Proof by contradiction: why √2 cannot be a ratio of integers

What to assume they know

• What "rational" means, and why the denominator cannot be zero — the definition of a rational number and the co-prime representative
• The Baudhāyana–Pythagoras Theorem on a right triangle
• Squaring and square roots of small numbers
• Even and odd numbers, and that an even number can be written as twice an integer
• Manipulating an equation by squaring both sides and by multiplying or dividing both sides by the same non-zero quantity
• The logical form "if this were true, then that would follow"

What they should be able to do

• Derive the length of a unit square's diagonal from the Baudhāyana–Pythagoras Theorem
• State what an irrational number is, in the chapter's terms
• State the structure of a proof by contradiction: assume the opposite, reason, and reach an impossibility
• Reproduce the eight printed steps of the proof in order
• Identify which step carries the geometry and which steps are pure algebra
• Justify the claim that an even square forces an even root, which the printed proof asserts without argument
• Identify the exact contradiction reached, and say which earlier assumption it contradicts
• Explain why the argument does not generalise to every square root, using a perfect square as the counter-case
• Adapt the argument to another non-square integer, as the chapter's prompts ask

Where it usually goes wrong

• "√2 is about 1.414, so it is a decimal and decimals are fractions." 1.414 is a rational approximation, not √2. The proof shows that no fraction, however large its parts, hits it exactly. The decimal signature of the difference is §3.6's business.
• "Nobody has found a fraction equal to √2 yet." This is the misreading the proof exists to prevent. It is not a search that has come up empty; it is a demonstration that the search cannot succeed.
• "Proof by contradiction is a trick." It is the only practical way to establish that something does not exist. You cannot check infinitely many fractions, so you show that supposing one exists breaks a rule you were relying on.
• "The contradiction is that 2q² = p² is impossible." That equation is not the problem — plenty of integer pairs satisfy it in the reals. The contradiction is with the co-primality chosen in Step 1. Students who cannot name which assumption broke have not followed the proof.
• "If p² is even then p might be even or odd." This is the step that needs the short odd-square argument. Give it; do not assert it.
• "All square roots are irrational." √81 is 9. Exercise Set 3.5, Q3 puts a perfect square first, deliberately, and an explanation that has just proved √2 irrational will produce students who answer that item wrongly unless warned.
• "The proof needs the geometry." It does not. The square supplies the motivation and the number; after d² = 2 the geometry is finished.
• "Simplest form is a tidiness convention, so contradicting it is not serious." It is the hinge of the whole argument. If any fraction can be reduced to lowest terms — and it can — then a writing that cannot be is impossible.

Questions to check understanding

• Prove that √2 is irrational, giving the steps in order
• Prove that √5 is irrational — set as end-of-chapter Q2 and a standard board item
• Prove that √3 is irrational
• State which assumption the contradiction contradicts
• Explain why an even square must have an even root
• Classify a list of surds and decimals as rational or irrational, including at least one perfect square
• Explain in words what a proof by contradiction establishes and why direct checking will not do
• Derive the diagonal of a unit square and say why the result cannot be written as a fraction

