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Mathematics
Number Systems
NCERT Solutions
NCERT Solutions
Number Systems
24 Solutions
Exercise:
All Exercises
Exercise 1.1
Exercise 1.2
Exercise 1.3
Exercise 1.4
Exercise 1.5
Q1
Exercise 1.1
Is zero a rational number? Can you write it in the form
p
q
\frac{p}{q}
q
p
, where
p
p
p
and
q
q
q
are integers and
q
≠
0
q \neq 0
q
=
0
?
Q2
Exercise 1.1
Find six rational numbers between 3 and 4.
Q3
Exercise 1.1
Find five rational numbers between
3
5
\frac{3}{5}
5
3
and
4
5
\frac{4}{5}
5
4
.
Q4
Exercise 1.1
State whether the following statements are true or false. Give reasons for your answers.
(i)
Every natural number is a whole number.
(ii)
Every integer is a whole number.
(iii)
Every rational number is a whole number.
Q1
Exercise 1.2
State whether the following statements are true or false. Justify your answers.
(i)
Every irrational number is a real number.
(ii)
Every point on the number line is of the form
m
\sqrt{m}
m
, where
m
m
m
is a natural number.
(iii)
Every real number is an irrational number.
Q2
Exercise 1.2
Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Q3
Exercise 1.2
Show how
5
\sqrt{5}
5
can be represented on the number line.
Q1
Exercise 1.3
Write the following in decimal form and say what kind of decimal expansion each has:
(i)
36
100
\frac{36}{100}
100
36
(ii)
1
11
\frac{1}{11}
11
1
(iii)
4
1
8
4 \frac{1}{8}
4
8
1
(iv)
3
13
\frac{3}{13}
13
3
(v)
2
11
\frac{2}{11}
11
2
(vi)
329
400
\frac{329}{400}
400
329
Q2
Exercise 1.3
You know that
1
7
=
0.
142857
‾
\frac{1}{7}=0 .\overline{142857}
7
1
=
0.
142857
. Can you predict what the decimal expansions of
2
7
,
3
7
,
4
7
,
5
7
,
6
7
\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}
7
2
,
7
3
,
7
4
,
7
5
,
7
6
are, without actually doing the long division? If so, how?
Q3
Exercise 1.3
Express the following in the form
p
q
\frac{p}{q}
q
p
, where
p
p
p
and
q
q
q
are integers and
q
≠
0
q \neq 0
q
=
0
.
(i)
0.
6
‾
0 .\overline{6}
0.
6
(ii)
0.4
7
‾
0.4 \overline{7}
0.4
7
(iii)
0.
001
‾
0 .\overline{001}
0.
001
Q4
Exercise 1.3
Express
0.99999
…
0.99999 \ldots
0.99999
…
in the form
p
q
\frac{p}{q}
q
p
. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
Q5
Exercise 1.3
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of
1
17
\frac{1}{17}
17
1
? Perform the division to check your answer.
Q6
Exercise 1.3
Look at several examples of rational numbers in the form
p
q
(
q
≠
0
)
\frac{p}{q}(q \neq 0)
q
p
(
q
=
0
)
, where
p
p
p
and
q
q
q
are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property
q
q
q
must satisfy?
Q7
Exercise 1.3
Write three numbers whose decimal expansions are non-terminating non-recurring.
Q8
Exercise 1.3
Find three different irrational numbers between the rational numbers
5
7
\frac{5}{7}
7
5
and
9
11
\frac{9}{11}
11
9
.
Q9
Exercise 1.3
Classify the following numbers as rational or irrational :
(i)
23
\sqrt{23}
23
(ii)
225
\sqrt{225}
225
(iii)
0.3796
(iv)
7.478478
…
7.478478 \ldots
7.478478
…
(v)
1.101001000100001
…
1.101001000100001 \ldots
1.101001000100001
…
Q1
Exercise 1.4
Classify the following numbers as rational or irrational:
(i)
2
−
5
2-\sqrt{5}
2
−
5
(ii)
(
3
+
23
)
−
23
(3+\sqrt{23})-\sqrt{23}
(
3
+
23
)
−
23
(iii)
2
7
7
7
\frac{2 \sqrt{7}}{7 \sqrt{7}}
7
7
2
7
(iv)
1
2
\frac{1}{\sqrt{2}}
2
1
(v)
2
π
2 \pi
2
π
Q2
Exercise 1.4
Simplify each of the following expressions:
(i)
(
3
+
3
)
(
2
+
2
)
(3+\sqrt{3})(2+\sqrt{2})
(
3
+
3
)
(
2
+
2
)
(ii)
(
3
+
3
)
(
3
−
3
)
(3+\sqrt{3})(3-\sqrt{3})
(
3
+
3
)
(
3
−
3
)
(iii)
(
5
+
2
)
2
(\sqrt{5}+\sqrt{2})^{2}
(
5
+
2
)
2
(iv)
(
5
−
2
)
(
5
+
2
)
(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})
(
5
−
2
)
(
5
+
2
)
Q3
Exercise 1.4
Recall,
π
\pi
π
is defined as the ratio of the circumference (say
c
c
c
) of a circle to its diameter (say
d
d
d
). That is,
π
=
c
d
\pi=\frac{c}{d}
π
=
d
c
. This seems to contradict the fact that
π
\pi
π
is irrational. How will you resolve this contradiction?
Q4
Exercise 1.4
Represent
9.3
\sqrt{9.3}
9.3
on the number line.
Q5
Exercise 1.4
Rationalise the denominators of the following:
(i)
1
7
\frac{1}{\sqrt{7}}
7
1
(ii)
1
7
−
6
\frac{1}{\sqrt{7}-\sqrt{6}}
7
−
6
1
(iii)
1
5
+
2
\frac{1}{\sqrt{5}+\sqrt{2}}
5
+
2
1
(iv)
1
7
−
2
\frac{1}{\sqrt{7}-2}
7
−
2
1
Q1
Exercise 1.5
Find :
(i)
64
1
2
64^{\frac{1}{2}}
6
4
2
1
(ii)
32
1
5
32^{\frac{1}{5}}
3
2
5
1
(iii)
125
1
3
125^{\frac{1}{3}}
12
5
3
1
Q2
Exercise 1.5
Find :
(i)
9
3
2
9^{\frac{3}{2}}
9
2
3
(ii)
32
2
5
32^{\frac{2}{5}}
3
2
5
2
(iii)
16
3
4
16^{\frac{3}{4}}
1
6
4
3
(iv)
125
−
1
3
125^{\frac{-1}{3}}
12
5
3
−
1
Q3
Exercise 1.5
Simplify:
(i)
2
2
3
⋅
2
1
5
2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}
2
3
2
⋅
2
5
1
(ii)
(
1
3
3
)
7
\left(\frac{1}{3^{3}}\right)^{7}
(
3
3
1
)
7
(iii)
11
1
2
11
1
4
\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}
1
1
4
1
1
1
2
1
(iv)
7
1
2
⋅
8
1
2
7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}
7
2
1
⋅
8
2
1
More from this chapter
Chapter overview
Important Points
Practice Questions
Flashcards