32)3 = 3
3 (iii) 4 Ex.2
3
2×3
3
(am )n amn
= 36 - 729
1 3 4
3
1
1 64 27 27 64
33 43
a n
Simplify each of the following 2
3
2 11 3 55 60 97 18 (i) (ii) 2 × 2 2 × 2 11 3 2 4
Sol.
1 an
2
(iii)
16 2n 1 4 2n 16 2n 2 2 2n 2
3
24 112 33 2 4 112 33 2 3 6 2 11 3 (i) 4 2 3 4 2 3 2 121 11 3 2 11 3 2 11 11 3 2 (ii) We have 255 260 297 218 255 60 297 18 2115 2115 0 (iii)
Ex.3
2 4 22 1 2 2 2n n 2
22
1 / ab
1 / bc
xb xc
2
n 3
2
2n 5 2n 2 n5
2.2
n 2
2.2
1 / ac
xc xd
1
.x . x
x a b 1 1
1 / ab
b c
1 1
1 1 c
x b a.x c b .x a Ex. 4
n 6
2n 5 2n 2 n5
2( 2
n 2
2
)
1 2
Assuming that x is a positive real number and a, b, c are rational numbers, show that : xa xb
Sol.
n 2
2 2 4
2n 5 2n 2
if
c a 1 / ac
1 1 1 1 1 1 a c b a c
xb
9n 32 (3n / 2 )2 ( 27)n 3m
3
2
x a b / ab .x(b c ).x c a / ac
3
x0 1
1 prove that m - n = 1. 27
MANISH KUMAR Sol.
MATHEMATICS
(3 ) 3 3 n
2
n 2 3 2
(33 )n
33m 23
32n 32 3n 33n
3m
3
3
2n 2 n
3
2
3 3n
23
3m
33n 2 33n
3
33m 23
.
33m 23 33n.8
3m
3 .8
1 27
33n 3m
1 27
1 27
1 27
33n 32 1 33n
1 27
1 27
1 33
33n3m 33
[On equating the exponent]
3n - 3m = -3 n - m = - 1 m - n = 1. Ex.5
Assuming that x is a positive real number and a,b,c are rational numbers, show that :
xa xb Sol.
x
b c a
xb xc
a b a b c
c a b
xc xa
b c a
1
c a b
. xb c . x c a x a b a b c a b .x b c b c ab c .x c a c a b c a = xa x
Ex.6 Sol.
a b c
2
b 2 ca bc
.xb
2
c 2 ab ac c 2 a 2 bc ba
x
a 2 b 2 ca bc b 2 c 2 ab bc a 2 b ba
x0 1 x 2 If a = b . by = c and c = a. prove that xyz = 1. We have, xyz x yz a = (a )
axyz = (b)yz
a
axyz = cz
a
xyz
=a
a
xyz
= a xyz = 1
x
xyz
y
[ x 2 b]
y yz
= (b )
[b y c ]
[c z a] 1
z
Ex. 7
If a = b = c and b2 = ac. prove that y =
Sol.
Let a = b = c = k. Than. a = k1 Now b2 = ac
x
y
z
/x
b=k
(k1/y)2 = k1/x × k1/z k2/y = k1/x+1/z
2 xy xz
1/y
and c = k
1/z
MANISH KUMAR Ex. 8 Sol.
2 1 1 y x z x-1
If 25 = 5 We have.
2x-1
MATHEMATICS
2 xz xz y
- 100. find the value of x.
25x-1 = 52x-1 - 100 (52)x-1 = 52x-1 - 100 52x-2 = 52x-1 = -100 52x-2 - 52x-2 .51 = - 100 52x-2 (1-5) = - 100 52x-2 (-4) = - 100 52x-2 = 25 52x-2 = 52 2x - 2 = 2 2x = 4 x=2
y
2 xz xz
MANISH KUMAR D.
MATHEMATICS
SURDS : Surd : - Any irrational number of the form
n
a is given a special name surd.
Where ‘a’ is called radicand, it should always be a rational number. Also the symbol radical sign and the index n is called order of the surd. 1
n
D-1.
a is read as nth root of ‘a’ and also be written as a n .
LAWS OF A SURDS :-
a
n
n
(i)
a n b n ab
n
(ii)
n
(iv)
n m
n
(i)
3
n.p
a b
ap
am
[Important for changing order of surds]
n.p
am.p
62 make its order 6 Then
3
(ii)
62
3
3
6
(iii)
3
8 23 2
(iv)
4
81 34 3
3
3 .2
62.2 6 4 6
6 make its order 15 Then
(v)
[Here order should be same]
a mm a m n a
a
n
or Ex.
a nb n
(iii)
(v)
n
an a
3 .5 1 .5
6
15
65
3
4
2 3 6 3 2 6 3 12
But
3
3 4 6 3 6
(Because order is not same)
st
1 make their order same & then you can multiple.
