CBSE Class 10 Maths: Real Numbers Exercise 1.4 - Decimal Expansions

Understanding Decimal Expansions of Rational Numbers

In Class 9, you learned that rational numbers can have either a terminating decimal expansion or a non-terminating repeating decimal expansion. For example, 1/2 = 0.5 (terminating) and 1/3 = 0.333... (non-terminating repeating). In Exercise 1.4 of Real Numbers, we'll learn a powerful way to predict the nature of a rational number's decimal expansion without actually performing the division. This method relies entirely on the prime factorization of the denominator. If a rational number, expressed in its simplest form p/q (where p and q are coprime integers and q ≠ 0), has a denominator q whose prime factorization contains only powers of 2, or only powers of 5, or both powers of 2 and 5, then its decimal expansion will be terminating. If, however, the prime factorization of q includes any other prime factor (like 3, 7, 11, etc.) besides 2 or 5, then its decimal expansion will be non-terminating and repeating. This theorem provides a quick and elegant way to classify rational numbers.

The Fundamental Theorem for Decimal Expansions

Terminating Decimal Expansion Theorem

Let x = p/q be a rational number, where p and q are coprime (i.e., the fraction is in its simplest form). If the prime factorization of q is of the form 2^n × 5^m, where n and m are non-negative integers, then x has a decimal expansion which terminates.

Non-terminating Repeating Decimal Expansion Theorem

Let x = p/q be a rational number, where p and q are coprime. If the prime factorization of q is not of the form 2^n × 5^m (i.e., it contains prime factors other than 2 or 5), then x has a decimal expansion which is non-terminating and repeating.

Steps to Determine Decimal Expansion Without Actual Division

1. Step 1: Simplify the Rational Number — First, express the given rational number in its simplest form, p/q. This means ensuring that p and q have no common factors other than 1 (i.e., HCF(p, q) = 1). This is crucial because sometimes common factors might hide the true nature of the denominator's prime factorization.
2. Step 2: Find the Prime Factorization of the Denominator — Next, find the prime factors of the denominator, q. Break down q into its prime components. For example, if q = 20, its prime factorization is 2² × 5.
3. Step 3: Examine the Prime Factors of q — Once you have the prime factorization of q, observe the prime factors. If q has only 2s or only 5s, or both 2s and 5s as its prime factors (i.e., of the form 2^n × 5^m), then the decimal expansion is terminating. If q has any other prime factor (like 3, 7, 11, etc.) besides 2 or 5, then the decimal expansion is non-terminating and repeating.
4. Step 4: State Your Conclusion — Based on the analysis in Step 3, clearly state whether the given rational number has a terminating or a non-terminating repeating decimal expansion.

Worked Examples: Classifying Decimal Expansions

• Example 1: Determine whether the decimal expansion of 13/3125 is terminating or non-terminating repeating. Step 1: The fraction 13/3125 is already in its simplest form as 13 is a prime number and 3125 is not a multiple of 13. Step 2: Find the prime factorization of the denominator, q = 3125. 3125 = 5 × 625 = 5 × 5 × 125 = 5 × 5 × 5 × 25 = 5 × 5 × 5 × 5 × 5 = 5^5. Step 3: The prime factorization of q is 5^5, which is of the form 2^n × 5^m (here n=0, m=5). Step 4: Since the denominator's prime factors are only 5s, the decimal expansion of 13/3125 is terminating.
• Example 2: Without performing long division, state whether 64/455 has a terminating or non-terminating repeating decimal expansion. Step 1: The fraction 64/455 is already in its simplest form because HCF(64, 455) = 1. (64 = 2^6; 455 = 5 × 7 × 13, no common factors). Step 2: Find the prime factorization of the denominator, q = 455. 455 = 5 × 91 = 5 × 7 × 13. Step 3: The prime factorization of q is 5 × 7 × 13. This contains prime factors 7 and 13, which are not 2 or 5. Step 4: Since the denominator contains prime factors other than 2 or 5, the decimal expansion of 64/455 is non-terminating and repeating.
• Example 3: Check if 15/1600 has a terminating or non-terminating repeating decimal expansion. Step 1: Simplify the fraction. HCF(15, 1600) = 5. Divide both numerator and denominator by 5. 15 ÷ 5 = 3 1600 ÷ 5 = 320 So, the simplified fraction is 3/320. Step 2: Find the prime factorization of the new denominator, q = 320. 320 = 32 × 10 = 2^5 × 2 × 5 = 2^6 × 5. Step 3: The prime factorization of q is 2^6 × 5, which is of the form 2^n × 5^m (here n=6, m=1). Step 4: Since the denominator's prime factors are only 2s and 5s, the decimal expansion of 15/1600 is terminating.

