# Question-1

**Write the following rational numbers in decimal form:**

(i)

(ii)

(iii) 3

(iv)

(v)

(vi)

(vii)

(viii)(i)

(ii)

(iii) 3

(iv)

(v)

(vi)

(vii)

(viii)

**Solution:**

(i) = 0.42

(ii)

**= 0.654**

(iii)

**3== 3.375**

(iv)

**= 0.833â€¦ = 0.8333**

(v) = 0.2

(vi)

**=**

(vii)

**=**

(viii)

**=**

# Question-2

**If a is a positive rational number and n is a positive integer greater than 1, prove that a**^{n}is a rational number.**Solution:**

We know that product of two rational number is always a rational number. Hence if a is a rational number then

a

^{2}= a x a is a rational number,

a

^{3}= a

^{2}x a is a rational number,

a

^{4}= a

^{3}x a is a rational number,

...

...

âˆ´ a

^{n}= a

^{n-1 }x a is a rational number.

# Question-3

**Find three rational numbers lying between 0 and 0.1. Find twenty rational numbers between 0 and 0.1. Give a method to determine any number of rational numbers between 0 and 0.1.****Solution:**

The three rational numbers lying between 0 and 0.1 are 0.01, 0.05, 0.09.

The twenty rational numbers between 0 and 0.1 are 0.001, 0.002, 0.003, 0.004, â€¦ 0.011, 0.012, â€¦ 0.099.

To determine any number of rational numbers between 0 and 0.1 insert 0 after the decimal.

# Question-4

**Complete the following:**

(i) Every point on the number line corresponds to a

(i) Every point on the number line corresponds to a

**_____________**

**number which may be either _____________ or**

**_____________.**

**(ii) The decimal form of an irrational number is neither**

**_____________**

**or**

**_____________.**

**(iii) The decimal representation of the rational number is**

**_____________.**

**(iv) 0 is**

**_____________**

**number. [Hint: a rational /an irrational]**

**Solution:**

(i) Every point on the number line corresponds to a

__real__number which may be either

__rational__or

__irrational.__

(ii) The decimal form of an irrational number is neither

__recurring__or

__terminating.__

(iii) The decimal representation of the rational number is

__0.296__

(iv) 0 is

__a rational__number.

# Question-5

**Which of the following rational numbers have the terminating decimal representation?**

(i) 3/5 (ii)7/20 (iii)2/13

(iv) 27/40 (v) 133/125 (vi) 23/7(i) 3/5 (ii)7/20 (iii)2/13

(iv) 27/40 (v) 133/125 (vi) 23/7

**[Making use of the result that a rational number p/q where p and q have no common factor(s) will have a terminating representation if and only if the prime factors of q are 2's or 5's or both.]****Solution:**

(i) The prime factor of 5 is 5. Hence 3/5 has a terminating decimal representation.

(ii) 20 = 4 Ã— 5 = 2

^{2}Ã— 5.

The prime factors of 20 are both 2's and 5's. Hence 7/20 has a terminating decimal.

(iii) The prime factor of 13 is 13. Hence 2/13 has non-terminating decimal.

(iv) 40 = 2

^{3}Ã— 5.

The prime factors of 40 are both 2's and 5's. Hence 27/40 has a terminating decimal.

(v) 125 = 5

^{3}

The prime factor of 125 is 5's. Hence 13/125 has a terminating decimal.

(vi) The prime factor of 7 is 7. Hence 23/7 has a non-terminating decimal representation.

# Question-6

**You have seen that****is not a rational number. Show that 2 +****is not a rational number.****Solution:**

Let 2 + be a rational number say r.

Then 2 + = r

= r â€“ 2

But, is an irrational number.

Therefore, r â€“ 2 is also an irrational number.

=> r is an irrational number.

Hence our assumption r is a rational number is wrong.

Hence, 2 + is not a rational number.

# Question-7

**Prove that 3****is not a rational number.****Solution:**

Let 3be a rational number say r.

Then 3= r

= (1/3)r

(1/3) r is a rational number because product of two rational number is a rational number.

=> is a rational number but is not a rational number.

Therefore our assumption that 3 is a rational number is wrong.

# Question-8

**Show that is not a rational number.****Solution:**

Let be a rational number, say where q â‰ 0.

Then

**=**

Since 1

^{3}= 1 , and 2

^{3}= 8, it follows that 1 < < 2

Then q > 1 because if q = 1 then will be an integer, and there is no integer between 1 and 2.

