π: from Āryabhaṭa's approximation to Mādhava's infinite series

What to assume they know

• Proof by contradiction: why √2 cannot be a ratio of integers — irrational numbers, and what it means for no fraction to work
• What "rational" means, and why the denominator cannot be zero — that adding, subtracting, multiplying and dividing fractions returns a fraction
• Circumference and diameter of a circle, and that their ratio is the same for all circles
• Adding and subtracting fractions with unlike denominators
• Reading a decimal to several places, and comparing two such decimals
• Density: averaging always finds another rational in between — density, so that "closer and closer" has somewhere to happen

What they should be able to do

• State what π measures about a circle
• Evaluate Āryabhaṭa's fraction as a decimal and compare it with π's true expansion
• Explain what asanna means and why Āryabhaṭa's own qualification matters
• State who proved π irrational and in which year
• Explain why no fraction, and no finite collection of fractions, can be exactly equal to π
• Write out the first several terms of Mādhava's series and compute its first partial sums
• Describe what an infinite sum means, in the chapter's terms — the value the running totals approach
• Observe that the partial sums alternate above and below the target, and that convergence here is slow
• Place π on the number line between two rationals, and find rationals arbitrarily close to it

Where it usually goes wrong

• "π is 22/7." The chapter never gives 22/7 as a value of π. It gives 3927/1250. 22/7 does not appear anywhere in this chapter — not in its prose and not on any of its figures. What Fig. 3.12 (p. 57) marks is −22/5, a different number entirely, and Fig. 3.7 (p. 51) carries −22/5, 26/7 and 32/7 but no 22/7. Do not let a half-remembered 22/7 substitute for the chapter's actual figure.
• "π is 3.14." That is a rounding used for arithmetic. π has no terminating decimal, which is what §3.6.3 shows.
• "Āryabhaṭa was wrong, and later mathematicians got it right." He was right about what he had — including right that it was not exact. Reporting a value with an honest label is a better piece of mathematics than reporting it without one.
• "An infinite series gives an approximation, so it is no better than a fraction." Any stopping point gives an approximation; the series is exact, and that difference between the object and its truncations is the topic's whole argument.
• "Adding infinitely many things must give infinity." The terms shrink and alternate, so the totals settle rather than run away. The chapter's definition — the value approached — is what makes the sum finite.
• "More terms always gets you closer." Term by term the totals overshoot and undershoot in turn, so a single extra term can move you further from π than the previous total was in the other direction. The narrowing is in the size of the swings, not in each individual step's direction.
• "Mādhava's series is the fast way to compute π." It is famously slow. Say so. An explanation that implies otherwise sets up a disappointment the moment a student tries five terms.
• "Lambert's proof made Āryabhaṭa's work obsolete." It explained why the search Āryabhaṭa had already doubted could not succeed. The chapter's own summary frames Āryabhaṭa as having suspected exactly this.

Questions to check understanding

• State what π is the ratio of
• Give Āryabhaṭa's fractional value and its decimal, and say what asanna means
• Name the mathematician who proved π irrational and the year
• Explain why π cannot be written as a fraction, and why finitely many fractions also fail
• Write the first four or five terms of Mādhava's series and compute the running totals
• Explain what it means to add infinitely many terms
• Find rationals lying between two close decimals that bracket π
• Compare a stated fractional approximation of π against the printed expansion and say to how many places it agrees

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated inputs. The chapter prints no answers.

