## Number Systems

## Chapter 1

## Exercise 1.3

**EX 1.3 QUESTION 1.**

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

**(i) 36/100**

**(ii)1/11**

**(iv) 3/13**

**(v) 2/11**

**(vi) 329/400**

**Solution:**

**EX 1.3 QUESTION 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:**

**EX 1.3 QUESTION 3.**

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

**(i) 0.**

**(ii) 0.4**

**(iii) 0.**

**Solution:**

(i) 0.

Let* x* = 0.

⇒* x *= 0.6666……….. (i)

multiplying equation (i) by 10 on both sides,

10*x* = 6.666…

10*x* = 6 + *x *[From equation (I)]

9*x* = 6

*x* = 6/9

*x* = 2/3

(ii) 0.4

Let* x* = 0.4

⇒* x *= 0.47777………….. (i)

multiplying equation (i) by 10 on both sides,

⇒* 10x *= 4.7777 ………….. (ii)

multipmultiplying equation (ii) by 100 on both sides,

100*x* = 47.7777…

100*x* = 43 + 47.7777* *

100*x = *43 + 10*x* [From equation (Ii)]

100*x* – 10 *x *= 43

90*x* = 43

*x* = 43/90

(iii) 0.

Let* x* = 0.

⇒* x* = 0.001001001…………(i)

multiplying equation (i) by 1000 on both sides,

1000*x =*1.001001001……

⇒ 1000*x = *1 + 0.001001001…..

⇒ 1000*x =*1 + *x * [from equation (i)]

⇒ 1000*x *– *x* = 1

⇒ 999*x = *1

⇒ *x = *1/ 999

**EX 1.3 QUESTION 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:**

Let *x*= 0.99999………..(i)

multiplying equation (i) by 10 on both sides,

10*x* = 9.99999

⇒ 10*x* = 9+ 0.99999

⇒ 10*x* = 9+ *x *[from equation (i)]

⇒ 10*x *–* x *= 9

⇒ 9*x = *9

⇒ *x = *9/9 = 1

⇒ *x = *1

The difference between 1 and 0.999999 is 0.000001 which is very close.

Hence, we can say that 0.99999 = 1.

**EX 1.3 QUESTION 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

1/17 =

There are maximum16 digits in the repeating block of the decimal expansion of 1/17.

**EX 1.3 QUESTION 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:**

- 2/5 = 0.4
- 1/10 = 0.1
- 3/2 = 1.5
- 7/8 = 0.875

The denominator of all the rational numbers is in front of 2^{m} x 5^{n} , where m and n are integers.

**EX 1.3 QUESTION 7.**

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

**Solution:**

- √3 = 1.732050807568
- √5 = 2.23606797
- √26 =5.099019513592

**EX 1.3 QUESTION 8.**

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

**Solution:**

Three different irrational numbers between **0.**** **and ** 0. **are

- 0.73073007300073000073…
- 0.75075007300075000075…
- 0.76076007600076000076…

**EX 1.3 QUESTION 9.**

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

(i) √23

(ii)√225

(iii) 0.3796

(iv) 7.478478…..

(v) 1.101001000100001

**Solution:**

(i) √23

√23 = 4.79583152331…

Irrational number.

(ii)√225

√225 = 15 = 15/1

Rational number.

(iii) 0.3796

Rational number.

(iv) 7.478478…..

Rational number.

(v) 1.101001000100001

Irrational number.