Now, 6 =

6 =

6q

^{2}=

q being an integer, 6q

^{2}is an integer, and since q > 1 and q does not have a common factor with p and consequently with p

^{3}.

So, is a fraction different from an integer.

Thus 6q

^{2}â‰ .

This contradiction proves the result.

# Question-9

**Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers.**

(i)

(ii) 3

(iii)

(iv)

(v) -

(vi)(i)

(ii) 3

(iii)

(iv)

(v) -

(vi)

**Solution:**

(i) = 2 is rational.

(ii) 3

**=**3

**=**3 Ã— 3

**=**9is irrational.

(iii)

**==**= 1.2 is rational.

(iv)

**= ==**is irrational.

(v)

**-=**- 0.8 is rational.

(vi) = 10 is rational.

# Question-10

**In the following equations, find which of the variables x, y, z etc. represent rational numbers and which represent irrational numbers:**

(i) x

(i) x

^{2 }= 5**(ii) y**

^{2 }= 9**(iii) z**

^{2 }= 0.04**(iv) u**

^{2 }= 17/4**(v) v**

^{2 }= 3**(vi) w**

^{3 }= 27**(vii) t**

^{2 }= 0.4

**Solution:**

(i) x

^{2 }= 5

âˆ´ x = is irrational.

(ii) y

^{2 }= 9

âˆ´ y = 3 is rational.

(iii) z

^{2 }= 0.04

âˆ´ z = 0.2 is rational.

(iv) u

^{2 }=

âˆ´ u =

**==**is irrational.

(v) v

^{2 }= 3

âˆ´ v =

**is irrational.**

(vi) w^{3 }= 27

w = = 3 is rational.

(vii) t^{2 }= 0.4âˆ´ t = ** = = **is irrational.

# Question-11

**Give an example to show that the product of a rational number and an irrational number may be a rational number.**

**Solution:**

A rational number 0 multiplied by an irrational number gives the rational number 0.

# Question-12

**State with reason which of the following are surds and which are not.**

(i)(i)

**Ã—**

(ii) Ã—

(iii) Ã—

(iv) Ã—

(v) 5Ã— 2

(vi) Ã—

(vii) Ã—

(viii) 6Ã— 9

(ix) Ã—

(x) Ã—(ii) Ã—

(iii) Ã—

(iv) Ã—

(v) 5Ã— 2

(vi) Ã—

(vii) Ã—

(viii) 6Ã— 9

(ix) Ã—

(x) Ã—

**.****Solution:**

(i) Ã—

**=**Ã—

**=**Ã— Ã— = 5 is a surd.

(ii) Ã—

**=**Ã—

**=**2Ã— Ã— Ã— = 4is a surd.

(iii) Ã— = Ã— =3Ã— = 9 is not a surd.

(iv) Ã— = 4 Ã— 2 = 8 is not a surd.

(v) 5Ã— 2= 5Ã— 2

**=**5 Ã— 2Ã— 2Ã— = 5 Ã— 2 Ã— 2 Ã— 2 Ã— = 40is a surd.

(vi) Ã—

**=**Ã— = 5 Ã— =

**5 Ã— 5 = 25 is not a surd.**

(vii) Ã— = 10 is a surd.

(viii) 6Ã— 9= 54 is a surd.

(ix) Ã— = Ã— = 2 Ã— 3= 2 Ã— Ã— Ã— 3 Ã— = 30 is a surd.

(x) Ã— = Ã—

**=**Ã— Ã— Ã— = 3 Ã— is a surd.

# Question-13

**Give two examples to show that the product of two irrational numbers may be a rational number.****Solution:**

Take a = (2+) and b =(2-); a and b are irrational numbers, but their product

(2+)(2-) = 4 - 3 = 1 is a rational number.

Take c = and d = -; c and d are irrational numbers, but their product = -3,

is a rational number.

# Question-14

**Find the value of****correct to two places of decimal.****Solution:**

We know that 2

^{2}= 4 < 5 < 9 = 3

^{2}

Taking positive square roots we get

2 < < 3.

Next, (2.2)

^{2}= 4.84 < 5 < 5.29 = (2.3)

^{2}

Taking positive square roots, we have

2.2 < < 2.3

Again, (2.23)

^{2}= 4.9729< 5< 5.0176 = ( 2.24)

^{2}

Taking positive square roots, we obtain

2.23 < < 2.24

Hence the required approximation is 2.24 as (2.24)

^{2}is nearest to 5 than (2.23)

^{2}.

