Q 1.

Is zero a rational number? Can you write it in the form. where p and q are integers and
q ≠ 0?

SOLUTION:

Yes, zero is a rational number. We can write it in the form
denominator q can also be taken as negative integer.

Q 2.

Find six rational numbers between 3 and 4.

SOLUTION:

We have,

Thus, the six rational numbers between 3 and 4 are

Q 3.

Find five rational numbers between and .

SOLUTION:

Since, we need to find five rational numbers, therefore, multiply numerator and denominator by 6.

Q 4.

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.

SOLUTION:

(i) True, ∴ The collection of all natural numbers and 0 is called whole numbers.
(ii) False, ∴ Negative integers are not whole numbers.
(iii) False, ∴ Rational numbers of the form p/q, q15≠15and q does not divide p completely are not whole numbers.

Q 5.

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 where m is a natural number.
(iii) Every real number is an irrational number.

SOLUTION:

(i) True; because all rational numbers and all irrational numbers form the group (collection) of real numbers.
(ii) False; because negative numbers cannot be the square root of any natural number.
(iii)False; because rational numbers are also a part of real numbers.

Q 6.

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.

SOLUTION:

No, if we take a positive integer, say 9, its square root is 3, which is a rational number.

Q 7.

Show how can be represented on the number line.

SOLUTION:

Draw a line on x-axis and take point O and A on it such that OA = 1 unit. Draw
BA ⊥ OA as BA = 1 unit. Join OB Now draw BB₁ ⊥ OB such that BB₁ = 1 unit. Join
OB₁ units. Next, draw B₃ B₃ ⊥ OB₃ such that
B₃B₃ = 1 unit. Join OB₂ units. Again draw B₃B₃ ⊥ OB₃ such that B₃B₃ = 1 unit. Join O B₃ units.

Take O as centre and OB₃ as radius, draw an arc which cuts x-axis at D. Point D represents the number on x-axis

Q 8.

Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point O and draw a line segment OP1 of unit length. Draw a line segment P1P2,
perpendicular to OP1 of unit length (see given figure). Now draw a line segment P2 163 perpendicular to OP2. Then draw a line segment P3 164 perpendicular to OP3. Continuing in this manner, you
can get the line segment Pn–1Pn by drawing a line segment of unit length perpendicular to OPn –1. In this manner, you will have created the points P2, P3, ., 16n, .., and joined them to create a
beautiful spiral depiciting can be represented on the number line.

SOLUTION:

Do it yourself

Q 9.

Write the following in decimal form and say what kind of decimal expansion each has :
(i)
(ii)
(iii)
(iv)
(v)
(vi)

SOLUTION:

(i) We have,
∴ The decimal expansion of is terminating.
(ii) Dividing 1 by 11, we have

Thus, the decimal expansion is non-terminating repeating.
(iii) we have
Thus, the decimal expansion is terminating.
(iv) Dividing 3 by 13, we get
we have
Here, the repeating block of digits is 230769

Thus, the decimal expansion of is non-terminating repeating.
(v) Dividing 2 by 11, we have
Here, the repeating block of digits is 18.

(vi) Dividing 329 by 400, we have

Thus, the decimal expansion of is terminating.

Q 10.

You know that Can you predict what the decimal expansions of are, without actually doing the long division? If so, how?

SOLUTION:

We are given that


Thus, without actually doing the long division we can predict the decimal expansions of the given rational numbers.

Q 11.

Express the following in the form where p and q are integers and q ≠ 0.
(i)
(ii)
(iii)

SOLUTION:

(i) Let, (1)
Multiplying (1) by 10 both sides, we get [As there is only one repeating digit]
10x = (0.666.) x 10
⇒ 10x = 6.6666.. (2)
Subtracting (1) from (2), we get
10x – x = 6.6666. – 0.6666.
⇒ 9x = 6

(ii) Let, (1)
Multiplying (1) by 10 both sides, we get [As there is only one repeating digit]
⇒ 10x = 10 x (0.4777.)
⇒ 10x = 4.777. (2)
Subtracting (1) from (2), we get
10x – x = 4.777.. – 0.4777..

(iii) Let, (1)
Multiplying (1) by 1000 both sides, we get
[As there are 3 repeating digits]
⇒ 1000 x = 1.001001.. (2)
Subtracting (1) from (2), we get
1000x – x = (1.001...) – (0.001...)

Q 12.

Express 0.99999... in the form Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

SOLUTION:

Let x = 0.99999.... . (i)
Multiplying (i) by 10 both sides, we get [As there is only one repeating digit] 10 x x = 10 x (0.99999.)
⇒ 10x = 9.9999 .. (ii)
Subtracting (i) from (ii), we get
10x – x = (9.9999.) – (0.9999.)

Thus, 0.9999. = 1
As 0.9999. goes on forever, there is no gap between 1 and 0.9999. .
Hence, both are equal.