2 8 2
(vi) D-2.
IDENTITY OF A SURD :(i) These are not a surd, 3
3
3
8 23 which is a rational number.
7 4 3 is a surd as 7-4
(ii)
3 is a perfect square of ( 2 - 3 )
7 4 3 , 9 4 5 , 9 4 5 ......
Ex :
2 3 because 2 +
(iii) (iv)
8 because
3
3 is not a perfect square.
1 3 because radicand is an irrational number.
n
is called the
MANISH KUMAR 3
(v)
4 is a surd as radicand is a rational number.
Ex :
3
(vi)
3
3 is a surd as
Ex :
3 3
3 , 4 5 6,.....
(vii) 2 +
5, 4 12 , 5 7 ...... 3
1 1 3 32 36 3 3
3 is a surd (as surd + rational number will give a surd)
3 2, 3 1, 3 3 1,....
Ex : D-3.
MATHEMATICS
TYPES OF SURDS :(a) Simplest form of a surds :-
Ex.
(i)
3
(ii)
135 it’s simplest form is 33 5
4
(iii)
1875 it’s simplest form is 54 3 4
6
8 23 2 Simplest form
(b) Quadratic surds : Surds of order 2 Ex.
2 . 3.....
(c) Biquadratic surds : Surd of order 4 Ex :
4
8
(d) Cubic surds : Surd of order 3 Ex :
3
3.3 15
(e) Like surds : Two or more surds are called like if they have or can be reduced to have the same irrational or surds factor. Ex :
2 .3 2
(f) Unlike surds : Two or more surds are called unlike, if they are not similar, (i.e. radicand s well as index are different). Ex :
5. 3. 6
(g) Pure surds : A Surds which has unity only as its rational factor, the other being irrational, is called pure surd. Ex :
3 . 15, 4 1875 , 6 8
(h) Mixed surds : A surd which as a rational factor other than unity, the other factor being irrational, is called a mixed surd. Ex : 2 3, 54 3 (i) Simple surds : A surd consisting of a single term is called a simple surd.
73 6 2 (j) Compound surds : An algebraic sum of two or more surds is called as compound surd. are simple surd Ex :
3, 34 5 .
Ex :
3 5 3 4, 3 3 5 etc are compound surds.
(k) Monomial surds : Single surd is called monomial surds.
MANISH KUMAR
MATHEMATICS
43 3..... 2 (1) Binomial surds : An algebraic sum of two simple surds or a rational number and a simple surds is Ex :
3
2 , 2.
known as a binomial surd. Ex. : 2 3, 3 2 ,2 3 3, 3 2 3 3 (m) Trinomial surds: An algebraic sum of three simple surds or the sum of a rational number and two simple surds is known as a trinomial surd. Ex : 1 2 3,2 3 5, 3 9 3 6 3 4 (n) Equiradical surds : - Surds of the same order are called equiradical surds. Ex :
2, a, 5
(o) Non-equiradical surds : Surds of the different orders are known as non-equiradical surds. Ex : Ex.1
Sol.
D-4.
2
3
4 ,4 b
Express following as a pure surds and mixed surds. (i)
33 128 4
(i)
33 33 27 128 3 3 3 128 3 128 3 54 4 64 4
(ii)
5
5
(ii)
96
5
96 5 32 3 25 3 25 5 3 25 3 5
OPERATION OF SURDS :(a) Addition and subtraction of surds :- Addition & subtraction of surds are possible only when order and radicand are same i.e. only for like surds.
Ex.(i)
Simplify the 15 16 216 96
15 6 62 6 16 6
15 6 6 6 4 6 (15 6 4) 6
13 6 (b) Multiplication and division of surds : Ex.
(i)
Ex.
(ii) 3
3 3
3
4 3 22 3 4 22 23 11 23 11 2 by
4
243
12
3 24
3
12 3
12
24 33 12 16 27 12 432
(c) Comparison of surds : It is clear that if x > y > 0 and n > 1 is a positive integers then 3
16 3 12, 5 36 5 25 and so on.
n
x ny.
MANISH KUMAR Ex.
MATHEMATICS
Which is greater in each of the following: (i)
3
6 and 5 8
(ii)
1 1 and 3 2 3
MANISH KUMAR Sol.
3
(i)
5
6 and
MATHEMATICS 8
L.C.M. of 3 and 5 is 15 3
6
3 .5
5
8
3 .5 3
65 15 7776 8 15 512
15 7776 15 512 Sol.
