Short Intro

In this article, we provide complete step-by-step solutions of Exercise 3.5 from Chapter 3 “The World of Numbers”. This exercise focuses on decimal expansion of rational numbers, terminating and repeating decimals, rational and irrational numbers, and cyclic number patterns.

Quick Information Box

Chapter: The World of Numbers
Exercise: 3.5
Class: 9 Mathematics
Main Topic: Decimal Expansion of Rational and Irrational Numbers
Sub-topics: Terminating decimals, repeating decimals, rational numbers, irrational numbers, cyclic numbers, 0.999… = 1

Concepts Used

1. Rational numbers can be written in the form p/q, where p and q are integers and q ≠ 0.
2. A rational number has either a terminating decimal or a repeating decimal.
3. If the denominator of a rational number in lowest form has only prime factors 2 and/or 5, then its decimal expansion terminates.
4. If the denominator has any prime factor other than 2 or 5, then its decimal expansion is non-terminating repeating.
5. Irrational numbers have non-terminating and non-repeating decimal expansions.
6. Some repeating blocks show cyclic properties.

Important Formulas

1. Terminating Decimal Test:
   If q = 2ᵐ × 5ⁿ, then p/q has a terminating decimal.
2. Repeating Decimal Test:
   If q has prime factors other than 2 and 5, then p/q has a repeating decimal.
3. Pure Repeating Decimal:
   0.(abc) = abc/999
4. Algebraic Method for 0.999…:
   Let x = 0.999…
   10x = 9.999…
   10x − x = 9
   9x = 9
   x = 1

Questions and Step-by-Step Solutions

Question 1

Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: 7/20, 4/15 and 13/250. Then check your answers by explicitly performing long divisions and expressing these rational numbers as decimals.

Solution

(i) 7/20

Step 1: Check the denominator.
20 = 2² × 5

Step 2: Since the denominator has only the prime factors 2 and 5, the decimal expansion will terminate.

Step 3: Convert into decimal.
7/20 = 35/100 = 0.35

Answer:
7/20 = 0.35, terminating decimal.

(ii) 4/15

Step 1: Check the denominator.
15 = 3 × 5

Step 2: Since the denominator contains 3, which is not 2 or 5, the decimal expansion will be non-terminating repeating.

Step 3: Convert into decimal.
4 ÷ 15 = 0.2666…

Answer:
4/15 = 0.2666… = 0.2(6), repeating decimal.

(iii) 13/250

Step 1: Check the denominator.
250 = 2 × 5³

Step 2: Since the denominator has only 2 and 5 as prime factors, the decimal expansion will terminate.

Step 3: Convert into decimal.
13/250 = 52/1000 = 0.052

Answer:
13/250 = 0.052, terminating decimal.

Final Answer:
7/20 = terminating
4/15 = repeating
13/250 = terminating

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Question 2

Perform the long division for 1/13. Identify the repeating block of digits. Does it show cyclic properties if you evaluate 2/13? Now compute 3/13, 4/13, etc. What do you notice?

Solution

First, divide 1 by 13.

1/13 = 0.076923076923…

Repeating block = 076923

Now calculate other fractions:

2/13 = 0.153846153846…
3/13 = 0.230769230769…
4/13 = 0.307692307692…
5/13 = 0.384615384615…
6/13 = 0.461538461538…
7/13 = 0.538461538461…
8/13 = 0.615384615384…
9/13 = 0.692307692307…
10/13 = 0.769230769230…
11/13 = 0.846153846153…
12/13 = 0.923076923076…

Observation:
The repeating blocks are related to each other. They are not all simple rotations of 076923, but they form two cyclic groups.

Group 1:
076923, 230769, 307692, 692307, 769230, 923076

Group 2:
153846, 384615, 461538, 538461, 615384, 846153

Conclusion:
The decimal expansion of 1/13 has repeating block 076923. The multiples of 1/13 show cyclic patterns in two groups.

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Question 3

Classify the following numbers as rational or irrational. Find explicit fractions in case they are rational.

(i) √81
(ii) √12
(iii) 0.33333…
(iv) 0.123451234512345…
(v) 1.01001000100001…
(vi) 23.560185612239874790120

Solution

(i) √81

√81 = 9

9 can be written as 9/1.

Answer:
√81 is rational.
Explicit fraction = 9/1

(ii) √12

√12 = √(4 × 3) = 2√3

Since √3 is irrational, 2√3 is also irrational.

Answer:
√12 is irrational.

(iii) 0.33333…

Let x = 0.33333…
10x = 3.33333…

Subtract:
10x − x = 3.33333… − 0.33333…
9x = 3
x = 3/9 = 1/3

Answer:
0.33333… is rational.
Explicit fraction = 1/3

(iv) 0.123451234512345…

The block 12345 repeats continuously.

So,
0.1234512345… = 0.(12345)

For a five-digit repeating block:
0.(12345) = 12345/99999

Simplify:
12345/99999 = 4115/33333

Answer:
0.123451234512345… is rational.
Explicit fraction = 4115/33333

(v) 1.01001000100001…

Here the number of zeros keeps increasing.
The pattern is not repeating as one fixed block.

