Download 1 NUMBER SYSTEMS CHAPTER

CHAPTER
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NUMBER SYSTEMS
Points to Remember :
1. Number used for counting 1, 2, 3, 4, ... are known as Natural numbers.
2. All natural numbers together with zero i.e. 0, 1, 2, 3, 4, ..... are known as whole numbers.
3. All natural numbers, zero and negative numbers together i.e. ...., –4, –3, –2, –1, 0, 1, 2, 3, 4, ... are known
as Integers.
p
2 5 4
,
4. Rational Numbers : Numbers of the form
where p, q both are integers and q  0. For e.g. ,
q
3 7 1
etc.
5. Every rational number have either terminating or repeating (recurring) decimal representation.
Terminating
Repeating (Recurring)
For eg. 2  0.4 , 13  3.25 etc.
5
4
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For e.g. 1  0.333.....  0.3
3
15
= 2.142857142857...
7
here, prime factors of denominator are
2 and 5 only.
B
 2.142857 etc.
6. There are infinitely many rational numbers between any two given rational numbers.
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7. Irrational Numbers : Numbers which cannot be written in the form of q , where p, q are integers and
q  0.
For e.g.
2 , 3 , 17 , , 0.202202220......,3 9 etc.
8. Real numbers : Collection of both rational and irrational numbers. For e.g.  3, 7 , 0 , 2 , 5 ,  etc.
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9. Every real number is represented by a unique point on the number line. Also, every point on the number
line represents a unique real number.
10. For every given positive real number x, we can find
x geometrically..
11. Identities related to square root :
Let p, q be positive real numbers. Then,
(i)
pq 
p. q
(iii) ( p  q ) ( p  q )  p  q
(ii)
p

q
p
;q  0
q
(iv) ( p  q ) 2  p  2 pq  q
12. Laws of Radicals : Let x, y > 0 be real numbers and p, q be rationals. then
(i) xp × xq = xp+q
(ii) x p  x q  x p  q
(iii) ( x p ) q  x pq
(iv) x p . y p  ( xy) p
MATHEMATICS–IX
NUMBER SYSTEMS
1
ILLUSTRATIVE EXAMPLES
Example 1. Find six rational numbers between 3 and 4.
—NCERT.
Solution.
We know that between two rational numbers a and b, such that a < b, there is a rational number
a b
.
2
1
7
A rational number between 3 and 4 is (3  4)  .
2
2
Now, a rational number between 3 and
A rational number between
1
7  1  6  7  13
7
 .
is  3    
2
2 2 2  4
2
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17
 1  7  8  15
7
 .
and 4 is   4   
22
 2 2  4
2
Also, a rational number between 3 and
1  13  1  12  13  25
13

is  3    
2
4  2 4  8
4
A rational number between
1  15
 1  15  16  31
15

and 4 is   4   
2
4
4

 2 4  8
A rational number between
1  31
 1  31  32  63
31

and 4 is   4   
2 8
8
 2  8  16
B
25 13 7 15 31 63

 


4
8
4 2 4
8 16
This can be represented on number line as follows :
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 3
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OR
(without using formula)
10 40
We have, 3  3  10  30 and 4  4 
10 10
10 10
We need to find six rational numbers between 3 and 4 i.e.
30
40
31 32 33 34 35 36
and
, which are ,
,
,
,
,
. Ans.
10
10
10 10 10 10 10 10
Example 2. Find four rational numbers between 1 and 4 .
4
3
Solution.
1 1 3 3
4 4 4 16
  
and   
4 4 3 12
3 3 4 12
 4 rational numbers between
2
3
16
1
4
4 5 6 7
and
i.e.
and
are
, , , .
12
12
4
3
12 12 12 12
NUMBER SYSTEMS
MATHEMATICS–IX
Example 3. Express 0.12 in the form of rational number,
Solution.
Let x  0.12  0.121212 .....
multiplying both sides by 100, we get
100 x = 12.1212.....
Subtracting (1) from (2),
100 x – x = 12.1212..... – 0.1212....
 99 x = 12  x 
Example 4. Represent
Solution.
p.
q
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...(1)
...(2)
12
4

