Why √2 is neither a terminating decimal nor a fraction

What to assume they know

• The isosceles right triangle: why its hypotenuse must be a√2 — √2 arrived at as the hypotenuse of a unit isosceles right triangle, and as the side of a square of area 2
• A square's area is side × side, so a longer side means a larger area — the monotonicity the whole bounding table relies on
• Multiplying decimals, and knowing how many places the product carries
• The last digit of a product depends only on the last digits of the factors
• Prime factorisation of a whole number, and that it is unique
• That a square number's prime factorisation uses every prime an even number of times — established in earlier work on square numbers
• Reading m/n as a fraction of two counting numbers

What they should be able to do

• Bound √2 between 1 and 2 by comparing the areas of squares, and name 1 the lower bound and 2 the upper bound
• Tighten the bound to tenths, hundredths and thousandths by squaring candidates, and read the chapter's three bounding boxes
• Explain why the tightening never terminates, using the last-digit argument
• State the conclusion that the decimal expansion of √2 does not stop
• Set up the fraction assumption, reach 2n² = m², and derive the contradiction from the parity of the exponent of the prime 2
• Say which of the two results is the stronger one, and why the chapter still gives both
• Distinguish a bound from a value, and a truncated decimal from the number itself
• Attribute the fraction proof as the chapter attributes it, to Euclid's Elements, dated in the chapter to about 300 BCE

Where it usually goes wrong

• "√2 = 1.41421356." That is a truncation and it is the number the calculator had room for. Its square is not 2. Compute it and show the shortfall.
• "Non-terminating means it is not a fraction." The single most important correction in this topic, and the reason the chapter needs two separate arguments. One third is a fraction whose decimal never stops. Section 11 exists to fix this.
• "So the two proofs are the same proof twice." They are not, and the relationship runs one way: any decimal that stops can be written as a fraction — 1.414 is 1414 over 1000 — so the fraction result already rules out a stopping decimal. The decimal argument is the weaker of the two. The chapter runs it first anyway, because it is the one a student can reach with the bounding table already in hand, and that is a fair reason.
• "If we compute far enough we will find the pattern and it will repeat." The chapter does not raise repetition at all, and neither should the explanation — repeating decimals are a later topic. What it establishes is that the expansion does not stop, and that no fraction produces it.
• "Every prime appears an even number of times." Only in squares. Twelve is two twice and three once. The student needs the fact in the right form or the proof of the fraction result evaporates.
• "1.414 < √2 < 1.415 means √2 = 1.414 to three places." True but beside the point. The bound is a statement about where the number is, and it does not become a value however tight it gets.
• "You can't have a length you can't write down." You can draw this one with a ruler in one stroke — it is the diagonal of a square of side 1. Being unwritable as a decimal or a fraction is a fact about notation, not about existence. This is worth saying explicitly, because the two proofs can otherwise leave a student feeling the number has been argued away.
• "2 has a last digit, namely 2, so the last-digit argument fails." The argument is about digits after the decimal point. Written as a decimal, 2 has none — or, equivalently, all zeros. Make the decimal point visible when this step is made.

Questions to check understanding

• Bound a given square root between consecutive whole numbers, showing the two squares that justify it
• Tighten a bound to one, then two decimal places, and say what each squaring established
• Decide whether a given decimal squares to a whole number, using only its last digit
• Complete the fraction proof from 2n² = m² to the contradiction, in writing
• Explain, in one or two sentences, why a decimal that never stops may still be a fraction, with an example
• Given a number, say which of the two chapter results applies to it, and which does not
• Attribute the fraction proof, and give the century the chapter gives for it
• State the chapter's bound on √2 to three decimal places from memory

Examples worth working on the board

Everything in the three bounding boxes and the two proofs is printed in the chapter; items marked derived are added here.