3
6 5 8
1
(ii)
2
and
3
1 3
L.C.M. of 2 and 3 is 6 3
1 and 2
6
6
1 8
so Ex.2 Sol.
and
6
6
6
1 3
2
1
1 1 as 8 < 9 9 8 9
1 6 1 8 9
Which is greater
1 3 1 2 3
7 3 or 5 1 ?
7 3 7 3 7 3 4 7 3 7 3 4 5 1 5 1 5 1 4 5 1 5 1 5 1 5 1
7 3
and
Now, we known that add
1
4 7 3
D-5.
3 1
7 3 5 1 7 3
So
7 5 and
3
1 5 1 4 5 1
7 3 5 1
5 1 7 3
RATIONALIZATION OF SURDS :(a) Rationalizing Factor : - If the product of two surds is a rational number then each of them is called the rationalizing factor (R.F.) of the other. (b) Rationalization : - The process of converting a surds to a rational number by using an appropriate multiplier is known as rationalization.
MANISH KUMAR
MATHEMATICS
MANISH KUMAR
MATHEMATICS
(i) Rationalizing factor of
a a a a
a is 3
(ii) Rationalizing factor of
3 3 a 2 3 a a2 a3 a
3
a is
a b a b a b (iv) Rationalizing factor of a + b is a b & vice versa a b a b a b (v) Rationalizing factor of a b is a ab b a b a ab b a b is
(iii) Rationalizing factor of
a b & vice versa
2
3
3
3
2
3
3
2
3
3
3
2
b = a + b which is rational. (vi) Rationalizing factor of a b c is a b c and (a b c 2 3 a
Ex. 1
3
3
Find the following rationalizing factors : (i)
Sol.
3
(ii)
10
(iii)
3
(i)
10
162 3
(iv)
4
16
[ 10 10 10 10 10] as 10 is rational number (ii)
162
Simplest form 9 2 Rationalizing factor of
2 is
Rationalizing factor of
162 is
3
(iii)
3
3
3
3
3
4 is
3
42
16
Simplest from of
3
16 is 23 2 3
Now rationalizing factor of
Rationalizing factor of
3
Ex.2
Find rationalizing factor of
Sol.
Simplest form of
D-6.
2
43 4
Rationalizing factor of (iv)
2
4
4 42
4
2 is
16 is 4
3
3
22
22
162
162 is 34 2
Now rationalizing factor of
4
Rationalizing factor of
162 is
4
2 is
4 4
23 23
WHEN CONJUGATE SURDS AND RECIPROCALS ARE SAME :(a) 2 +
3 it’s conjugate is 2 -
3 , its reciprocal is 2+
3 & vice versa.
(b) 5 - 2 6 , it’s conjugate is 5 + 2 6 , its reciprocal is, 5 - 2 6 & vice versa. Ex.1
3
Express the following surd with a rational denominator.
ab ).
3
2
MANISH KUMAR
MATHEMATICS
8 15 1 5 3
8
15 1 5 3
8
15 1 5 3 15 1 5 3
15 1 5 3 8 15 1 5 3 15 1 5 3 8 15 1 5 3 15 1 2 15 5 3 2 15 8 15 1 5 3 2
Ex.2
2
8
15 1
If
32 5 a b 2 . where a and b are rational. 3 2
5 3
Find the values of a and b
3 2 2
L.H.S. =
3 2
3 2 2 3 2 9 3 3 2 3 2
2 6 2 4 13 9 2 92 7 7
13 9 2 ab 2 7 7 equating the rational and irrational parts
13 9 .b 7 7
We get a Ex.3
If x =
1 . find the value of x3 - x2 - 11x + 3 2 3
as x =
1 2 3
x-2=-
=2-
3
3
squaring both sides, we get
(x - 2)2 = 3
2
x2 + 4 - 4x = 3 x2 - 4x + 1 = 0
Now x3 - x2 - 11x + 3 = x3 - 4x2 + x + 3x2 - 12x + 3 x(x2 - 4x + 1) + 3 (x2 - 4x + 1) x x0 + 3 (0) 0 + 0 =0 Ex.4
If x = 1 +
Sol.
x=1+
2 3 , prove that x4 - 4x3 - 4x2 + 16 - 8 = 0
2 3
Squaring both sides
( x - 1)2 -
2 3
2
MANISH KUMAR
MATHEMATICS
x2 + 1 - 2x = 2 + 3 + 2 6 x2 - 2x - 4 = 2 6 squaring both sides
2
(x2 - 2x - 4)2 = 2 6
x4 + 4x2 + 16 - 4x3 + 16x - 8x2 = 24 x4 - 4x3 - 4x2 + 16x + 16 - 24 =0 x4 - 4x3 - 4x2 + 16x - 8 =0 D-7.