Ans.
99 33
3 on the number line.
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Let XOX be number line with O as origin. Let OA = 1 unit. Draw AB  OA such that AB = 1 unit.
Join OB. Then using Pythagoras theorem, In OAB
OB  OA 2  AB2  12  12  2 units
Again, draw DC  OB such that BC = 1 unit. Join OC.
then, OC  OB 2  BC 2  ( 2 ) 2  (1) 2  3 units .
With O as centre and OC as radius, draw on arc, meeting OX at
B
P. Then OC = OP = 3 units.
Example 5. State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
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(ii) Every point on the number line is of the form
m , where m is a natural number..
(iii) Every real number is an irrational number
—NCERT
(i) True, since collection of real numbers is made up of rational and irrational numbers.
(ii) False, no negative number can be the square root of any natural number.
(iii) False, for example 5 is a real but not irrational.
Example 6. Write the following in decimal form and say what kind of decimal expansion each as:
36
1
3
329
1
2
(i)
(ii)
(iii) 4
(iv)
(v)
(vi)
—NCERT
100
8
13
400
11
11
Solution.
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Solution.
(i) 36  0.36  Terminating decimal expansion.
100
1
, by long division, we have :
(ii) Consider
11
0.090909
11 1.000000
99
100
99
100
99
1
MATHEMATICS–IX
NUMBER SYSTEMS
3

1
 0.090909......  0.09, which is non-terminating and repeating decimal expansion.
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1 4  8  1 33
(iii) 4 
 . By long division, we have
8
8
8
4.125
8 33.000
32
10
8
20
16
40
40
0
 33  4.125, terminating decimal expansion.
8
(iv) Consider,
3
by long division, we have
13
0.23076923....
13 3.0000000
26
40
39
100
91
90
78
120
117
30
26
40
39
1
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B
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3
 0.23076923.......  0.230769, which is non-terminating and repeating decimal expan13
sion.

4
NUMBER SYSTEMS
MATHEMATICS–IX
(v) Consider,
2
, by long division, we have
11
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0.181818
11 2.00000
11
90
88
20
11
90
88
20
11
90
88
2

2
 0.181818.....  0.18 , which is non-terminating and repeating decimal expansion.
11
329
, by long division, we have
400
(vi) Consider,
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0.8225
400 329.0000
3200
900
800
1000
800
2000
2000
0
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
B
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329
 0.8225, which is terminating decimal expansion.
400
Example 7. What can be the maximum number of digits be in the repeating block of digits in the decimal
expansion of
MATHEMATICS–IX
1
? Perform the division to check your answer..
17
NUMBER SYSTEMS
—NCERT
5
Solution.
0.588235294117647....
17 1.0000000000000000
85
150
136
140
136
40
34
60
51
90
85
50
34
160
153
70
68
20
17
30
17
130
119
110
102
80
68
120
119
1
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Thus,
B
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1
 0.588235294117647
7
 The maximum number of digits in the quotient while computing
Example 8. Look at several examples of rational numbers in the form
Solution.
6
1
are 15.
17
p
(q  0) , where p and q are integers
q
with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?
—NCERT
Let us consider various such rotational numbers having terminating decimal representation.
NUMBER SYSTEMS
MATHEMATICS–IX
1
3
5
 0.5 ;  0.75;  0.625
2
4
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39
11
43
 1.56;
 0.088;
 0.215 etc.
25
125
200
from the examples shown above, it can be easily observe that, ‘‘If the denominator of a rational
number in standard form has no prime factors other than 2 or 5 or both, then and the only then it
can be represented as a terminating decimal.’’
Example 9. Visualise 3.765 on the number line, using successive magnification.
—NCERT
Solution.
We know that 3.765 lies between 3 and 4. We divide portion of number line between 3 and 4 in 10
equal parts i.e. 3.1, 3.2, ....., 3.9 and then look at the interval [3.7, 3.8] through a magnifying glass
and observe that 3.765 lies between 3.7 and 3.8 (see figure).
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Now, we imagine that each new intervals [3.1, 3.2], [3.2, 3.3], ...... , [3.9, 4] have been sub-divided
into 10 equal parts. As before, we can now visualize through the magnifying glass that 3.765 lies
in the interval [3.76, 3.77]. (see figure).
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B
Again, 3.765 lies between 3.76 and 3.77. So, let us focus on this portion of the number line, and
imagine to divide it again into 10 equal parts. The first mark represents 3.761, second mark represents 3.762, and so on. So, 3.765 is the 5th mark in these subdivisions.
Example 10. Recall,  is defined as the ratio of the circumference (say c) of a circle to its diameter (say d).
That is,   c . This seem to contradict the fact that  is irrational. How will you resolve this
d
contradiction?
—NCERT
Solution.
There is no contradiction. Remember that when you measure a length with a scale or any other
device, you only get on approximate rational value. So, you may not realise that either c or d is
irrational.
Example 11. Simplify the following :
(i) 3 7  4 7
(iii) ( 5  3 ) 2
MATHEMATICS–IX
(ii) ( 7  3 ) ( 7  3 )
(iv) 8 30  2 5
NUMBER SYSTEMS
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Solution.
(i) 3  4  ( 7 ) 2  12  7  84
(ii) ( 7 ) 2  ( 3 ) 2  7  3  4
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(iii) ( 5 ) 2  ( 3 ) 2  2( 5 )( 3 )  5  3  2 5  3  8  2 15
(iv)
8
30