Examples worth working on the board

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

• Fig. 3.10 (p. 53), read from the printed page. A filled yellow square with a black diagonal running corner to corner. The side is labelled 1 twice — once along the bottom edge and once up the right edge — and the diagonal carries the label √2. Two further 1s and a 2 appear in the extracted text stream near this figure; on the printed page the square itself carries the side labels and the diagonal label, and nothing else. It is a small inline figure set into the text column, not a full-width diagram.
• The geometry (§3.5, p. 53). Inputs: a square whose side is exactly 1 unit, and the theorem in the form 1² + 1² = d². The chapter concludes d² = 2 and therefore d = √2. Verified. Note that this is the only geometrical step in the topic; everything after it is arithmetic.
• The historical setting (§3.5, p. 53). Baudhāyana composed his Śhulbasūtra, described as a manual for constructing geometric fire altars, around 800 BCE, and met lengths that no fraction would give. The Greeks met the same difficulty a few centuries later. Hippasus, of the Pythagorean school, around 400 BCE, produced the first proof.
• Think and Reflect (p. 53). Asks whether √2 can be written as a ratio, and answers that it cannot, pointing back to Class 8 and forward to §3.5.1. Use it as the explanation's hinge between the geometry and the proof.
• The eight printed steps (§3.5.1, pp. 54–55). Hand these over in order, since the explanation's sections 5 to 9 follow them:
  • Assume √2 is rational, so it equals p/q in simplest form, with the two parts co-prime and the lower part non-zero.
  • Square both sides: 2 = p²/q².
  • Multiply through by q²: 2q² = p².
  • Since p² is twice an integer, p² is even; and since an even square has an even root, p is even. Write p = 2k with k an integer.
  • Substitute: 2q² = (2k)², so 2q² = 4k².
  • Divide by 2: q² = 2k².
  • So q² is twice an integer, hence even, hence q is even.
  • Both p and q are even, so both share the factor 2 — contradicting Step 1, where the writing was chosen to share no factor. The chapter closes on p. 55 by concluding that since the reasoning is sound the assumption must fail, so √2 admits no fractional writing.
• The gap in Step 4, and how to close it (not in the book). The chapter asserts that an even square must have an even root and does not prove it. The proof is short and belongs in the explanation: if p were odd, write p = 2m + 1; then p² = 4m² + 4m + 1, which is odd. So an even p² rules out an odd p. Verified. Without this, the proof's central step is an unsupported claim, and Class 9 students notice.
• The perfect-square trap (Exercise Set 3.5, Q3(i) and Q3(ii), p. 61). Inputs: √81 and √12, to be classified as rational or irrational. Verified: √81 = 9, which is rational, so the argument pattern of §3.5.1 must fail somewhere for 81. Be careful where. It is tempting to say the deduction step breaks because 81 is not twice anything, but that only describes what happens if you transcribe the printed words with 81 swapped in, and it is the wrong general lesson: adapt the argument properly, using 3 in place of 2, and the deduction step goes through perfectly well, since 81 dividing p² does force 9 to divide p. It also goes through for an even perfect square like 4 or 36, where the parity step is untouched. What actually fails, for every perfect square, is the ending: the chain never collides with the lowest-terms assumption. It closes quietly at q = 1 and p = 9, and a proof by contradiction with no contradiction proves nothing. √12 is irrational. Use this pair to show that the proof is about the specific number under the root, not about roots in general.
• The extension prompts. Think and Reflect, p. 55: prove √3 irrational the same way, and ask whether the approach also works for √5, √7 and √10. End-of-chapter Q2, p. 64: prove √5 is irrational. Verified: the argument runs through for 3, 5, 7 and 10 with "even" replaced by "divisible by the relevant prime factor", and the version for 10 needs a little more care because 10 is not prime. Flag that the chapter asks the question and supplies no worked version, so the explanation is choosing what to show.

Figures to have open

• Fig. 3.10 (p. 53) redrawn as a unit square with its diagonal, both side labels and the diagonal label. The chapter's own figure; simple enough to redraw and it should not be reproduced from the page.
• A three-slot frame for the structure of a contradiction argument, reusable when the explanation reaches the extension prompts. Standard schematic.
• A step ladder holding the eight printed steps, so that section 9 can point back at Step 1 while Step 8 is shown. This is the figure the proof most needs and the chapter prints the steps as running text with no diagram.
• A dot or block picture of an odd number squared, showing the leftover unit, to carry the section 7 argument visually. Standard schematic.
• 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, printed heading "Irrational Numbers", opens on p. 53 with Fig. 3.10 and a Think and Reflect box.
• §3.5.1, whose printed heading names the proof of irrationality and then carries the surd as a mathematical symbol, runs pp. 53–55. The eight steps sit on pp. 54–55 and the conclusion at the top of p. 55.
• Think and Reflect, p. 55, extending the method to other surds.
• Exercise Set 3.5, Q3(i) and Q3(ii), p. 61. End-of-chapter exercises, Q2, p. 64.
• Chapter Summary, p. 67, fourth bullet, restates the irrationality of √2, credits Hippasus with around 400 BCE, and names the method.