Have you ever wondered what happens when you divide 1 by 3 on your calculator? You get 0.333333… and the threes just keep going forever. This is what we call a decimal expansion – the way a number looks when written as a decimal.
What is a Decimal Expansion?
A decimal expansion is simply how we write a number using the decimal system. Every number can be expressed as a decimal. Some end after a few digits, while others go on forever.
For example:
- 0.5 is the decimal expansion of 1/2
- 0.333… is the decimal expansion of 1/3
- 3.14159… is the decimal expansion of π
Decimal Expansion of Rational Numbers
A rational number is any number that can be written as a fraction p/q, where p and q are integers and q ≠ 0.
The NCERT Class 9 textbook tells us that rational numbers have only two types of decimal expansions:
1. Terminating Decimals
2. Non-terminating Recurring Decimals
What is a Terminating Decimal?
A terminating decimal is a decimal number that stops after a certain number of digits. It doesn’t go on forever.
Examples:
- 0.5 (stops after one digit)
- 0.75 (stops after two digits)
- 0.125 (stops after three digits)
- 4.25 (stops after two decimal places)
Think of terminating decimals like finishing a race – they have a clear end point. When you convert fractions like 1/4, 3/8, or 7/20 to decimals, they all terminate.
What is a Non-Terminating Repeating Decimal?
A non-terminating repeating decimal is a decimal that goes on forever but has a pattern of digits that keeps repeating over and over again.
Examples:
- 0.333333… (the digit 3 repeats forever)
- 0.272727… (the digits 27 repeat forever)
- 0.142857142857… (the six digits 142857 repeat forever)
We often write these with a bar over the repeating part: 0.3̅, 0.2̅7̅, 0.1̅4̅2̅8̅5̅7̅
Think of non-terminating repeating decimals like a song that keeps playing the same chorus over and over – it never ends, but there’s a clear pattern that repeats.
Easy Way to Remember:
- Terminating decimals are like a short story – they have an ending.
- Non-terminating repeating decimals are like a broken record – they play the same pattern forever.
Both types represent rational numbers (numbers that can be written as fractions).
Decimal Expansion of Irrational Numbers
Irrational numbers are numbers that cannot be written as simple fractions. Their decimal expansions have special features:
- They never terminate (never end)
- They never repeat in a pattern
Famous irrational numbers include:
- √2 = 1.4142135623730950488…
- π = 3.14159265358979323846…
- e = 2.71828182845904523536…
You can keep calculating these numbers to millions of decimal places, and you’ll never find a repeating pattern.
Converting Recurring Decimals to Fractions
The textbook shows us a clever way to convert recurring decimals back to fractions:
For example, to convert 0.333… to a fraction:
- Let x = 0.333…
- Multiply by 10: 10x = 3.333…
- Subtract: 10x – x = 3.333… – 0.333…
- Simplify: 9x = 3
- Solve: x = 3/9 = 1/3
This works for any recurring decimal!
NCERT Solutions for Class 9 Maths Exercise 1.3
Question 1: Write the following in decimal form and say what kind of decimal expansion each has:
(i) 36/100
Answer: 36/100 = 0.36 This is a terminating decimal expansion.
(ii) 1/11
Answer: 1/11 = 0.090909… To find this, perform the division 1 ÷ 11:
- 1 ÷ 11 = 0.090909… This is a non-terminating recurring decimal where the digits “09” repeat. We write it as 0.0̅9̅.
(iii) 4 1/8
Answer: 4 1/8 = 4 + 1/8 = 4 + 0.125 = 4.125 This is a terminating decimal expansion
(iv) 3/13
Answer: 3/13 = 0.230769230769… Performing the division:
- 3 ÷ 13 = 0.230769230769… This is a non-terminating recurring decimal where the digits “230769” repeat. We write it as 0.2̅3̅0̅7̅6̅9̅.
(v) 2/11
Answer: 2/11 = 0.181818… Performing the division:
- 2 ÷ 11 = 0.181818… This is a non-terminating recurring decimal where the digits “18” repeat. We write it as 0.1̅8̅.
(vi) 329/400
Answer: 329/400 = 0.8225 This is a terminating decimal expansion because 400 = 2⁴ × 5², and the denominator has only 2 and 5 as prime factors.
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?
Answer: Yes, we can predict these decimal expansions by multiplying the decimal expansion of 1/7 by the respective numerators:
1/7 = 0.142857142857…
Thus:
- 2/7 = 2 × (1/7) = 2 × 0.142857… = 0.285714285714…
- 3/7 = 3 × (1/7) = 3 × 0.142857… = 0.428571428571…
- 4/7 = 4 × (1/7) = 4 × 0.142857… = 0.571428571428…
- 5/7 = 5 × (1/7) = 5 × 0.142857… = 0.714285714285…
- 6/7 = 6 × (1/7) = 6 × 0.142857… = 0.857142857142…
Notice that all these decimal expansions have the same six digits (142857) repeating, but starting from different positions in the cycle.
Hint: Looking at the remainders when finding 1/7 helps us understand this pattern. The remainders in the division process repeat in a cycle, which causes the digits in the quotient to repeat in the same pattern.
Question 3: Express the following in the form p/q, where p and q are integers and q ≠ 0:
(i) 0.6̅
Answer: 0.6̅ = 0.666666… Let x = 0.666666… Then 10x = 6.666666… Subtracting: 10x – x = 6.666… – 0.666… 9x = 6 x = 6/9 = 2/3 Therefore, 0.6̅ = 2/3
(ii) 0.4̅7̅
Answer: 0.4̅7̅ = 0.474747… Let x = 0.474747… Then 100x = 47.474747… Subtracting: 100x – x = 47.474747… – 0.474747… 99x = 47 x = 47/99 Therefore, 0.4̅7̅ = 47/99
(iii) 0.00̅1̅
Answer: 0.00̅1̅ = 0.001001001… Let x = 0.001001001… Then 1000x = 1.001001001… Subtracting: 1000x – x = 1.001001… – 0.001001… 999x = 1 x = 1/999 Therefore, 0.00̅1̅ = 1/999
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.
Answer: Let x = 0.99999… Then 10x = 9.99999… Subtracting: 10x – x = 9.99999… – 0.99999… 9x = 9 x = 9/9 = 1 Therefore, 0.99999… = 1
Yes, this result might be surprising! It shows that 0.99999… is exactly equal to 1, not just approximately equal.
This makes sense because:
- The difference between 1 and 0.99999… is infinitesimally small
- If we try to find this difference: 1 – 0.99999… = 0.00000…1, but since the 9’s go on forever, we can never place the 1
- If two numbers differ by 0, they must be equal
- Therefore, 0.99999… = 1
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.
Answer: The maximum number of digits in the repeating block can be at most 16 (which is 17-1).
Performing the division 1 ÷ 17: 1/17 = 0.0588235294117647058823529411764…
Looking at the decimal expansion, we notice the digits “0588235294117647” repeat. This is a 16-digit repeating block, confirming our prediction.
The theoretical explanation is based on the fact that when dividing 1 by a prime number p, the maximum length of the repeating block is p-1 or a factor of p-1.
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?
Answer: After examining several examples, we notice that rational numbers p/q have terminating decimal expansions if and only if the prime factorization of q contains only the factors 2 and/or 5.
For example:
- 1/4 = 0.25 (q = 4 = 2²)
- 3/5 = 0.6 (q = 5 = 5¹)
- 7/8 = 0.875 (q = 8 = 2³)
- 3/20 = 0.15 (q = 20 = 2² × 5)
This is resulting in a terminating decimal.
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