# Question-1

**Is zero a rational number? Can you write it in the form, Where**

**and***p**q*are integers and*q***â‰ 0?****Solution:**

Yes.

0 = etc.

Also denominator

*can be taken as a negative integer*

**q**# Question-2

**Find six rational numbers between 3 and 4.**

**Solution:**

Let us take the 3 and 4 as rational numbers with denominator 6 + 1 = 7

Then the 6 rational numbers between 3 and 4 are

# Question-3

**Find five rational numbers between and .**

**Solution:**

3/5=0.6 , 4/5=0.8

It can be observed that 0.6<0.61<0.65<0.69<0.7<0.75<0.8

Scince the decimals 0.61,0.65,0.69,0.7 and 0.75are terminating, they are ratinal numbers.

0.61,0.65,0.69,0.7 and 0.75 are five rational numbers lying between 3/5 and 4/5

The five rational numbers between and are

# Question-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.

(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, since the collection of whole number contains all the natural numbers.

(ii) False, for example â€“ 2 is not a whole number.

(iii) False, for example is a rational number but not a whole number.

# Question-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

(iii) Every real number is an irrational number.

(i) Every irrational number is a real number.

(ii) Every point on the number line is of the form, Where

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

**Solution:**

(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 2 is real but not irrational.

# Question-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, For example is a rational number.

# Question-7

**Show how can be represented on the number line.**

**Solution:**

Consider a unit square OABC and transfer in onto the number line making sure that the vertex O coincides with zero.

Then OB = =

Construct BD of unit length perpendicular to OB.

Then OD = =

Construct DE of unit length perpendicular to OD.

Then OE = = = 2

Construct EF of unit length perpendicular to OE.

Then OF = =

Using a compass, with centre O and radius OF, draw an arc which intersects the number line in the point R. Then R corresponds to .

Representation of

# Question-8

**Write the following in decimal form and say what kind of decimal expansion each has:(i) (ii) (iii)4 (iv) (v) (vi)**

**Solution:**

(i) 0.36. It is a terminating decimal.

(ii) . It is a non terminating repeating decimal.

(iii) 4.125. It is a terminating decimal.

(iv) is a non terminating repeating decimal.

(v) is a non terminating repeating decimal.

(vi) 0.8225 is a terminating decimal.

# Question-9

**You know that = . Can you predict what the decimal expansions of ,,,, are, without actually doing the long division? If so, how? [Hint : Study the remainders while finding the value of carefully.]**

**Solution:**

Yes, it can be done as follows

# Question-10

**Express the following in the form , Where p and q are integers and q0.**

**(i) 0.6 (ii)0.47 (iii)0.****001****Solution:**

(i) Let x = 0.666 ...........(1)

.............(2)

(2) - (1)

â‡’ 9x = 6

x =

(ii)

Let x = = 0.47777â€¦.. ...........(1)

Multiplying both sides by 10 (since one digit is repeating),

we get 10x = 4.7777â€¦â€¦. ............(2)

(2)-(1)

â‡’ 10x â€“ x = 4.3

â‡’ 9x = 4.3

â‡’ x = =

Thus, =

Here p = 43, q = 90(â‰ 0)

(iii)

Let x = = 0.001001001â€¦ ...........(1)

Multiplying both sides by 1000(since three digits are repeating),

we get 1000x = 1.001001â€¦ ............(2)

(2)-(1)

â‡’ 1000x â€“ x = 1

â‡’ 999x = 1

â‡’ x =

Thus, =

Here P = 1, q = 999 (â‰ 0)

# Question-11

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

**Solution:**

Let

*x*= 0.99999â€¦ ...............(1)

Multiplying both sides by 10(since one digit it repeating ), we get

10

*x*= 9.9999â€¦. .............(2)

(2) - (1) â‡’ 10

*x â€“ x =*9 â‡’ 9

*x =*9 â‡’

*x*= = 1

Thus 0.99999 â€¦â€¦.. = 1 =

Here

*p =*1,

*q =*1

Since 0.99999â€¦.. goes on for ever, so there is no gap between 1 and 0.99999â€¦.. and hence they are equal

# Question-12

**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:**

The maximum number of digits in the repeating block of digits in the decimal expansion of can be 16.

Thus, =

By long Division, the number of digits in the repeating block of digits in the decimal expansion of = 16. âˆ´ The answer is verified

# Question-13

**Look at several examples of rational numbers in the form, Where**

*and***p***are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property***q****must satisfy?***q***Solution:**

The property that

**must satisfy in order that the rational numbers in the form , where**

*q***and**

*p***are integers with no common factors other than 1, have terminating decimal representation (expansions) is that the prime factorization of**

*q***has only powers of 2 or powers of 5 or both, i.e.,**

*q***must be of the form 2**

*q*^{m}Ã— 5

^{n};

**= 0,1,2,3,â€¦â€¦..,**

*m***= 0,1,2,3,â€¦â€¦â€¦**

*n*# Question-14

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

**Solution:**

0.01001000100001...

0.20200220003200002...

0.003000300003...

# Question-15

**Find three different irrational numbers between the rational numbers and .**

**Solution:**

Thus, = 0.714285 â€¦â€¦.. =

Thus, = 0.8181â€¦â€¦.. =

Three different irrational numbers between the rational numbers and can be taken as

0.75075007500075000075â€¦â€¦.

0.767076700767â€¦â€¦.

0.808008000800008â€¦..

# Question-16

**Classify the following numbers as rational or irrational:**

(i) (ii) (iii)0.3796 (iv) 7.478478â€¦ (v) 1.101001000100001â€¦

(i) (ii) (iii)0.3796 (iv) 7.478478â€¦ (v) 1.101001000100001â€¦

**Solution:**

Thus, = 4.795831523â€¦. âˆ´ The decimal expansion is non terminating non-recurring. âˆ´ is an irrational number.

(ii)

âˆ´ = 15 =

âˆ´ is a rational number.

(iii) 0.3796 âˆ´ The decimal expansion is terminating âˆ´ 0.3796 is a rational number

(iv) 7.478478â€¦â€¦.

7.478478 â€¦â€¦.. = 7. âˆ´ The decimal expansion is non-terminating recurring. âˆ´ 7.478478â€¦â€¦â€¦.. is a rational number.

(v) 1.101001000100001â€¦â€¦â€¦â€¦â€¦ âˆ´ The decimal expansion is non-terminating non-recurring. âˆ´ 1.101001000100001â€¦â€¦ is an irrational number.

# Question-17

**Visualise 3.765 on the number line, using successive magnification.**

**Solution:**

# Question-18

**Visualise on the number line, up to 4 decimal places.**

**Solution:**

# Question-19

**Classify the following numbers as rational or irrational:**

(i) 2 â€“ (ii) (3 +) â€“ (iii) (iv) (v) 2Ï€

(i) 2 â€“ (ii) (3 +) â€“ (iii) (iv) (v) 2Ï€

**Solution:**

(i) 2 - 2 is a rational number and is an irrational number. âˆ´ 2 â€“ is an irrational number. âˆ´ The difference of a rational number and an irrational number

is irrational.

(ii) (3 + ) â€“

(3 + ) â€“ = 3 +â€“ = 3 which is a rational number.

(iii)

= = 0.285714285714... which is a rational number.

(iv) 1(â‰ 0) is a rational and is an irrational number. âˆ´ is an irrational number

âˆ´ The quotient of a non-zero rational number with an irrational number is irrational.

(v) 2Ï€ 2 is a rational number and Ï€ is an irrational number. âˆ´ The product of a non zero with an irrational number is irrational âˆ´ 2Ï€ is an irrational number

# Question-20

**Simplify each of the following expressions:**

(i) (3 + ) (2 +) (ii) (3 + ) (3 â€“) (iii) ( + )

(iv) ( - )( + )

(i) (3 + ) (2 +) (ii) (3 + ) (3 â€“) (iii) ( + )

^{2}(iv) ( - )( + )

**Solution:**

(i) (3 +)(2 +)

(3 +) (2 +) = 3(2 +) + | Left Distributive law of multiplication over addition

= (3)(2) + 3+((2))+ ()()

= 6 + 3+2 + |âˆ´ =

= 6 + 3+ 2+

(ii) (3+) (3-)(3 +)(3 â€“)

= (3)

^{2}â€“ ()

^{2}

= 9 â€“ 3 = 6

(iii) ( + )

^{2}= ( + )(+)

= ()

^{2}+ 2+()

^{2}

= 5 + 2 + 2 |âˆ´ =

= 7 + 2

(iv) ( -)(+( â€“ (+

= ()

^{2}â€“ (

^{2 }

= 5 â€“ 2 = 3

# Question-21

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

**Solution:**

Actually

**=**is only an approximate value of Ï€ and also a non-terminating decimal.

# Question-22

**Represent on the number line.**

**Solution:**

Mark the distance 9.3 from a fixed point A on a given line to obtain a point B such that

AB = 9.3 units. From B mark a distance of 1 unit and mark the new point as C. Find the mid-point of AC and mark that point as O. Draw a semi-circle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semi-circle at D. Then BD = .

# Question-23

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

(i) = Ã—

= |Multiplying and dividing by

(ii)

**|**

=

=

**Multiplying and dividing by**

=

**= +**

(iii)

**=Ã—**| Multiplying and dividing by

**= =**

(iv)

**= Ã—**| Multiplying and dividing by

**= =**

# Question-24

**Find:**

**Solution:**

i) (64)

^{1/2}=(8

^{2})

^{1/2}

= 8^{2 }^{Â´}^{ Â½}

= 8^{1}

= 8

(ii) (32)^{1/5} = (2^{5})^{1/5}

= 2^{5 }^{Â´}^{ 1/5}

^{ }= 2^{1}

= 2

(iii) (125)^{1/3} = (5^{3})^{1/3}

= 5^{3 }^{Â´}^{ 1/3}

= 5^{1}

= 5

# Question-25

**(i) (ii) (iii) (iv)****Solution:**

(i) = Ã—

= |Multiplying and dividing by

(ii)

**=**|

**Multiplying and dividing by**

=

**= +**

(iii)

**=Ã—**| Multiplying and dividing by

**= =**

(iv)

**= Ã—**| Multiplying and dividing by

**= =**

# Question-26

**Represent on the number line.**

**Solution:**

Mark the distance 9.3 from a fixed point A on a given line to obtain a point B such that

AB = 9.3 units. From B mark a distance of 1 unit and mark the new point as C. Find the mid-point of AC and mark that point as O. Draw a semi-circle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semi-circle at D. Then BD = .