So, it is non-terminating and non-repeating.

Answer:
1.01001000100001… is irrational.

(vi) 23.560185612239874790120

This decimal is terminating because it has a finite number of decimal digits.

Any terminating decimal is rational.

There are 21 digits after the decimal point.

So,
23.560185612239874790120
= 23560185612239874790120 / 1000000000000000000000

On simplification:
= 589004640305996869753 / 25000000000000000000

Answer:
23.560185612239874790120 is rational.
Explicit fraction = 589004640305996869753 / 25000000000000000000

Final Classification:
(i) Rational
(ii) Irrational
(iii) Rational
(iv) Rational
(v) Irrational
(vi) Rational

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Question 4

The number 0.9, which means 0.99999…, is a rational number. Using algebra, explain why 0.9 is exactly equal to 1.

Solution

Let
x = 0.99999…

Multiply both sides by 10:
10x = 9.99999…

Now subtract the first equation from the second equation:

10x − x = 9.99999… − 0.99999…

9x = 9

Divide both sides by 9:
x = 1

But x = 0.99999…

Therefore,
0.99999… = 1

Answer:
0.9 repeating is exactly equal to 1.

Explanation:
0.999… is not slightly less than 1. It is exactly equal to 1 because the infinite repeating decimal has no gap left between itself and 1.

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Question 5

We have seen that the repeating block of 1/7 is a cyclic number. Try to find more numbers n whose reciprocals 1/n produce decimals with repeating blocks that are cyclic.

Solution

We know:
1/7 = 0.142857142857…

Repeating block = 142857

Now check its multiples:

142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142

The same digits rotate cyclically. So 7 gives a cyclic number.

Now let us find more such numbers.

Example 1: n = 17

1/17 = 0.0588235294117647…

Repeating block = 0588235294117647

This block also shows cyclic behaviour when multiplied by 1, 2, 3, …, 16.

So, n = 17 is another example.

Example 2: n = 19

1/19 = 0.052631578947368421…

Repeating block = 052631578947368421

This also produces a cyclic pattern.

So, n = 19 is another example.

Example 3: n = 23

1/23 = 0.0434782608695652173913…

This also has a long repeating block with cyclic behaviour.

Answer:
Some values of n whose reciprocals produce cyclic repeating blocks are:

7, 17, 19, 23

Note:
Such cyclic behaviour is usually seen when the decimal expansion of 1/n has a repeating block of length n − 1.

Common Mistakes

1. Students often think every non-terminating decimal is irrational. This is wrong. A non-terminating repeating decimal is rational.
2. Students forget to reduce the fraction to its lowest form before applying the terminating decimal test.
3. Many students classify 0.999… as less than 1, but it is exactly equal to 1.
4. Students confuse repeating pattern with increasing pattern. For example, 1.01001000100001… is not repeating, so it is irrational.
5. In recurring decimals, students sometimes write only a few digits and forget to use “…” or a repeating notation.

Exam Tips

1. Always reduce p/q to lowest form before checking the denominator.
2. If the denominator has only 2 and/or 5, the decimal terminates.
3. If the denominator has any prime factor other than 2 or 5, the decimal repeats.
4. A terminating decimal is always rational.
5. A repeating decimal is always rational.
6. A non-terminating and non-repeating decimal is irrational.
7. For 0.999… = 1 type questions, use the algebraic method.

Practice MCQs

MCQ 1

Which of the following has a terminating decimal expansion?
A. 4/15
B. 7/20
C. 1/13
D. 2/3

Answer: B. 7/20

MCQ 2

The decimal expansion of 1/13 is:
A. Terminating
B. Non-terminating non-repeating
C. Non-terminating repeating
D. Irrational

Answer: C. Non-terminating repeating

MCQ 3

0.33333… is equal to:
A. 3/10
B. 1/3
C. 33/100
D. 3/100

Answer: B. 1/3

MCQ 4

Which number is irrational?
A. √81
B. 0.333…
C. √12
D. 23.56

Answer: C. √12

MCQ 5

0.999… is equal to:
A. 0.9
B. 0.99
C. Slightly less than 1
D. 1

Answer: D. 1

FAQ Section

Q1. What is a terminating decimal?

A decimal that ends after a finite number of digits is called a terminating decimal. Example: 0.35.

Q2. What is a repeating decimal?

A decimal in which one digit or a group of digits repeats forever is called a repeating decimal. Example: 0.333…

Q3. How do we know whether p/q is terminating?

First reduce p/q to lowest form. If the denominator has only 2 and/or 5 as prime factors, the decimal is terminating.

Q4. Is every repeating decimal rational?

Yes, every repeating decimal can be converted into a fraction, so it is rational.

Q5. Is 0.999… really equal to 1?

Yes. Algebraically, if x = 0.999…, then 10x = 9.999…, so 9x = 9 and x = 1.

Q6. Why is 1.01001000100001… irrational?

Because the number of zeros keeps increasing and no fixed block repeats forever. Therefore, it is non-terminating and non-repeating.

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