• The area comparison that gives the first bound (Part II, §2.3, p.38). A square of side 1 has area 1; a square of side √2 has area 2; so 1 is below √2. Written the second way: 1² = 1 and (√2)² = 2. Then a square of side 2 has area 4, which exceeds 2, so 2 is above √2. Hence 1 < √2 < 2. The chapter names 1 the lower bound and 2 the upper bound.
• Bounding box 1 (Part II, §2.3, p.38, left box of three). 1.1² = 1.21; 1.2² = 1.44; 1.3² = 1.69; 1.4² = 1.96; 1.5² = 2.25. Verdict: 1.4 < √2 < 1.5. On the printed page the two decisive lines, 1.4² and 1.5², are printed in a contrasting colour; the other three are not. The colour is doing work — it marks the pair that brackets 2.
• Bounding box 2 (Part II, §2.3, p.38, middle box). 1.41² = 1.9881; 1.42² = 2.0164. Verdict: 1.41 < √2 < 1.42. Both lines are colour-emphasised.
• Bounding box 3 (Part II, §2.3, p.38, right box). 1.411² = 1.990921; 1.412² = 1.993744; 1.413² = 1.996569; 1.414² = 1.999396; 1.415² = 2.002225. Verdict: 1.414 < √2 < 1.415. The last two lines are colour-emphasised.
• The last-digit argument (Part II, §2.3, pp.38–39). Suppose a decimal that stops, beginning 1.414…, squares to 2. Its final digit is not zero. Then the square's final decimal digit is not zero either. The chapter demonstrates with one case: a final digit of 4 gives a square whose final digit is 6, and it prints this as a row of empty boxes with a decimal point, an ellipsis, and a lone 6 in the last position. But 2 written out as a decimal has nothing but zeros after the point. So no terminating decimal squares to 2, and the expansion of √2 must run on without end.
• What the argument actually needs, and the chapter shows only one case of it (derived, and worth showing). The last digits of the nine squares 1², 2², …, 9² are 1, 4, 9, 6, 5, 6, 9, 4, 1. Not one of them is zero. That is the fact the argument turns on, and it is checkable in nine lines. The chapter checks the case 4 → 6 and leaves the rest.
• The fraction proof (Part II, §2.3, p.39, marked Try This). Suppose √2 = m/n with m and n counting numbers. Squaring, 2 = m²/n², so 2n² = m². In a square number every prime is used an even number of times. On the left, the prime 2 is used an even number of times inside n² plus once more, making it odd; on the right it is used an even number of times. That is impossible, so no such fraction exists. The chapter credits the proof to Euclid, Elements, about 300 BCE, and says it will be taken further in a later class.
• The value the chapter offers (Part II, §2.3, p.39). √2 = 1.41421356…, with the ellipsis. This is a truncation, not a value, and the ellipsis is the whole content of the line.
• Where the number came from. Section 1 should re-show §2.3's own figure — the unit square PEAR with square REST of area 2 built on its diagonal (Part II, §2.3, p.37) — so the student remembers that this number was not invented, it was measured off a drawing they made themselves.
• The SUMMARY lines (Part II, p.54). The summary records that √2 sits between 1.414 and 1.415, that it is no terminating decimal, and that it is no fraction of two positive integers.
• A test of the two results against a familiar number (derived, and the key worked contrast for section 11). One third is a fraction, and its decimal 0.333… does not terminate. So a decimal running on forever does not by itself make a number un-fractionable. Run the chapter's decimal argument on 1/3 and it correctly concludes only that 1/3 does not stop — it never claimed more.

Figures to have open

• The three bounding boxes as the chapter groups them, with the decisive rows emphasised as the printed page emphasises them (Part II, §2.3, p.38). This is the chapter's own figure and sections 4 and 5 are built on it; redraw as a schematic table.
• A number line that zooms three times: 1 to 2, then 1.4 to 1.5, then 1.41 to 1.42, with the candidate squares shown. Standard schematic, not printed in the chapter, and it is what makes the tightening feel like tightening rather than arithmetic.
• The empty-boxes schematic for the last digit of a square, with the final digit appearing (Part II, §2.3, p.38, foot). The chapter prints this; redraw it.
• A nine-row table of last digits of 1² to 9², to supply what the chapter demonstrates in one case. Standard schematic; an addition made here.
• The unit square with diagonal √2, reused from §2.3 (Part II p.37), for sections 1 and 12.
• No photograph is needed. The chapter prints no artwork in this material.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 2, "The Baudhāyana-Pythagoras Theorem", §2.3, the subheading "Decimal Representation of √2", Part II pp.38–39. The fraction material sits on Part II p.39 under a Try This marker; the three bounding boxes are on Part II p.38.
• The rest of §2.3 — the isosceles hypotenuse and the "General Solution" examples — belongs to The isosceles right triangle: why its hypotenuse must be a√2.
• Attribution printed in the chapter: Euclid, Elements, about 300 BCE, with a note that the proof will be revisited in a later class.
• The chapter's SUMMARY at Part II p.54 carries both results and the three-decimal bound.
• Where the number came from: Part II p.37, the square REST of area 2 built on the diagonal of the unit square PEAR.