4 6
2
5
Example 12. Find
Solution.
5.3 geometrically..
Draw AB = 5.3 units and extend it to C such that BC
= 1 unit. Find mid-point O of AC. With O as centre,
and OA as radius, draw a semicircle. Draw
BD  AC, interesting semicircle at D. Then BD
=
5.3 units. With B as centre and BD as radius,
draw an arc, intersecting AC produced at E. Then,
BE  BD  5.3 units.
Example 13. Find value of a and b, where a  b 2 
Solution.
3 2
We have,
3 2
3 2

3 2
3 2
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
(3) 2  ( 2 ) 2
.
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9  2  6 2 11  6 2

92
7
11 6
2  ab 2

7 7

B
3 2
3 2

(3  2 ) 2

3 2
11
6
and b 
7
7
Example 14. Simplify the following :

a
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2
4
(i) 3 5  3 5
Solution.
2 4

5
(i) 3 5
1 1

4
(ii) 7 3
3
2 4
5
7
6
 35
4 3
12
8
2
5
2
5
1
4 1
(iii) (3 )
(iv) (32) 2/5
1
 7 12
(iii) 34 x(–1)  34 
(iv) (25 )
1
(ii) 7 3  7 4
2
5
1
3
4

1
81
 22 
1
2
2

1
4
NUMBER SYSTEMS
MATHEMATICS–IX
1
Example 15. Simplify :
Solution.
1  x b a  x c  a

1
1
Given expression 
b
1
x
x
 a
a
x
x
a
b
x x x
xa
xa
x a  xb  xc
a
1
c
1  x b  c  x a c
1

1


c
1

1  x a b  x c  b


x
x
 b
b
x
x
b
1
1
b
a
x x x
xb
c
xb
x a  xb  xc


x
xa

x c xc
1
c
x  xb  x a
xc
xc
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x a  xb  xc
PRACTICE EXERCISE

x a  xb  x c
x a  xb  x c
1
1. Represent each of the following rational numbers on the number line :
3
(i)
5
2.
7
(ii)
4
(i) Represent
2 and
1
2
and .
5
3
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(ii) Find 5 rational numbers between
3.
B
(iii) – 3.6
(i) Find 4 rational numbers between
3
4
and .
4
3
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1

