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🔢 Chapter 3: The World of Numbers • Exercise Set 3.5

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Exercise Set 3.1 4Exercise Set 3.2 4Exercise Set 3.3 8Exercise Set 3.4 5Exercise Set 3.5 5End-of-Chapter Exercises 14All Exercises 40
Q1

Exercise Set 3.5

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 the long divisions and expressing these rational numbers as decimals.

  • 7/20:
    Denominator factorized: 20 = 2² × 5. Since prime factors are only 2 and 5, it is terminating.
    Long division check: 7 ÷ 20 = 0.35
  • 4/15:
    Denominator factorized: 15 = 3 × 5. It contains prime factor 3 (other than 2 or 5), so it is repeating.
    Long division check: 4 ÷ 15 = 0.26̅ (0.2666...)
  • 13/250:
    Denominator factorized: 250 = 2 × 5³. Since prime factors are only 2 and 5, it is terminating.
    Long division check: 13 ÷ 250 = 0.052
Q2

Exercise Set 3.5

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?

  • 1/13: Performing long division gives 1/13 = 0.076923̅ with a 6-digit repeating block: 076923.
  • Computing Multiples:
    • 2/13 = 0.153846̅ (repeating block: 153846)
    • 3/13 = 0.230769̅ (cyclic permutation of 076923!)
    • 4/13 = 0.307692̅ (cyclic permutation of 076923!)
    • 9/13 = 0.692307̅ and 10/13 = 0.769230̅ (also cyclic shifts of 076923).
  • Observation: The decimal expansion of k/13 splits into two distinct 6-digit cyclic families: one family generated by cyclic shifts of 076923 (for remainders 1, 3, 4, 9, 10, 12) and another generated by cyclic shifts of 153846 (for remainders 2, 5, 6, 7, 8, 11).
Q3

Exercise Set 3.5

3. Classify the following numbers as rational or irrational:

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

Find the explicit fractions in case they are rational.

  • (i) √81: Rational. Since √81 = 9 = 9/1.
  • (ii) √12: Irrational. Since √12 = 2√3 and 3 is not a perfect square.
  • (iii) 0.33333...: Rational. Non-terminating repeating decimal = 1/3.
  • (iv) 0.123451234512345...: Rational. Repeating 5-digit block 12345 = 12345/99999 = 4115/33333.
  • (v) 1.01001000100001...: Irrational. The sequence of zeros grows indefinitely, so it is non-terminating and non-repeating.
  • (vi) 23.560185612239874790120: Rational. Terminating decimal with 21 decimal places = 23560185612239874790120 / 10²¹ (or 2356018561223987479012 / 10²⁰ in lowest terms).
Q4

Exercise Set 3.5

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

  1. Let x = 0.99999... — (Equation 1)
  2. Multiply both sides by 10:
    10x = 9.99999... — (Equation 2)
  3. Subtract Equation 1 from Equation 2:
    10x - x = 9.99999... - 0.99999...
    9x = 9
  4. Divide both sides by 9:
    x = 9/9 = 1

Thus, 0.9̅ is exactly equal to 1. This demonstrates the non-uniqueness of decimal representations for rational numbers.

Q5

Exercise Set 3.5

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.

Reciprocals 1/n produce full cyclic numbers when n is a full-reptend prime (primes whose decimal periods have maximum possible length n - 1):

  • n = 17: 1/17 = 0.0588235294117647̅ (16-digit cyclic number). Multiplying the 16-digit block by 1 through 16 cyclically shifts the same digits.
  • n = 19: 1/19 = 0.052631578947368421̅ (18-digit cyclic number).
  • n = 23: 1/23 = 0.0434782608695652173913̅ (22-digit cyclic number).
  • n = 29: Produces a 28-digit full cyclic repeating block.