π is irrational, and what that rules out

What to assume they know

• Why C/D is the same number for every circle — that circumference over diameter is one constant
• Cornering π: from inscribed polygons to Mādhava's exact series — the historical fractions offered for π, and Mādhava's series
• A rational number as a quotient of two integers with non-zero denominator
• Long division producing a decimal expansion, and the two things that can happen: it stops, or it repeats
• From Class 8, that √2 is irrational
• The symbols = and ≈, used with intent

What they should be able to do

• State what a rational number is, in the form the chapter uses, and give examples in each of the three shapes the chapter offers
• Describe what the decimal expansion of a fraction always does, and demonstrate it on 1/3, 1/11 and 1/7
• State that π's expansion does neither, and identify this as the observable symptom of irrationality rather than a proof of it
• Explain the chapter's argument that no fraction can be the best approximation to π, and supply the reason it works
• Use ≈ and ≠ correctly for π against 22/7, and for √2 against 1.414
• Distinguish irrational from three things it is often confused with: unknown, incomputable, and not exactly expressible
• Name Lambert and the year 1761, and say why the chapter defers the proof
• Explain why Pi Day and Pi Approximation Day fall on the dates they do

Where it usually goes wrong

• "Irrational means we do not know it." We know π to hundreds of trillions of digits (p. 123). Irrationality is a fact about the number, not a report on our ignorance.
• "Irrational means it cannot be written down exactly." Mādhava's series writes it exactly, two pages earlier in the same chapter. What cannot be done is writing it as one integer over another.
• "π = 22/7." The chapter goes out of its way to print both π ≈ 22/7 and π ≠ 22/7 on the same page (p. 124), because this is the error it expects. 22/7 is the value the exercise sets instruct you to use; it is not π.
• "The digits look random, so nobody could prove anything about them." Lambert proved irrationality in 1761 without knowing much about the digits at all. The digits are the symptom; the proof is elsewhere.
• "There must be a closest fraction, we just have not found it." If a fraction were closest, halving its gap to π by truncating π's own expansion would produce a nearer one. The absence of a best fraction is a consequence, not a research frontier.
• "Irrational means not constructible, so this is why you cannot square the circle." This is the most seductive error available here, and it is wrong. √2 is irrational and perfectly constructible. Squaring the circle fails for a stronger reason than irrationality, and the chapter does not give that reason — see Notes.
• "Every non-terminating decimal is irrational." 1/3 does not terminate. It is non-terminating and repeating that keeps a number rational; irrational means neither.

Questions to check understanding

• Decide whether a given number is rational, and justify by exhibiting a quotient of integers or by naming the obstruction
• Long-divide a given fraction and state the length of its repeating block
• Complete a statement with =, ≈ or ≠, and say why the other two are wrong
• Given two approximations to π, say which is nearer and quantify the error
• Short answer: explain why no fraction is the closest fraction to π
• Explain why 14 March and 22 July are both associated with π
• Common trap to set deliberately: ask whether a non-terminating decimal must be irrational

Examples worth working on the board

Inputs, not answers. Values marked Verified are worked out here; the chapter prints no answers and this volume has no appended answer key.

• The three fractions of §6.3 (p. 123). 1/3, 1/11 and 1/7, printed with their expansions to show the repeating rhythm. Verified: 1/3 = 0.333… with a block of length 1; 1/11 = 0.0909… with a block of length 2; 1/7 = 0.142857 142857… with a block of length 6.
• The eleven decimal places (p. 123). The chapter prints 3.14159265358 as what Mādhava reached. Verified: π = 3.14159265358979…, so those eleven places are correct.
• The mnemonic (Fun Fact, p. 124). The chapter gives a seven-word English sentence whose word lengths are counted off as digits, and prints the resulting sequence 3, 1, 4, 1, 5, 9, 2. Verified: those are the first seven digits of π. The explanation may use the printed sentence as a single named artefact; it must not reproduce surrounding prose around it.
• The p. 124 cartoon. A locomotive whose carriages are the digits of π running off into the distance past a level crossing, with two children waiting at the barrier. Its content is the digit string, and the digits shrink as they recede — a good visual for "it does not stop". Redraw rather than reproduce.
• The four printed statements of intent (p. 124). π ≈ 22/7 and π ≠ 22/7; √2 ≈ 1.414 and √2 ≠ 1.414. Verified: 22/7 = 3.142857…, which differs from π at the third decimal place; 1.414² = 1.999396, which is not 2.
• The chapter's no-best-fraction argument (p. 124). As printed: since π is irrational, given any fraction close to π another fraction is closer. Verified, and worth working: 22/7 is out by about 1.3 × 10⁻³, and 355/113 by about 2.7 × 10⁻⁷, which is already a beating by a factor of nearly five thousand. The general reason is that a fraction differing from π cannot be π, so the gap is a positive number, and a decimal truncation of π taken far enough is a fraction inside that gap.
• 355/113 (p. 124). Named on this page as the much better approximation, and described on p. 122 as beaten by no fraction with denominator under 15 000.
• The two dates (pp. 124–125). 14 March, written 3-14 in North American order, and 22 July, written 22-7 in Indian order.
• What the chapter says about the proof (p. 123). Āryabhaṭa's and Zu Chongzhi's writings suggest they took π to be irrational; Lambert established it in 1761; and the proof needs mathematics reserved for much later study.
• The comparison. √2 is irrational and is nonetheless the exact diagonal of a unit square, drawable with a straight edge and compasses. π is irrational and worse than that — it satisfies no polynomial equation with whole-number coefficients at all — which is the actual reason a circle cannot be squared with those tools. Mādhava's series (p. 122) writes π exactly. So irrationality forbids one very specific thing and nothing else.

Figures to have open

• A long-division worked for 1/7 with the remainders written in a column, so the reason a block must repeat is visible rather than asserted. Standard schematic; the chapter prints the expansions but not the division.
• A number-line zoom: π marked, then 22/7 and 355/113 placed at successive magnifications so the second is seen to be far nearer. Not in the book.
• A digit train receding — redraw from the p. 124 cartoon rather than reproducing it. Not essential to the argument, but it is the chapter's own image for endlessness and children remember it.
• A unit square with its diagonal marked √2 beside a circle with a square of equal area sketched in dashes and struck through. Not in the book, and the whole load of section 9 rests on it.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics (NCF-SE 2023), Chapter 6, §6.3 "π Is Irrational" (pp. 123–124), continuing into the Fun Fact panel on p. 124 and its last line, which runs onto p. 125.
• The definition of a rational number restated on p. 123, with its three printed examples.
• Chapter Summary (p. 154) states flatly that π is an irrational number.
• Back-reference inside the book: rational numbers, decimal expansions and the repeating-block argument are Chapter 3, "The World of Numbers", pp. 41–67 — in particular the remainder argument for why a fraction's expansion must repeat.
• Mādhava's exact series, needed for section 9, is on p. 122.
• §6.3 carries no figure. The only artwork on its pages is the unnumbered cartoon on p. 124 — checked on the printed pages.