Important Exam Tips for Exercise 1.4

Always remember to simplify the rational number (p/q) to its simplest form first, before finding the prime factorization of the denominator. If you miss this step, you might incorrectly conclude that a terminating decimal is non-terminating. For example, 10/20 simplifies to 1/2. If you factorize 20 directly (2² × 5), it fits the criteria for terminating. But if you had a number like 6/15, simplifying it to 2/5 reveals the 5 in the denominator. If you only looked at 15 (3×5), you might incorrectly conclude it's non-terminating repeating because of the '3'. Make sure to write down the prime factorization clearly in your solutions to score full marks. Also, clearly state the theorem being applied.

Practice Questions with Solutions

• Q: Without performing actual division, determine whether the decimal expansion of 17/8 is terminating or non-terminating repeating. A: Step 1: The fraction 17/8 is already in its simplest form. Step 2: Find the prime factorization of the denominator, q = 8. 8 = 2 × 2 × 2 = 2^3. Step 3: The prime factorization of q is 2^3, which is of the form 2^n × 5^m (here n=3, m=0). Final answer: Since the denominator's prime factors are only 2s, the decimal expansion of 17/8 is terminating.
• Q: Classify the decimal expansion of 29/343 as terminating or non-terminating repeating. A: Step 1: The fraction 29/343 is already in its simplest form (29 is prime, 343 = 7^3, no common factors). Step 2: Find the prime factorization of the denominator, q = 343. 343 = 7 × 49 = 7 × 7 × 7 = 7^3. Step 3: The prime factorization of q is 7^3. This contains prime factor 7, which is not 2 or 5. Final answer: Since the denominator contains prime factors other than 2 or 5, the decimal expansion of 29/343 is non-terminating and repeating.
• Q: What type of decimal expansion does 23/(2^3 × 5^2) have? A: Step 1: The fraction 23/(2^3 × 5^2) is already in its simplest form (23 is prime, not a factor of the denominator). Step 2: The prime factorization of the denominator, q = 2^3 × 5^2, is already given. Step 3: The prime factorization of q is of the form 2^n × 5^m (here n=3, m=2). Final answer: Since the denominator's prime factors are only 2s and 5s, the decimal expansion of 23/(2^3 × 5^2) is terminating.
• Q: Determine if the rational number 77/210 has a terminating or non-terminating repeating decimal expansion. A: Step 1: Simplify the fraction. HCF(77, 210) = 7. Divide both by 7. 77 ÷ 7 = 11 210 ÷ 7 = 30 The simplified fraction is 11/30. Step 2: Find the prime factorization of the new denominator, q = 30. 30 = 2 × 3 × 5. Step 3: The prime factorization of q is 2 × 3 × 5. This contains the prime factor 3, which is not 2 or 5. Final answer: Since the denominator contains prime factors other than 2 or 5, the decimal expansion of 77/210 is non-terminating and repeating.
• Q: Without actual division, verify if 6/15 has a terminating decimal expansion. A: Step 1: Simplify the fraction. HCF(6, 15) = 3. Divide both by 3. 6 ÷ 3 = 2 15 ÷ 3 = 5 The simplified fraction is 2/5. Step 2: Find the prime factorization of the new denominator, q = 5. 5 = 5^1. Step 3: The prime factorization of q is 5^1, which is of the form 2^n × 5^m (here n=0, m=1). Final answer: Since the denominator's prime factors are only 5s, the decimal expansion of 6/15 is terminating.

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