Q1. 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.
Answer:
Solution:
For a rational number p/q (in lowest terms), the decimal terminates if q has only prime factors 2 and/or 5.
7/20: 20 = 2² × 5 (only 2 and 5) → terminating decimal.
4/15: 15 = 3 × 5 (has prime factor 3) → repeating decimal.
13/250: 250 = 2 × 5³ (only 2 and 5) → terminating decimal.
Checking by long division:
- 7/20 = 0.35 (terminating)
- 4/15 = 0.2666... = 0.26 (repeating)
- 13/250 = 0.052 (terminating)
Answer: 7/20 and 13/250 have terminating decimals; 4/15 has a repeating decimal.
Q2. 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?
Answer:
Solution:
Long division for 1/13:
1 ÷ 13 = 0.076923076923... = 0.076923
The repeating block is 076923.
Cyclic properties:
- 2/13 = 0.153846153846... = 0.153846
- 3/13 = 0.230769230769... = 0.230769
- 4/13 = 0.307692307692... = 0.307692
- 5/13 = 0.384615384615... = 0.384615
- 6/13 = 0.461538461538... = 0.461538
- 7/13 = 0.538461538461... = 0.538461
- 8/13 = 0.615384615384... = 0.615384
- 9/13 = 0.692307692307... = 0.692307
- 10/13 = 0.769230769230... = 0.769230
- 11/13 = 0.846153846153... = 0.846153
- 12/13 = 0.923076923076... = 0.923076
Observation: The decimals for 1/13 through 12/13 are cyclic permutations of the same digits (076923, 153846, 230769, etc.). The digits cycle in a fixed pattern.
Answer: The repeating block for 1/13 is 076923. The other fractions show cyclic permutations of the digits.
Q3. Classify the following numbers as rational or irrational: (i) √81 (ii) √12 (iii) 0.33333... (iv) 0.123451234512345... (v) 1.01001000100001... (Notice the pattern: Is it repeating a single block?) (vi) 23.560185612239874790120... Find the explicit fractions in case they are rational.
Answer:
Solution:
(i) √81 = 9, which is rational (can be written as 9/1).
(ii) √12 = 2√3, which is irrational (not a perfect square).
(iii) 0.33333... = 1/3, which is rational (repeating decimal).
(iv) 0.123451234512345... = 0.12345 (repeating block 12345), which is rational.
Explicit fraction: Let x = 0.12345. Then 100000x = 12345.12345...; subtract: 99999x = 12345; x = 12345/99999 = 4115/33333.
(v) 1.01001000100001... The pattern shows increasing zeros between the ones, but it is not repeating a single block. This is irrational.
(vi) 23.560185612239874790120... This is a non-terminating, non-repeating decimal; it is irrational.
Answer: (i) Rational, (ii) Irrational, (iii) Rational (1/3), (iv) Rational (4115/33333), (v) Irrational, (vi) Irrational.
Q4. The number 0.9 (which means 0.99999...) is a rational number. Using algebra (let x = 0.9, multiply by 10, and subtract), explain why 0.9 is exactly equal to 1.
Answer:
Solution:
Let x = 0.99999... = 0.9
Multiplying by 10: 10x = 9.99999... = 9.9
Subtract the first equation from the second:
10x − x = 9.99999... − 0.99999...
9x = 9
x = 1
Therefore, 0.99999... = 1.
Answer: 0.9 = 1 exactly.
Q5. 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.
Answer:
Solution:
Numbers whose reciprocals produce cyclic numbers include:
- 1/7 = 0.142857 (cyclic)
- 1/13 = 0.076923 (cyclic)
- 1/17 = 0.0588235294117647 (cyclic)
- 1/19 = 0.052631578947368421 (cyclic)
- 1/23 = 0.0434782608695652173913 (cyclic)
These numbers are typically prime numbers (but not all primes produce cyclic decimals, e.g., 1/3 = 0.333... is not cyclic).
Answer: Some numbers with cyclic reciprocals are 7, 13, 17, 19, 23, etc.