Q 13.

What can the maximum number of digits be in the repeating block of digits in the decimal expansion of Perform the division to check your answer.

SOLUTION:

Since, the number of entries in the repeating block of digits is less than the divisor.
In . the divisor is 17.
M∴ The maximum number of digits in the repeating block is 16. To perform the long division, we have

The remainder 1 is the same digit from which we started the division.

Thus, there are 16 digits in the repeating block in the decimal expansion of Hence, our answer is verified

Q 14.

Look at several examples of rational numbers in the form where p and q are integers with no common factors other than 1 and having terminating decimal representations
(expansions). Can you guess what property q must satisfy?

SOLUTION:

Let us look decimal expansion of the following terminating rational numbers:

We observe that the prime factorisation of q (i.e. denominator) has only powers of 2 or powers of 5 or powers of both.

Q 15.

Write three numbers whose decimal expansions are non-terminating non-recurring.

SOLUTION:

Q 16.

Find three different irrational numbers between the rational numbers

SOLUTION:

We have,


Three irrational numbers between and are
(i) 0.750750075000750.
(ii) 0.767076700767000767.
(iii) 0.78080078008000780.

Q 17.

Classify the following numbers as rational or irrational.
(i)
(ii)
(iii)0.3796
(iv) 7.478478.
(v) 1.101001000100001.

SOLUTION:

(i) ∴ 23 is not a perfect square.
∴ is an irrational number.
(ii) ∴ 225 = 15 x 15 = 15²
∴ 225 is a perfect square.
Thus, is a rational number.
(iii) ∴ 0.3796 is a terminating decimal.
∴ It is a rational number.
(iv) 7.478478. =
Since, is a non-terminating recurring (repeating) decimal.
∴ It is a rational number.
(v) Since, 1.101001000100001. is a non-terminating, non-repeating decimal number.
∴ It is an irrational number.

Q 18.

Visualise 3.765 on the number line, using successive magnification.

SOLUTION:

3.765 lies between 3 and 4.

Q 19.

Visualise on the number line, using successive magnification.

SOLUTION:

Q 20.

Classify the following numbers as rational or irrational.
(i)
(ii)
(iii)
(iv) (v) 2π

SOLUTION:

(i) Since, it is a difference of a rational and an irrational number.
∴ is an irrational number.
(ii) which is a rational number.
(iii) Since which is a rational number.
(iv) ∴ The quotient of rational and irrational number is an irrational number.
∴ is an irrational number.
(v) ∴ 2π = 2 x π = Product of a rational and an irrational number is an irrational number.
∴ 2π is an irrational number.

Q 21.

Simplify each of the following expressions
(i)
(ii)
(iii)
(iv)

SOLUTION:

(i)

Thus,
(ii)∴
= 32 – 3 = 9 – 3 = 6
(iii)


(iv)
= 5 – 2 = 3

Q 22.

Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, his seems to contradict the fact that π is irrational. How will you resolve this
contradiction ?

SOLUTION:

When we measure the length of a line with a scale or with any other device, we only get an approximate rational value, i.e. c and d both are irrational.
is irrational and hence p is irrational. Thus, there is no contradiction in saying that π is irrational.

Q 23.

Represent on the number line.

SOLUTION:

Draw a line segment AB = 9.3 units and extend it to C such that BC = 1 unit.
Find mid point of AC and mark it as O.
Draw a semicircle taking O as centre and AO as radius. Draw BD ⊥ AC. Draw an arc taking B as centre and BD as radius which
cuts line at E.
BE = BD = units.

Q 24.

Rationalise the denominators of the following:
(i)
(ii)
(iii)
(iv)

SOLUTION:

(i)
(ii)
(iii)
(iv)

Q 25.

Find:
(i) 64½
(ii) 32^1/5
(iii)125^1/3

SOLUTION:

(i) ∵ 64 = 8 x 8 = 8²
∴ (64)^1/2 = (8²)^1/2 = 8^{(2 x 1/2)} = 8
[(a^m)ⁿ = a^{m x n}]
(ii) 32 = 2 x 2 x 2 x 2 x 2 = 2⁵
∴ (32)^(1/5) = (2⁵)^{(5 x 1/5)} = 2^{(5 x 1/5)} = 2
[(a^m)ⁿ = a^{m x n}]
(iii) 125 = 5 x 5 x 5 = 5³
∴ (125)^(1/3) = (5³)^(1/3) = 5^{3 x 1/3} = 5
[(a^m)ⁿ = a^{m x n}]

Q 26.

Find:
(i) 9^3/2
(ii) 32^2/5
(iii) 16^3/4
(iv) 125^–1/3

SOLUTION:

(i) 9 = 3 x 3 = 3²

Q 27.

Simplify:
(i) 2^2/3 · 2^1/5
(ii)
(iii)
(iv)7^1/2 · 8^1/2

SOLUTION: