Real Numbers 1.1 |
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Real Numbers 1.2 |
Question 1:
What is the rational numbers? Give 10 examples of rational numbers?
Solution:
The number that can be written in the
Examples of rational numbers:
(1)
(7)
Solution 2:
Question 2:
Represent each of the following rational numbers on the number line:
I – 5
II — 3
III —
IV–
V – 1.3
VI — -24
VII —
SOLUTION :
(i) Let
X =
Rational number lying between x and y:
=
(ii) Let:
X =
Rational number lying between x and y:
=
(iii) Let:
X = 1.3 and y = 1.4
Rational number lying between x and y:
=
(iv) Let:
X = 0.75 and y = 1.2
Rational number lying between x and y;
=
(v) Let:
X= -1 and y =
Rational number lying between x and y:
=
(vi) Let:
X =
Rational number lying between x and y:
=
Question 3:
Find three rational numbers lying between
Solution:
Let:
X =
We know:
So, three rational numbers lying between x and y are:
(x + d), (x + 2d) and ( x + 3d)
=
=
Question 4:
Find five rational numbers lying between
Solution:
Let:
X= 25, y = 34 and n= 5
We know:
So, five rational numbers between x and y are:
(x + d) ,(x + 2d), (x + 3d), (x + 4d) and (x + 5d)
=
=
Question 5:
Insert 6 rational numbers between 3 and 4.
Solution:
Let:
X= 3, y =4 and n=6
We know:
d =
So, six rational numbers between x and y are:
(x + d) ,(x + 2d), (x + 3d), (x + 4d), (x + 5d) and (x + 6d)
=
=
Question 6: Insert 16 rational numbers between 2.1 and 2.2 .
Solution:
Let x = 2.1, y = 2.2 and n= 16
We know,
d =
So, 16 rational numbers between 2.1 and 2.2 are:
(x + d), (x + 2d), . . . . . (x + 16d)
= [ 2.1 + 0.005], [ 2.1 +2( 0.005)]. . . . . . . . . [2.1 + 16(0.005)]
= 2.105, 2.11, 2.115, 2.12, 2.125 , 2.13, 2.135, 2.14, 2.145, 2.15, 2.155, 2.16, 2.165, 2.17, 2.175 and 2.18
Question 7: Without actual division find which of the following rational are terminating decimals
1380 724 512 835 16125
Solution:
(i) Denominator of
And,
80 = 2^{4} x 5
Therefore, 80 has no other factors than 2 and 5.
Thus,
(ii) Denominator of
And 24 = 2^{3}x 3
So, 24 has a prime factor 3, which is other than 2 and 5.
Thus,
(iii) Denominator of
And,
12 = 2^{2}x 3
So, 12 has a prime factor 3, which is other than 2 and 5.
Thus,
(iv) Denominator of
And,
35 = 7 x 5
So, 35 has a prime factor 7, which is other than 2 and 5.
Thus,
(v) Denominator of
And,
125 = 5^{3}
Therefore, 125 has no other factors than 2 and 5.
Thus,
Question 8: Convert each of the following into a decimal.
58 916 725 1124 2512
Solution:
(i)
By actual division, we have:
(ii)
By actual division, we have:
(iii)
By actual division, we have:
(iv)
By actual division, we have:
(v)
By actual division, we have:
Question 9: Express each of the following as a fraction in simplest form.
- 0.
3¯¯¯ - 1.
3¯¯¯ - 0.
34¯¯¯¯¯ - 3.
14¯¯¯¯¯ - 0.
324¯¯¯¯¯¯¯¯ - 0. 1
7¯¯¯ - 0.5
4¯¯¯ - 0.1
63¯¯¯¯¯
Solution:
(i) 0.
Let x = 0.
Therefore, X = 0.3333 – – – – – – – – – – – – – – – (1)
10x = 3.3333 – – – – – – – – – – – – – – (2)
On subtracting (1) from (2) we get:
9x = 3
= x =
Therefore, 0.
(ii) 1.
Let x = 1.
Therefore, X = 1.33333 – – – – – – – – – – – – – (i)
10x = 13.33333 – – – – – – – – – – – (ii)
On subtracting (i) and (ii) we get:
9x = 12
= x =
Therefore, 1.
(iii) 0.
Let x = 0.
Therefore, X = 0.3434… – – – – – – – – – – – – (1)
100x = 34.343434… – – – – – – – – – – (2)
On Subtracting (1) and (2) we get:
99x = 34
= x =
Therefore, 0.
(iv) 3.
Let x = 3.
Therefore, X = 3.1414.. – – – – – – – – – – – (1)
100x = 314.1414… – – – – – – – – – – (2)
On subtracting (1) from (2), we get:
99x = 311
= x =
Therefore, 3.
(v) 0.
Let x = 0.
Therefore, X = 0.324324… – – – – – – – – – – – – – – – (1)
1000x = 324.324324.. – – – – – – – – – – – – (2)
On subtracting (1) from (2), we get:
999x = 324
= x =
Therefore, 0.
(vi) 0. 1
Let x = 0.1
Therefore, X = 0.1777.. – – – – – – – – – – – – – (1)
10x = 1.7777.. – – – – – – – – – – – – – (2)
On subtracting (1) from (2) we get:
90x = 16
= x =
Therefore, 0.1
(vii) 0.5
Let x = 0.5
Therefore, X = 0.54444…
10x = 5.4444.. – – – – – – – – – – – – – – – (1)
100x = 54.4444… – – – – – – – – – – – – – – (2)
On subtracting (1) from (2) we get:
90x = 49
= x =
Therefore, 0.5
(viii) 0.1
Let x = 0.1
Therefore, X = 0.1636363…
10x = 1.6363.. – – – – – – – – – – – – – – – (1)
1000x = 163.6363… – – – – – – – – – – – – – – (2)
On subtracting (1) from (2) we get:
990x = 162
= x =
Therefore, 0.1
Question 10: Write, whether the given statement is true or false. Give reason.
- Every natural number is a whole number.
- Every whole number is a natural number.
- Every integer is a rational numbers.
- Every rational number is a whole number.
- Every terminating decimal is a rational number.
- Every repeating decimal is a rational number.
- 0 is a rational number
Solution:
(i) True
Natural numbers start from 1 to infinity and whole numbers start from 0 to infinity; hence, every natural number is a whole number.
(ii) False
0 is a whole number not a natural number, so every whole number is not a natural number.
(iii) True
Every integer can be expressed in the
(iv) False
Because whole numbers consist only of number of the form
(v) True
Every terminating decimal can be easily expressed in the
(vi) True
Every terminating decimal can be easily expressed in the
(vii) True
0 can be expressed in the form
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