c
.
(iv) 4.53
3 on the some number line
(ii) Represent
5 on the number line.
4. Without actual division, find which of the following rationals are terminating decimals.
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(i)
13
24
(ii)
19
125
(iii)
6
35
(iv)
17
80
(iii)
11
30
(iv)
37
175
(v)
31
200
5. Find the decimal expansions of the following :
(i) –
16
9
(ii)
22
7
6. Find the decimal representation of
1
1
. Deduce from the decimal representation of , without actual
7
7
calculations, the decimal representation of
2 3 4
5
, , and .
7 7 7
7
7. Express each of the following recurring decimals in the form of a rational number,
(i) 0.7
MATHEMATICS–IX
(ii) 0.123
(iii) 0.45 6
NUMBER SYSTEMS
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q
(iv) 3.456
9
8. (i) Find three irrational numbers between
2
4
and .
3
5
(ii) Insert three irrational numbers between
2 and
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3.
9. Give an example of two irrational numbers whose :
(i) difference is a rational number.
(ii) difference is an irrational number.
(iii) sum is a rational number.
(iv) sum is an irrational number.
(v) quotient is a rational number.
(vi) quotient is an irrational number.
(vii) Product is a rational number.
(viii) Product is an irrational number.
10. Simplify each of the following :
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(i) (7  3 ) (7  3 )
(ii) ( 5  3 ) 2
(iii) 12 20  3 5
(iv) (2 3  3 2 ) 2
(v) (4 2  3) (4 2  3)
(vii) 3 20  3 5  2 2  4 18
B
11. Represent the following on the number line :
(i)
(ii)
2.4
(iii)
5.7
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12. Rationalize the denominator :
1
(i)
2
(ii)
2 3
3 5
(iii)
13. Rationalize the denominator:
3
1
(i)
(ii)
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52 3
14. Simplify the following :
(i)
4 3
4 3

4 3
4 3
6 3 2
15. If a and b are rational numbers and if
3 2 5
3 2 5
16. If x and y are the rational numbers and
5 1
5 1
6.8
2 3
2 3
(iv)
9.2
1
(iv)
5 3
1
(iii)
(ii)
5 33 5
3 5
3 5

1
(iv)
2 3 3 2
3 5
3 5
 a  b 5 , find a and b.

5 1
5 1
 x  y 5 , find x and y.
17. Evaluate the following :
10
(i) 53  52
(ii) 58  55
(v) (3 8 )1/2
(vi) 3 –2  4 –2
(iii) (32 ) 2
NUMBER SYSTEMS
(iv) (64) 2/3
MATHEMATICS–IX
18. Find the valueof x if :
1
(ii) 32 x 1   
9
(i) 23 x 1  1
1
(iii)  
6
7 3 x
2
(iv)  
3
 6
x 3
3x  2
2
 
3
2 3 x
3
 
2
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x 1
19. Find whether the product of irrational numbers (5  2 ), (3  7 ), (3  7 ) and (5  2 ) is a rational or
irrational number.
20. (i) Given
1
2  1.414 , find value of
2 1
(ii)
3  1.732, find value of
21. (i) Prove that
2 3
2 3
.
2 is not a rational number..
(ii) Prove that
2  3 is an irrational number..
22. Simplify the following :
(i) 0.2  0.3  0.4
(ii) 0.42  0.34
23. Simplify the following :
(i)
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2 n 1  2 n
2 n  2 n 1
B
(iii) 2.13  1.16
(ii)
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(iv) 4.16 1.25
x 1 y y 1 z z 1 x
24. Assuming that x is positive real number and a, b, c are rational numbers, show that :
 xb 
(i)  c 
x 
a
 xc 
 
 xa 
 
b
 xa

 xb

c

 1


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25. If 2 x  3 y  6  z , show that
(ii) If x  5  2 6 , find the value of
1
x2
x
27. Simplify the following :
1

2

1
1 2  3
MATHEMATICS–IX
a b
 xb

 xc





b c
 xc

 xa





ca
1
1
.
1 .
x
(ii)
2 3
5  3 2 5
28. Rationalize the denominator of following :
(i)




1 1 1
  0
x y z
2
26. (i) If x  4  15 , find the value of x 
(i)
 xa
(ii)  b
x
(ii)
3 2
6 3

