Why long division must either stop or loop

What to assume they know

• What "rational" means, and why the denominator cannot be zero — what a rational number is
• Long division of a whole number by a whole number, carried past the decimal point
• What a remainder is, and that it is always smaller than the divisor
• Bar notation for a repeating block, from earlier classes
• Proof by contradiction: why √2 cannot be a ratio of integers — irrational numbers, so the test has something to detect

What they should be able to do

• Carry out long division past the decimal point and record the remainder at each step
• List the possible remainders when dividing by a given whole number, and say how many there are
• Explain why a remainder must eventually recur, and give the largest number of steps that can pass before it does
• Explain why a recurring remainder forces the digits to recur, by identifying the remainder as the state that determines everything after it
• Identify which of the two outcomes a given division reaches, and write the result with a bar where needed
• State an upper bound on the length of the repeating block from the divisor alone
• Conclude that a non-terminating, non-repeating decimal cannot be rational, and say which direction of reasoning that is

Where it usually goes wrong

• "Some fractions produce decimals that go on for ever without any pattern." None do. The counting argument rules it out for every fraction, and that is the whole content of the section.
• "It repeats because the digits happen to come round." The digits come round because the remainders do. Watching the digit column alone makes the repetition look like luck; watching the remainder column makes it look inevitable.
• "A remainder of 0 is just another remainder." It is the one that ends the process. Distinguishing it is what splits the two outcomes, and the chapter asks about it in brackets for exactly that reason.
• "The block can be as long as you like." It is capped by the divisor: at most q − 1 digits. Students who do not know the bound cannot check their own answers.
• "1/7 has a six-digit block, so 1/13 will too — thirteen is bigger." 1/13 also has a six-digit block, well under its bound of 12. Block length is not a simple function of the divisor's size, and Q2 is built on this.
• "A decimal with an obvious pattern is repeating." Not necessarily — a pattern and a fixed recurring block are different things, and Irrational decimals: an expansion with no stop and no repeating block turns on the difference.
• "This proves rationals repeat, so repeating decimals must be rational." That is the converse, and it needs the separate argument in Converting a terminating or repeating decimal back to p/q. The chapter provides both, in different places; do not let one stand in for the other.

Questions to check understanding

• Perform a long division and state the decimal expansion, using a bar where needed
• List the possible remainders for a stated divisor and explain the significance of zero
• Explain why the decimal expansion of a rational number must stop or repeat
• Identify a repeating block and state its length
• State the maximum possible block length for a given divisor
• Given a decimal that neither stops nor repeats, say what can be concluded about the number and why
• Compare the block lengths of a family of fractions with the same denominator

Examples worth working on the board

Values marked verified are an added long division on the chapter's stated inputs. The chapter prints no answers beyond the values it states in its examples.

• Example 2 (§3.6.1, p. 57). 3/8 = 0.375, with the chapter adding a question in brackets about which rationals terminate — answered in Predicting the expansion from the denominator's prime factors. Verified: the remainders in dividing 3 by 8 run 6, 4, 0, and the division stops when the third one arrives.
• Example 3 (§3.6.1, p. 57). 5/11 = 0.454545…, written with a bar over the two digits 45. Verified: the remainders alternate 5, 6, 5, 6, so the second distinct remainder is already the last new one and the block has length 2.
• The chapter's own argument (§3.6.1, p. 58, first paragraph). Inputs as printed: dividing 1 by 7, the only possible remainders are 1, 2, 3, 4, 5 or 6, with the chapter asking in brackets why 0 is not among them; because the supply is limited, some remainder must appear a second time; and once one does, the division loops. This is the argument the whole topic elaborates, and it is printed in four sentences. Verified: the answer to the bracketed question is that a remainder of 0 would stop the division rather than continue it, so it cannot be one of the values that keeps a non-terminating expansion going.
• 1/7 worked out (drawing on §3.6.2, p. 61 for the result). Verified: 1/7 = 0.142857142857… with a six-digit block, and the remainders run 3, 2, 6, 4, 5, 1, at which point 1 has returned and the cycle closes.
• The bound (not in the book). Dividing by q, there are q − 1 non-zero remainders available, so a repeating block can be at most q − 1 digits long. Verified against the chapter's own cases: for 11 the bound is 10 and the block is 2; for 7 the bound is 6 and the block is 6, which attains the bound. The chapter states the finiteness of the supply but never draws the bound.
• Think and Reflect (p. 57). Asks for the expansions of 10/3 and 11/12 and what the student notices about the repetition. Verified: 10/3 = 3.333… with a one-digit block; 11/12 = 0.91666…, which terminates in neither sense — it has a two-digit non-repeating head and then a one-digit block. That second one is the chapter's quiet introduction of the general repeating case handled in Converting a terminating or repeating decimal back to p/q, and it is worth flagging that the prompt mixes the two kinds.
• Exercise Set 3.5, Q2 (p. 62). Perform the long division for 1/13, identify the block, then look at 2/13, and then compute 3/13 and 4/13. Verified: 1/13 = 0.076923… with a six-digit block, and 13 − 1 = 12, so here the block is half the maximum rather than equal to it. This matters for Cyclic numbers: the hidden symmetry inside 1/7 and should be flagged, not smoothed over.
• No figure accompanies this material. §3.6.1 on pp. 57 and 58 is set as text with worked examples. A remainder-column table is added here to build.

Figures to have open

• A digits-and-remainders two-column table that can be filled in step by step. This is the central figure of the topic and the chapter prints nothing like it. Standard schematic.
• A row of labelled slots for the possible remainders, so that the second arrival in a slot is visible. Standard schematic; this is what makes the counting argument land.
• A rewind device for section 6 — the division shown replaying from the earlier identical state. Standard schematic.
• A small table setting divisor, bound and actual block length side by side, filled for 8, 11, 7 and 13. Built from the values listed above.
• No textbook figure is available or needed.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics, printed Chapter 3 on the world of numbers. Section §3.6, printed heading "Real Numbers: Decimals and Cyclic Patterns", opens on p. 57; §3.6.1, printed heading "Rational Decimals: Terminating and Repeating", follows on the same page with Examples 2 and 3.
• The remainder argument occupies the first paragraph of p. 58.
• Think and Reflect, p. 57.
• Exercise Set 3.5, Q2, p. 62.
• Chapter Summary, p. 67, sixth bullet, states the terminating-or-repeating property as the signature of a rational number.
• Forward pointers: the prediction rule is at §3.6.1, p. 58 (Predicting the expansion from the denominator's prime factors); the converse conversion is at §3.6.1, pp. 58–61 (Converting a terminating or repeating decimal back to p/q).