Exercise 3.5 - The World of Numbers

Question 1

Terminating or Repeating Decimals

Without performing long division, determine which of the following rational numbers have terminating decimals and which have repeating decimals: 7/20, 4/15 and 13/250. Then verify by converting them to decimals.

हिंदी में समझें:Rule: In lowest form, a rational number p/q has a terminating decimal exactly when q has no prime factor other than 2 and 5. (i) 7/20: 20 = 2² × 5. So it terminates. 7/20 = 0.35. (ii) 4/15: 15 = 3 × 5. The factor 3 is present, so it repeats. 4/15 = 0.2666… . (iii) 13/250: 250 = 2 × 5³. So it terminates. 13/250 = 0.052.

Step-by-Step Solution:

Visual: Diagram को देखकर decimal pattern, remainder cycle और rational/irrational classification को समझें।

Rule: In lowest form, a rational number p/q has a terminating decimal exactly when q has no prime factor other than 2 and 5. (i) 7/20: 20 = 2² × 5. So it terminates. 7/20 = 0.35. (ii) 4/15: 15 = 3 × 5. The factor 3 is present, so it repeats. 4/15 = 0.2666… . (iii) 13/250: 250 = 2 × 5³. So it terminates. 13/250 = 0.052.

✅ Final Answer: 7/20 = 0.35 (terminating); 4/15 = 0.2666… (repeating); 13/250 = 0.052 (terminating).

Exam Tip: Terminating/repeating questions में denominator को lowest form में लाकर prime factors check करें। Long division में जब same remainder दोबारा मिले, repeating cycle identify हो जाता है।

Question 2

Long Division of 1/13

Perform the long division for 1/13. Identify the repeating block of digits.

हिंदी में समझें:Start with 1 ÷ 13. Since 1 < 13, write 0. and bring down 0 to get 10. 10 < 13, so write 0 and bring down another 0: 100. Then 100 ÷ 13 = 7 remainder 9. Continue: 90 ÷ 13 = 6 remainder 12; 120 ÷ 13 = 9 remainder 3; 30 ÷ 13 = 2 remainder 4; 40 ÷ 13 = 3 remainder 1. The remainder 1 is the starting remainder, so the digits repeat from here.

Step-by-Step Solution:

Visual: Diagram को देखकर decimal pattern, remainder cycle और rational/irrational classification को समझें।

Start with 1 ÷ 13. Since 1 < 13, write 0. and bring down 0 to get 10. 10 < 13, so write 0 and bring down another 0: 100. Then 100 ÷ 13 = 7 remainder 9. Continue: 90 ÷ 13 = 6 remainder 12; 120 ÷ 13 = 9 remainder 3; 30 ÷ 13 = 2 remainder 4; 40 ÷ 13 = 3 remainder 1. The remainder 1 is the starting remainder, so the digits repeat from here.

✅ Final Answer: 1/13 = 0.076923076923…; repeating block = 076923.

Exam Tip: Terminating/repeating questions में denominator को lowest form में लाकर prime factors check करें। Long division में जब same remainder दोबारा मिले, repeating cycle identify हो जाता है।

Question 3

Rational or Irrational

Classify the following numbers as rational or irrational and give a reason: √81, √2, 7/11, 0.125 and π.

हिंदी में समझें:√81 = 9, which is an integer, so it is rational. √2 cannot be written as p/q, so it is irrational. 7/11 is already in p/q form, so it is rational. 0.125 is a terminating decimal, so it is rational; 0.125 = 1/8. π has a non-terminating, non-repeating decimal expansion, so it is irrational.

Step-by-Step Solution:

Visual: Diagram को देखकर decimal pattern, remainder cycle और rational/irrational classification को समझें।

√81 = 9, which is an integer, so it is rational. √2 cannot be written as p/q, so it is irrational. 7/11 is already in p/q form, so it is rational. 0.125 is a terminating decimal, so it is rational; 0.125 = 1/8. π has a non-terminating, non-repeating decimal expansion, so it is irrational.

✅ Final Answer: Rational: √81, 7/11, 0.125. Irrational: √2, π.

Exam Tip: Terminating/repeating questions में denominator को lowest form में लाकर prime factors check करें। Long division में जब same remainder दोबारा मिले, repeating cycle identify हो जाता है।

Question 4

Why 0.999… = 1

The number 0.9̅ (0.99999…) is rational. Using algebra, let x = 0.999…, multiply by 10 and subtract to explain why 0.9̅ is exactly equal to 1.

हिंदी में समझें:Let x = 0.999… Multiply both sides by 10: 10x = 9.999… Subtract the original equation: 10x − x = 9.999… − 0.999… 9x = 9 Therefore x = 1. But x was defined as 0.999…, so 0.999… = 1 exactly.

Step-by-Step Solution:

Visual: Diagram को देखकर decimal pattern, remainder cycle और rational/irrational classification को समझें।

Let x = 0.999… Multiply both sides by 10: 10x = 9.999… Subtract the original equation: 10x − x = 9.999… − 0.999… 9x = 9 Therefore x = 1. But x was defined as 0.999…, so 0.999… = 1 exactly.

✅ Final Answer: 0.999… = 1 exactly.

Exam Tip: Terminating/repeating questions में denominator को lowest form में लाकर prime factors check करें। Long division में जब same remainder दोबारा मिले, repeating cycle identify हो जाता है।

Question 5

Cyclic Repeating Blocks

We have seen that the repeating block of 1/7 is a cyclic number. Try to find more denominators n for which 1/n produces a decimal with a repeating block that shows a cyclic pattern.

हिंदी में समझें:Start by checking reciprocals of primes or other denominators that give long repeating periods. A standard example is 13: 1/13 = 0.076923076923… and its block 076923 has cyclic shifts in related multiples. Another example is 17: 1/17 = 0.0588235294117647… For this question, the key idea is to calculate several reciprocals and look for repeating blocks that are rearrangements of the same digits under multiplication.

Step-by-Step Solution:

Visual: Diagram को देखकर decimal pattern, remainder cycle और rational/irrational classification को समझें।

Start by checking reciprocals of primes or other denominators that give long repeating periods. A standard example is 13: 1/13 = 0.076923076923… and its block 076923 has cyclic shifts in related multiples. Another example is 17: 1/17 = 0.0588235294117647… For this question, the key idea is to calculate several reciprocals and look for repeating blocks that are rearrangements of the same digits under multiplication.

✅ Final Answer: Examples include n = 7 and n = 13; n = 17 gives another long repeating block.

Exam Tip: Terminating/repeating questions में denominator को lowest form में लाकर prime factors check करें। Long division में जब same remainder दोबारा मिले, repeating cycle identify हो जाता है।