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**Exercise 1.1**

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

**Answer**

Yes. Zero is a rational number as it can be represented as 0/1 or 0/2 or 0/3 etc

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

**Answer**

There are infinite rational numbers in between 3 and 4.3 and 4 can be represented as 24/8 and 32/8 respectively.

Therefore, six rational numbers between 3 and 4 are 25/8, 26/8, 27/8, 28/8, 29/8, 30/8.

**3. Find five rational numbers between 3/5 and 4/5**.

**Answer**

There are infinite rational numbers in between 3/5 and 4/5

3/5 = 3×6/5×6 = 18/30

4/5 = 4×6/5×6 = 24/30

Therefore, five rational numbers between 3/5 and 4/5 are

19/30, 20/30, 21/30, 22/30, 23/30.

**4. State whether the following statements are true or false. Give reasons for your answers.**

**(i) Every natural number is a whole number.**

**Answer**

True, since the collection of whole numbers contains all natural numbers.

**(ii) Every integer is a whole number.**

**Answer**

False, as integers may be negative but whole numbers are always positive.**(iii) Every rational number is a whole number.**

**Answer**

False, as rational numbers may be fractional but whole numbers may not be.

For example: 1/5 is a rational number but not a whole number.

**Exercise 1.2****1. State whether the following statements are true or false. Justify your answers.****(i) Every irrational number is a real number.**

**Answer**

True, since the collection of real numbers is made up of rational and irrational numbers.**(ii) Every point on the number line is of the form √ m, where m is a natural number.**

**Answer**

False, since positive number cannot be expressed as square roots.

**(iii) Every real number is an irrational number.**

**Answer**

False, as real numbers include both rational and irrational numbers. Therefore, every real number cannot be an irrational number.

**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.**

**Answer**

No, the square roots of all positive integers are not irrational. For example √4 = 2.

**3. Show how √5 can be represented on the number line.**

**Answer**

**Step 1:** Let AB be a line of length 2 unit on number line.

**Step 2:** At B, draw a perpendicular line BC of length 1 unit. Join CA.

**Step 3:** Now, ABC is a right angled triangle. Applying Pythagoras theorem,

AB^{2 } + BC^{2} = CA^{2}

⇒ 2^{2} + 1^{2} = CA^{2}

⇒ CA^{2} = 5

⇒ CA = √5

Thus, CA is a line of length √5 unit.

**Step 4: **Taking CA as a radius and A as a centre draw an arc touching the number line. The point at which number line get intersected by arc is at √5 distance from 0 because it is a radius of the circle whose centre was A.

Thus, √5 is represented on the number line as shown in the figure.

**Exercise 1.3****1. Write the following in decimal form and say what kind of decimal expansion each has:**

**(i) 36/100**

= 0.36 (Terminating)**(ii) 1/11**

0.09090909... = (Non terminating repeating)

**(iii) **

= 33/8 = 4.125 (Terminating)

**(iv) 3/13**

= 0.230769230769... (Non terminating repeating)

**(v) 2/11**

= 0.181818181818... (Non terminating repeating)

**(vi) 329/400**

= 0.8225 (Terminating)**2. You know that 1/7 = ****.Can you predict what the decimal expansion of 2/7, 3/7, 4/7, 5/7, 6/7 are without actually doing the long division? If so, how?**

[Hint: Study the remainders while finding the value of 1/7 carefully.]

**Answer**

Yes. We can be done this by:

**3. Express the following in the form p/q where p and q are integers and q ≠ 0.**

(i)

(ii)

(iii)

**Answer**

(i) = 0.666...

Let *x* = 0.666...

10*x* = 6.666...

10*x* = 6+ *x*

9*x* = 6*x* = 2/3

(ii) = 0.4777...

= 4/10 + 0.777/10

Let *x* = 0.777…

10*x* = 7.777…

10*x* = 7+*x*

*x* = 7/9

4/10 0.777.../10 = 4/10 7/90

= (36+7)/90 = 43/90

(iii) = 0.001001...

Let *x* = 0.001001...

1000*x* = 1.001001…

1000*x* = 1 + *x*

999*x* = 1

*x* = 1/999

**4. Express 0.99999…in the form p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.**

**Answer**

Let *x* = 0.9999…

10*x* = 9.9999…

10*x* = 9 + *x*

9*x* = 9*x* = 1

The difference between 1 and 0.999999 is 0.000001 which is negligible. Thus, 0.999 is too much near 1, Therefore, the 1 as answer can be justified.

**5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17? Perform the division to check your answer.**

**Answer**

1/17

There are 16 digits in the repeating block of the decimal expansion of 1/17.

Division Check:

**6. Look at several examples of rational numbers in the form p/q (q ≠ 0) 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?**

We observe that when q is 2, 4, 5, 8, 10... then the decimal expansion is terminating. For example:

1/2 = 0.5, denominator

7/8 = 0.875, denominator

4/5 = 0.8, denominator

We can observe that terminating decimal may be obtained in the situation where prime factorisation of the denominator of the given fractions has the power of 2 only or 5 only or both.

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

**Answer**

Three numbers whose decimal expansions are non-terminating non-recurring are:

0.303003000300003...

0.505005000500005...

0.7207200720007200007200000…

**8. Find three different irrational numbers between the rational numbers 5/7 and 9/11.**

**Answer**

5/7 =

9/11 =

Three different irrational numbers are:

0.73073007300073000073…

0.75075007300075000075…

0.76076007600076000076…

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

(i) √23

(ii) √225

(iii) 0.3796

(iv) 7.478478

(v) 1.101001000100001…

**Answer**

(i) √23 = 4.79583152331...

Since the number is non-terminating non-recurring therefore, it is an irrational number.

(ii) √225 = 15 = 15/1

Since the number is rational number as it can represented in *p*/*q *form.

(iii) 0.3796

Since the number is terminating therefore, it is an rational number.

(iv) 7.478478 =

Since the this number is non-terminating recurring, therefore, it is a rational number.

(v) 1.101001000100001…

Since the number is non-terminating non-repeating, therefore, it is an irrational number.

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