• What π measures (§3.5.3, p. 56). π is what you get when a circle's circumference is divided across its diameter. Any circle will do, and that is the fact that makes π a number rather than a measurement.
• Āryabhaṭa's value (§3.5.3, p. 56). Printed as the fraction 3927/1250, with the chapter giving its decimal as 3.1416, dating Āryabhaṭa to 499 CE, calling the value highly accurate, and reporting that he labelled it an asanna and indicated that an exact fraction probably could not be found. Verified: 3927 ÷ 1250 = 3.1416 exactly, a terminating decimal, because 1250 factors as 2 × 5⁴ — which is Predicting the expansion from the denominator's prime factors's criterion arriving early, and worth the cross-reference.
• How close it is (the explanation's comparison, using the chapter's own numbers). Set 3.1416 against the expansion the chapter prints at §3.6.3, p. 61: π = 3.1415926535897932384…. Verified: the two agree to three decimal places and part company at the fourth, so Āryabhaṭa's value is high by roughly 0.0000073.
• Lambert (§3.5.3, p. 56). The chapter states that π is irrational, formally proven by Lambert in 1761. The Chapter Summary, p. 67, adds that he was Swiss and gives his first name as Johann, and adds that Āryabhaṭa had suspected the irrationality in 499 CE.
• Why finitely many fractions cannot do it (an added argument, from printed ingredients). Two steps, both licensed by §3.4, p. 48: any sum, difference, product or quotient of fractions is again a fraction, so any finite combination of fractions is a fraction; and no fraction equals π. Therefore the sum must be endless. Verified as valid. The chapter asserts that one cannot use a single fraction and must use an infinite sum, and does not argue it; this is the argument.
• Mādhava's series (§3.5.3, p. 56). Printed as π = 4 × (1 − 1/3 + 1/5 − 1/7 + …). The bracket's terms are the reciprocals of the odd numbers 1, 3, 5, 7, 9, 11 and onwards, with signs alternating from positive. Verified partial sums, as inputs for the figure to reproduce: one term gives 4; two terms give 4 − 4/3 = 8/3, about 2.667; three give 8/3 + 4/5 = 52/15, about 3.467; four give 52/15 − 4/7 = 304/105, about 2.895; five give about 3.339. The pattern to show: the totals straddle π, alternately over and under, and the gap narrows slowly.
• The slowness is the point worth making (not in the book). Verified: after five terms the running total is still wrong in the first decimal place, while Āryabhaṭa's single fraction was already right to three. A student watching this should be told plainly that the series is not a faster route to a decimal; it is an exact description, which is a different virtue.
• What an infinite sum means (§3.5.3, pp. 56–57). Printed: the value approached as more and more terms are added, starting from the first, with the chapter contrasting this against adding 2, 100 or a lakh of terms and deferring the formal treatment to higher classes.
• π on the number line (Fig. 3.12, p. 57, read from the printed page). π is marked above the axis between the integers 3 and 4 and just left of 7/2, which is also marked above the axis. Verified: π ≈ 3.1416 and 7/2 = 3.5, so the drawn order is right. There is no figure of a circle anywhere in §3.5.3.
• Rationals crowding π (Exercise Set 3.4, Q5, p. 53, brought forward). Find three rationals between 3.1415 and 3.1416. Verified: π lies inside that interval, so this exercise is quietly producing rationals on both sides of an irrational number — a good closing beat, and the chapter does not point it out.

Figures to have open

• A convergence plot for section 9: a horizontal line at π with the partial sums dropped in one by one as points. The chapter prints no such figure and this is the single most valuable visual in the topic, because it turns "closer and closer" into something watchable.
• Three circles unrolled against their diameters for section 1. Standard schematic; the chapter prints no circle in §3.5.3, which I confirmed on p. 56.
• A one-axis timeline carrying 499 CE, the 14th century and 1761. Standard schematic.
• Fig. 3.12 (p. 57) redrawn for the closing beat, with π and 7/2 both above the axis and the integers below, as printed.
• An annotated fraction frame for 3927/1250 showing the division and the denominator's prime factors, so the cross-reference to §3.6.1's criterion lands.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics, printed Chapter 3 on the world of numbers. Section §3.5.3, printed heading "The Story of Pi (π) and Madhava's Infinite Series", runs pp. 56–57, with the series and the meaning of an infinite sum spanning the page turn.
• The decimal expansions of π and √2 are printed later, at §3.6.3, p. 61.
• Fig. 3.12, p. 57.
• Exercise Set 3.4, Q5, p. 53, used here as a closing beat.
• Chapter Summary, p. 67, fourth bullet, which names Lambert in full, calls him Swiss, and reports Āryabhaṭa's suspicion in 499 CE.