4 3
6 2

2 3
6 2
2
1 3  5
NUMBER SYSTEMS
11
29. Prove that :
(i)
1
1

1 2
1
(ii)
1

2 3
2 3 5

1
 .... 
3 4
2
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8 9
1

2 3 5
2
2
30. Represent the following on the number line :
(i) 13
(ii) 17
(iii) 2 3
(iv) 1  2
PRACTICE TEST
M.M : 30
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Time : 1 hour
General Instructions :
Q. 1-4 carry 2 marks, Q. 5-8 carry 3 marks and Q. 9-10 carry 5 marks each.
1. Find three rational numbers between
2. Represent
3
4
and .
4
3
5 on the number line.
3. Rationalise the denominator :
1
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52
4. Find decimal representation of
5. Simplify the following :
(i) ( 5  2 ) 2
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16
.
45
6. If a and b are rational numbers and
7. Evaluate the following :
7
3
 11  4  11  4
(i)     
2
2
4 3
4 3
B
(ii) (3 2  2 3 )(3 2  2 3 )
 a  3b , find a and b.
 1 
(ii)  
 32 
2 / 5
8. Express 0.246 as a rational number in the simplest form.
9. Represent
10. Simplify :
12
4.2 on the number line. Also, give step of constructions.
6
2 3

3 2
6 3

4 3
6 2
NUMBER SYSTEMS
MATHEMATICS–IX
ANSWERS OF PRACTICE EXERCISE
2. (i)
4 5 6 7
, , ,
15 15 15 15
(ii)
4. (ii), (iv) and (v)
6.
J
A
10 11 12 13 14
, , , ,
12 12 12 12 12
5. (i)  1.7
(ii) 3.142857
(iii) 0.36
(iv) 0.21142856
4
5
1
2
3
 0.142857 ,  0.284714,  0.428571,  0.571428,  0.714285
7
7
7
7
7
7
61
137
1711
(ii)
(iii)
(iv)
9
495
300
495
8. (i) 0.68010010001 ..., 0.69010110111 .... and 0.7101001000...
7. (i)
9. (i) 3  5 and 5  3
J
A
(ii) 1.501001000 ... and 1.601001000...
(ii) 3 5 and 5
(iii) 3  5 and 3  5
(iv) 4 5 and  2 5
(v)
(vi)
(vii) 2 3 and 3
(viii) 3 2 and 4 3
10. (i) 46
(ii) 8 – 2 15
1
12. (i) 2  3 (ii) (3  5 )
2
20 and 5
20 and 6
(iii) 8
(iv) 30  12 6
(iii) 7  4 3
1
(iv) ( 5  3 )
2
B
(v) 23
(vi) 3 5  10 2
13. (i)
1
1
1
(5 3  3 5 ) (iv)  1 (2 3  3 2 )
( 5  2 3 ) (ii) ( 6  3 2 ) (iii)
4
30
7
6
14. (i)
38
(ii)  3 5
13
17. (i) 3125 (ii) 125 (iii)
18. (i)
1
3
(ii)
5
8
T
I
15. a 
29
12
,b 
11
11
1
1
(iv)
(v)
81
16
(iii)
5
2
(iv)
M
A
1
2 (vi) 144
4
5
19. Rational
20. (i) 0.414
23. (i) 2 (ii) 1
26. (i) 62 (ii) 12
28. (i)
16. x = 3, y = 0
(ii) 13.928
7
422
1540
(iii)
(iv)
90
231
297
27. (i) 0 (ii) 0
22. (i) 1 (ii)
1
2
( 2  2  6 ) (ii)
(7  3 3  5  2 15 )
4
11
ANSWERS OF PRACTICE TEST
1.
10 11 12
, ,
12 12 12
6.
a
19
8
,b 
13
13
MATHEMATICS–IX
3.
52
7. (i)
4.  0.35
111
11
(ii) –4 8.
450
2
5. (i) 7  2 10 (ii) 6
10. 0
NUMBER SYSTEMS
13