Lecture 4: Converting Recurring Decimals to Rational Numbers

📘 Class 9 – CBSE Mathematics

Chapter 1: Number System

Lecture 4: Converting Recurring Decimals to Rational Numbers + Exercise 1.3

🔹 Examples: Converting Recurring Decimals to Rational Numbers

✅ Example 7

Convert: $0.3333\ldots = 0.\overline{3}$

Let $x = 0.3333\ldots$

Multiply both sides by 10:

$$10x = 3.3333\ldots$$

Now subtract:

$$10x - x = 3.3333\ldots - 0.3333\ldots = 3 \Rightarrow 9x = 3 \Rightarrow x = \frac{1}{3}$$

✅ Example 8

Convert: $1.272727\ldots = 1.\overline{27}$

Let $x = 1.272727\ldots$

Multiply both sides by 100 (since 2 digits repeat):

$$100x = 127.272727\ldots$$

Now subtract:

$$100x - x = 127.2727\ldots - 1.2727\ldots = 126 \Rightarrow 99x = 126 \Rightarrow x = \frac{126}{99} = \frac{14}{11}$$

✅ Example 9

Convert: $0.2353535\ldots = 0.2\overline{35}$

Let $x = 0.2353535\ldots$

Multiply by 100:

$$100x = 23.535353\ldots$$

Now subtract:

$$100x - x = 23.5353\ldots - 0.2353\ldots = 23.3 \Rightarrow 99x = 23.3 = \frac{233}{10} \Rightarrow x = \frac{233}{990}$$

📝 Exercise 1.3 – Solved

Q1. Convert to decimal form and classify the decimal expansion

Fraction	Decimal	Type
(i) 36/100	0.36	Terminating
(ii) 1/11	0.\overline{09}	Non-terminating recurring
(iii) 4/8	0.5	Terminating
(iv) 3/13	0.\overline{230769}	Non-terminating recurring
(v) 2/11	0.\overline{18}	Non-terminating recurring
(vi) 329/400	0.8225	Terminating

Q2. Predict the decimal expansions:

Given:
$\frac{1}{7} = 0.\overline{142857}$

Use cyclic patterns:

• $\frac{2}{7} = 0.\overline{285714}$

• $\frac{3}{7} = 0.\overline{428571}$

• $\frac{4}{7} = 0.\overline{571428}$

• $\frac{5}{7} = 0.\overline{714285}$

• $\frac{6}{7} = 0.\overline{857142}$

📌 These are cyclic permutations of the digits in $\frac{1}{7}$.

Q3. Express in the form $\frac{p}{q}$:

(i) $0.6 = \frac{6}{10} = \frac{3}{5}$
(ii) $0.47 = \frac{47}{100}$
(iii) $0.001 = \frac{1}{1000}$

Q4. Express $0.99999\ldots$ in the form $\frac{p}{q}$

Let $x = 0.99999\ldots$
Then $10x = 9.99999\ldots$
Subtract: $10x - x = 9 \Rightarrow 9x = 9 \Rightarrow x = 1$

✅ So, $0.99999\ldots = 1$

Q5. Repeating block length in $\frac{1}{7}$

Do long division:
$\frac{1}{7} = 0.\overline{142857}$

🔹 Number of digits in the repeating block = 6

Q6. Property of q for rational number $\frac{p}{q}$ to be terminating

Decimal of $\frac{p}{q}$ terminates iff q (in lowest terms) has no prime factor other than 2 or 5.

✅ Example:

• $\frac{1}{8} = 0.125$: q = 8 = $2^3$

• $\frac{1}{6} = 0.16666...$: q = 6 = $2 \times 3$ → Not terminating

Q7. Three numbers with non-terminating, non-recurring decimals:

• π = 3.1415926535...

• √2 = 1.41421356...

• e = 2.718281828...

Q8. Three irrational numbers between $\frac{5}{7}$ and $\frac{9}{11}$:

Convert bounds to decimal:

• $\frac{5}{7} ≈ 0.714$

• $\frac{9}{11} ≈ 0.818$

Now pick irrational numbers in between:

• 0.718281828... (e/4)

• 0.75√2

• 0.8π/3

Q9. Classify as rational or irrational

Number	Type
(i) $\sqrt{23}$	Irrational
(ii) $\sqrt{225} = 15$	Rational
(iii) 0.3796	Rational
(iv) 7.478478...	Rational
(v) 1.10100100010000...	Irrational

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