# Question-15

**Prove that****-****is irrational.****Solution:**

Let - be a rational number, say r

Then - = r

On squaring both sides we have

( - )

^{2}= r

^{2}

3 - 2 + 2 = r

^{2}

5 - 2 = r

^{2}

- 2 = r

^{2}- 5

= -(r

^{2}- 5)/2

Now -(r

^{2}- 5)/2 is a rational number and is an irrational number.

Since a rational number cannot be equal to an irrational number. Our assumption that

- is rational is wrong.

# Question-16

**Prove that****+****is an irrational number.****Solution:**

Let + be a rational number, say r

Then + = r

On squaring both sides,

( + )

^{2}= r

^{2}

3 + 2 + 5 = r

^{2}

8 + 2 = r

^{2}

2 = r

^{2}- 8

= (r

^{2}- 8)/2

Now (r

^{2}- 8)/2 is a rational number and âˆš15 is an irrational number.

Since a rational number cannot be equal to an irrational number. Our assumption that

+ is rational is wrong.

# Question-17

**Examine, whether the following numbers are rational or irrational:**

(i) ((i) (

**âˆš2 + 2)**

(ii) (2 - âˆš2)^{2}(ii) (2 - âˆš2)

**Ã—****(2 + âˆš2)****(iii) (âˆš2 + âˆš****3)**

(iv)^{2}(iv)

**Solution:**

(i) (âˆš2 + 2)

^{2}= (âˆš2)

^{2}+ 2âˆš2x2 + (2)

^{2}= 2 + 4âˆš2 + 4 = 6 + 4âˆš2 .

\ It is an irrational number.

(ii) (2 - âˆš2) Ã— (2 + âˆš2) = (2)

^{2}- (âˆš2)

^{2}= 4 - 2 = 2.

\ It is a rational number.

(iii) (âˆš2 + âˆš3)

^{2}= (âˆš2)

^{2}+ 2âˆš2 Ã— âˆš3 + (âˆš3)

^{2}= 2 + 2âˆš6 + 3 = 5 + 2âˆš6

âˆ´ It is an irrational number.

(iv) = = âˆš2

\ It is an irrational number.

# Question-18

**Prove that**

(a) 2 +is not a rational number and

(b)(a) 2 +is not a rational number and

(b)

**is not a rational number.****Solution:**

(a) If possible, let 2 += a , where a is rational.

Then, (2 + )

^{2 }= a

^{2}

7 + 4= a

^{2}

= -------(i)

Now, a is rational â‡’ is rational.

is rational [from (i)]

This is a contradiction.

Hence, 2 + is not a rational number.

(b) If possible, let

**=**, where p and q are integers,

having no common factors and q â‰ 0.

Then,

**()**

^{3}= ()

^{3}

â‡’ 7q

^{3}= p

^{3}------(i)

â‡’ p is a multiple of 7

â‡’ p is multiple of 7.

Let p = 7m, where m is an integer.

Then, p

^{3}= 343 m

^{3}------(ii)

â‡’ 7q

^{3}= 343 m

^{3}[from (i) and (ii)]

â‡’ q

^{3}= 49 m

^{3}â‡’ q

^{3}is a multiple of 7.

â‡’ q is a multiple of 7.

Thus, p and q are both multiples of 7, or 7 is a factor of p and q.

This contradicts our assumption that p and q have no common factors.

Hence is not a rational number.

# Question-19

**Examine whether the following numbers are rational or irrational:**

(i) (3 +)

(ii) (3-)(3+)

(iii)(i) (3 +)

^{ }(ii) (3-)(3+)

(iii)

**Solution:**

(i) (3 +)

^{}= 9 + 2 + 6 = 11 + 6, which is irrational.

(ii) (3 -)(3 + ) = (3)

^{ }- ()

^{}= 9 â€“ 3 = 6, which is rational.

(iii)

**=**

**Ã—**= = , which is irrational.

# Question-20

**Find two irrational numbers lying between****and .****Solution:**

Irrational numbers lying between and is ,i.e = 6

^{(1/4)}

Irrational numbers lying between and 6

^{(1/4)}is = 2

^{(1/4)}Ã— 6

^{(1/8)}

Hence two irrational numbers lying between and are 6

^{(1/4)}and 2

^{(1/4)}Ã— 6

^{(1/8)}.

# Question-21

**Express as a decimal fraction.****Solution:**

Therefore

**= 0.109375**

# Question-22

**Express as a decimal fraction.**

**Solution:**

Therefore = 0.096.

# Question-23

**Express as a decimal fraction.**

**Solution:**

Therefore

**34.692307**