Q1.
Try to find the decimal expansions of 10/3 and 11/12. What do you observe about the repetition of the digits after the decimal point?
Answer
10/3 by long division
10 ÷ 3 = 3, remainder 1
10 ÷ 3 = 3, remainder 1 (again)
10/3 = 3.3333… = 3.3
10 ÷ 3 = 3, remainder 1
10 ÷ 3 = 3, remainder 1 (again)
10/3 = 3.3333… = 3.3
11/12 by long division
110 ÷ 12 = 9, remainder 2
20 ÷ 12 = 1, remainder 8
80 ÷ 12 = 6, remainder 8
80 ÷ 12 = 6, remainder 8 (repeats)
11/12 = 0.91666… = 0.916
110 ÷ 12 = 9, remainder 2
20 ÷ 12 = 1, remainder 8
80 ÷ 12 = 6, remainder 8
80 ÷ 12 = 6, remainder 8 (repeats)
11/12 = 0.91666… = 0.916
What we observe. Both are non-terminating and repeating, but the repetition begins in different places:
- In 10/3 the repeating block starts immediately after the decimal point — a pure repeating decimal. Its denominator 3 has no factor of 2 or 5.
- In 11/12 two digits, 9 and 1, come first and only then does 6 repeat — a general repeating decimal. Its denominator 12 = 2² × 3 mixes a 3 with powers of 2.
Why it happens: the 2² in 12 can be cleared into the power of 10 (that produces the non-repeating digits), but the leftover 3 can never be cleared, so a remainder must recur and the tail loops for ever. When the denominator has no 2s or 5s at all, as in 3, there are no non-repeating digits to make, and the loop starts at once.
Check it yourself: the number of non-repeating digits equals the larger of the powers of 2 and 5 in the denominator. For 12 = 2² × 5⁰ that is 2 — exactly the two digits 9 and 1.