### Exercise 1.3

Answer:

**Q3: Express the following in the form p/q, where p and q are integers and q ≠ 0.**

Answer:

(i) |

(ii) |

(iii) |

**Q4: 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...

10x = 9.9999...

10x = 9 + 0.9999... = 9 + x

⇒ 10x -9x = 9

⇒ 9x = 9 ⇒ x = 1

Consider the case 1 - 0.999999 = 0.000001 (negligible difference). This means 0.999... approaches 1. Therefore 1 as an answer is justified.

It also shows that any terminating decimal can be represented as a non-terminating and recurring decimal expansion with an endless blocks of 9s.

e.g 6 = 5.9999...

It also shows that any terminating decimal can be represented as a non-terminating and recurring decimal expansion with an endless blocks of 9s.

e.g 6 = 5.9999...

**Q5: 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 = 0.05882352941176470588235294117647...

i.e. it is 16 repeating digits. Please try the division yourself.

**Q6: 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?**

Answer: We observe when denominator q is 2, 4, 5, 8, 10..., the decimal expansion is terminating. Consider the following examples:

(i) 7/8 = 0.875. Terminating Decimal. Here, denominator q = 8 ie. 2

^{3}
(ii) 4/5 = 0.8 (Terminating Decimal). Here, denominator q = 5 ie. 5

^{1}
(iii) 22/25 = 0.88 (Terminating Decimal). Here, denominator q = 25 ie. 5

^{2}
(iv) 41/100 = 0.41 (Terminating Decimal). Here, denominator q = 100 = 25 x 4 ie. 5

^{2}x 2^{2}
By studying similar patterns, we can conclude that for terminating decimal, denominator q has prime factors of the form 2

^{m}x 5^{n}where (m = 0,1,2,3... and n = 0,1,2,3,4...)**Q7: Write three numbers whose decimal expansions are non-terminating non-recurring.**

Answer:

- 0.303003000300003...
- 0.515115111511115...
- 0.92092009200092000920000...

**Q8: Find three different irrational numbers between the rational numbers 5/7 and 9/11.**

Answer: 5/7 = 0.714285714285... = 0.(714285)...

and 9/11 = 0.818181...

Three irrational numbers between 5/7 and 9/11 are:

- 0.76076007600076...
- 0.781781178111781111...
- 0.790790079000790000...

**Q9: 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... (non-terminating, non-recurring)

⇒ Irrational Number

(ii) √(225) = 15 = 15/1 = (p/q form, q ≠ 0) ⇒ Rational Number

(iii) 0.3796 Decimal expansion is terminating ⇒ Rational Number

(iv) 7.478478 = 7.(478)... (.478 is recurring pattern). Number is non-terminating but recurring.

⇒ Rational Number

(v) 1.10100100010000 … Decimal expansion is non-terminating, non-repeating.

⇒ Irrational Number

**Q11: Express the decimal expansion 0.137454545... as rational number.**

Answer:

Let x = 0.137454545…

⇒ 10

^{3}x = 137.4545… (I)

and 10

^{5}x = 13745.4545… (II)

Subtracting I from II,

10

^{5}x - 10

^{3}x = 13608

(10

^{5}- 10

^{3})x = 13608

99000x = 13608

⇒ x = 13608/99000 = 1701/12375 ...(answer)

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