SQUARE ROOTS OF BINOMIAL QUADRATIC SURDS:-
(a) Since
x y
2
( x y ) 2 xy &
x y
2
(b) square root of x + y + 2 xy x y
(c) Square root of (x + y)-2 xy x y
(d) Square root of a2 + b + 2a b a b
(e) and square root of a2 + b - 2a b a b Ex.1
7 4 3 ( 2) 2
2 3
2
3
2
x y 2 xy
22 3
2 3
52 6 52 6
Ex.2 Simplify
52 6 52 6
52 6 3 2& 52 6 3 2
Since
52 6 52 6 5 2 6 52 6 Ex.3 If x
a 2b a 2b a 2b a 2b
as x =
3 2 3 2 2 3 2 3 2 2
a 2b a 2b
a 2b a 2b a 2b
2
a 2b a 2b
a 2 4b 2
2bx - a =
a 2 4b 2
2
3
a 2b a 2b
a 2b a 2b 2 (a 2b)(a 2b)
2a a 2 4b 2 a a 2 4b 2 x= 4b 2b 2bx = a +
3
prove that b2x2 - abx + b2 = 0
a 2b a 2b
a 2b x=
2
4b
3 3
1 6 3
MANISH KUMAR squaring both sides (2bx - a)2 = a 2 4b 2 4b2x2 + a2 - 4abx = a2 - 4b2 4b2x2 - 4abc + 4b2 = 0 b2x2 - abx + b2 = 0
MATHEMATICS
MANISH KUMAR
MATHEMATICS
EXERCISE - 01 OBJECTIVE TYPE QUESTIONS 1.
How many prime number are there between 0 and 30:(A) 9
2.
(B) 10
5 and 2 5
411 16
423 1000
(D) 6
421 16
(C)
431 16
(D)
441 16
(B)
27 99
(C)
27161 9900
(D)
27191 9000
(B)
479 1000
(C)
423 990
(D)
419 990
(C)
9
(D)
16
1 3
(B)
2
1
1
1 1 1/ 3
The number
If x
4
is equal to :-
3 3 3 3
(A) Rational
(B) 11/7
(C) 3
(D) 1
(B) Irrational
(C) Both
(D) Can’t say
(C) 3
(D)
is :-
1 1 3 then x 3 3 equal :x x
(A) 6 3
11.
1/4
Which of the following is not a rational number :-
(A) 1/3
10.
(C) 6
The value of 0.423 is :-
1
9.
1/3
(B)
27161 999
(A)
8.
(B) 6
The rational form of 2.7435 is :-
(A) 7.
(D) None of these
The rational form of -25.6875 is :-
(A) 6.
5 and 2 7
(C)
2 3 is :-
1/2
(A) 5.
5 and 2 5
(B)
The exponential form of (A) 6
4.
(D) 11
Two irrational numbers between 2 and 2.5 are :(A)
3.
(C) 8
The value of 5.2 :-
(B) 3 3
3
MANISH KUMAR (A)
45 9
MATHEMATICS (B)
46 9
(C)
47 9
(D) None
MANISH KUMAR
x x x b c 2
a b
12.
MATHEMATICS c a 2
( x a .b b .x c ) 4 (A) -1
13.
(B) 0
( 0.6) 0 (0.1) 1
The value of
(3 / 2 ) (3 / 2 ) 1
3
(A) 3/2 14.
15.
If 9
(B)
=3
2x-1
If a
1 32 2
18.
2( 2 n 3 )
If 2
,b
1
n+1
(B) -2
= 32 and 2
The value of
(A)
(D) None of these
(C) 1.5
(D) 0
(C) 36
(D) 37
2 3 is equal to :-
x+y
+
1 8
9 2n 8
(C)
(D) 1
16 then a2 + y2 :-
(A) 9 19.
7 48
then the value of a2 + b2 is :-
3 2 2
n+1
2x-y
(C)
(B) 35
2 n 4 2( 2 n )
(A) 2
7 32
(B) 2.5
(A) 34 17.
(D) -1/2
- 486 then the value of x is :-
(A) 3.5 16.
(C) 2/3
1 1 1 4, then the value of x is :2 x 4 y 4z
7 16 x-1
(D) None
is :-
1
(B) -3/2
If 2x = 4y = 8z and
(A)
1 3
(C) 1
(B) 10
( 25) 5 / 2 ( 243) 2 / 5 (16) 3 / 4 (8) 5 / 3
5625 128
(C) 11
(D) 13
is :-
(B)
5615 256
(C)
5625 256
(D) None
a
20.
