__Exercise 1.3 Page: 14__

__Exercise 1.3 Page: 14__

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

(i) 36/100

Solution:

= 0.36 (Terminating)

(ii)1/11

Solution:

Solution:

= 4.125 (Terminating)

(iv) 3/13

Solution:

(v) 2/11

Solution:

(vi) 329/400 Solution:

= 0.8225 (Terminating)

**2. You know that 1/7 = 0.142857. Can you predict what the decimal expansions 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.]**

Solution:

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

(i)**Â **

Solution:

Assume that Â *x*Â = 0.666â€¦

Then,10*x*Â = 6.666â€¦

10*x*Â = 6Â +Â *x*

9*x*Â = 6

*x*Â = 2/3

**(ii)Â ****0.4\overline{7}0.47**

Solution:

0.4\overline{7} = 0.4777..0.47=0.4777..

= (4/10)+(0.777/10)

Assume thatÂ *x*Â = 0.777â€¦

Then, 10*x*Â = 7.777â€¦

10*x*Â = 7 +Â *x*

*x*Â = 7/9

(4/10)+(0.777../10) = (4/10)+(7/90) ( x = 7/9 and x = 0.777â€¦0.777â€¦/10 = 7/(9Ã—10) = 7/90 ) = (36/90)+(7/90) = 43/90

Solution:

Assume that Â *x*Â = 0.001001â€¦

Then, 1000*x*Â = 1.001001â€¦

1000*x*Â = 1 +Â *x*

999*x*Â = 1

*x*Â = 1/999

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

Solution:

Assume thatÂ *x*Â = 0.9999â€¦..Eq (a)

Multiplying both sides by 10,

10*x*Â = 9.9999â€¦. Eq. (b)

Eq.(b) â€“ Eq.(a), we get

(10*x*Â = 9.9999)-(*x*Â = 0.9999â€¦)

9*x*Â = 9

*x*Â = 1

The difference between 1 and 0.999999 is 0.000001 which is negligible.

Hence, we can conclude that, 0.999 is too much near 1, therefore, 1 as the 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.**

Solution:

1/17

Dividing 1 by 17:

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

**6. Look at several examples of rational numbers in the form p/q (q â‰ 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?**

Solution:

We observe that when q isÂ 2, 4, 5, 8, 10â€¦ Then the decimal expansion is terminating. For example:

1/2 = 0.Â 5, denominatorÂ qÂ = 2^{1}

7/8 = 0.Â 875, denominatorÂ qÂ =2^{3}

4/5 = 0.Â 8, denominatorÂ qÂ = 5^{1}

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

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

Solution:

We know that all irrational numbers are non-terminating non-recurring. three numbers with decimal expansions that are non-terminating non-recurring are:

- âˆš3 = 1.732050807568
- âˆš26 =5.099019513592
- âˆš101 = 10.04987562112

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

Solution:

hree different irrational numbers are:

- 0.73073007300073000073â€¦
- 0.75075007300075000075â€¦
- 0.76076007600076000076â€¦

**9. Â Classify the following numbers as rational or irrational according to their type:**

(i)âˆš23

Solution:

âˆš23 = 4.79583152331â€¦

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

(ii)âˆš225

Solution:

âˆš225 = 15 = 15/1

Since the number can be represented inÂ p/q form, it is a rational number.

**(iii) 0.3796**

Solution:

Since the number,0.3796, is terminating, it is a rational number.

**(iv) 7.478478**

Solution:

The number,7.478478, is non-terminating but recurring, it is a rational number.

**(v) 1.101001000100001â€¦**

Solution:

Since the number,1.101001000100001â€¦, is non-terminating non-repeating (non-recurring), it is an irrational number.