**Number Systems**

**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) **

**Solution:**

Assume that *x* = 0.666…

Then,10*x* = 6.666…

10*x* = 6 + *x*

9*x* = 6

*x* = 2/3

**(ii) 0.4\overline{7}0.47**

**Solution:**

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

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

Assume that *x* = 0.777…

Then, 10*x* = 7.777…

10*x* = 7 + *x*

*x* = 7/9

(4/10)+(0.777../10) = (4/10)+(7/90) ( x = 7/9 and x = 0.777…0.777…/10 = 7/(9×10) = 7/90 )

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

**Solution:**

Assume that *x* = 0.001001…

Then, 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.**

**Solution:**

Assume that *x* = 0.9999…..Eq (a)

Multiplying both sides by 10,

10*x* = 9.9999…. Eq. (b)

Eq.(b) – Eq.(a), we get

(10*x* = 9.9999)-(*x* = 0.9999…)

9*x* = 9

*x* = 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 ≠ 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 is 2, 4, 5, 8, 10… Then the decimal expansion is terminating. For example:

1/2 = 0. 5, denominator q = 2^{1}

7/8 = 0. 875, denominator q =2^{3}

4/5 = 0. 8, denominator q = 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:

- √3 = 1.732050807568
- √26 =5.099019513592
- √101 = 10.04987562112

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

**Solution:**

Three different irrational numbers are:

- 0.73073007300073000073…
- 0.75075007300075000075…
- 0.76076007600076000076…

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

**(i)√23**

**Solution:**

√23 = 4.79583152331…

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

**(ii)√225**

**Solution:**

√225 = 15 = 15/1

Since the number can be represented in 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) 1.101001000100001…**

**Solution:**

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