1 a 1 a a 1 a 1 The value of ( x = )
(A) x
21.
4 3
(C) x
a
(D) 1/x
1/2
(C) x
1/3
(D) x
a
x2 =
(A) x
22.
(B) 1/x
(B) x
The value of 5 3 3 12 2 75 on simplifying is :-
1/6
MANISH KUMAR (A) 5 3
MATHEMATICS (B) 6 3
(C)
3
(D) 9 3
MANISH KUMAR 23.
If
MATHEMATICS
3 = 1.732,
(A) 10.905 24.
(B) 134
2 3
If a =
2 3
,b
2 3 2 3
If x =
(B) -14
1 2 3
(B) 1
(B)
2
Simplify
5 3
1 3 2
If
(D) 154
(C) 8 3
(D) 3
(C) 0
(D) 3
52
74 3
3
2
(D) None of these
5
:(C) 10
(D) 100
(C) a = 15, b = 35
(D) a = 35, b = 25
(C) 3
(D) None of these
3b the value of a and b is :-
4a +
The value of 3
(C)
5
(B) 0
5 3
(A)
12
3
(A) a = 47, b = 27 30.
(C) 144
The surd 34 3 5 3 4 5 in its simplest form is equal to :-
(A) 1 29.
(D) None
find the value of a3 - 2x2 - 7x + 5 is :-
(A) 212 5 28.
(C) 11.905
then the value of a + b is :-
(A) 2 27.
is :-
The product of 4 6 and 3 24 is :-
(A) 14 26.
3 3
(B) 11.904
(A) 124 25.
6
5 = 2.236, then the value of
(B) a = 27, b = 47
24 3 81 .3 192 is :(B)
3
3
ANSWERS KEY Que.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
MANISH KUMAR Ans. Que. Ans.
B 16 A
MATHEMATICS
A 17 D
C 18 B
A 19 D
C 20 A
D 21 D
A 22 D
B 23 B
B 24 C
A 25 A
C 26 D
EXERCISE - 02 SUBJECTIVE TYPE QUESTIONS 3 4 and . 5 5
1.
Find 5 rational number between
2.
Represent 3.765 on the number line.
3.
Find the decimal representation of 22/7.
4.
Express the following decimal in the form (a) 0.32
p . q
( b) 0.123
5.
Prove that
6.
Find two rational and two irrational numbers between 0.15 and 0.16.
7.
Virsualize 4.26 on the number line, upto 4 decimal places.
8.
81 Simplify 16
9.
Prove that
10.
If
11.
If a = x
12.
If x is a positive real number and the exponents are rational numbers, show that : (i)
3 2 2 3 2 q+r
3 4
25 3 / 2 5 3 . 9 2
a 1 a 1 b 1
1 x
p
y ,b=x
Simplify
b a
a
a 1
a 1 b 1
2b 2 b2 a2
a b 2 , where a and b are rationals. Find the values of a and b.
r+p
1
xb (ii) c x 13.
3 is irrational.
x
xc xa
c a
b
xa xb
6 2 3 6
q
y ,c=x
p+q
r
q-r
y . Prove that a
1 1 x
a b
x
c b
1 x
c
6
r-p
1
1
3 2
×b
4 3 6 2
.
bc
x ac
p-q
×c
1
= 1.
C 27 A
B 28 B
A 29 C
A 30 A
MANISH KUMAR 5 2
MATHEMATICS , find the value of 3x2 + 4xy - 3y3.
If x =
15.
x = 3 + 2 2 , find the value of x4 +
16.
If x =
3 2 2 find the values of :-
(i) x
1 x
17.
5 2
(ii) x 2
5 2
7 5
7 5 7 5
1
x4
.
1 x2
a 7 5 b.
2
If x = 2 3 2 3 show that x3 - 6x = 6.
NUMBER SYSTEM
●
1
In the following determine rational numbers a and b.
7 5
18.
and y =
5 2
14.
Subjective type answers
ANSWER KEY
EXERCISE – 2 (IX) - CBSE
MANISH KUMAR 1.
19 20 21 22 23 , , , and 30 30 30 30 30
MATHEMATICS 3. 3.14285714
6. 0.155 and 0.1525 irrational number 0.1549 and d0.1524 10. a
13 9 and b 7 7
15. 1154
4. (a)
29 1221 , (b) 90 9900
8. 1
56 10 3
13. 0
14. 4
16. (i) 2 2 (ii) 6
17. a 0, b
1 11
