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Chapter 3 · The World of Numbers

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

यह वीडियो हिंदी में भी · Watch in Hindi

The gaps rationals cannot fill9 min

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9 min.

Also recorded in Hindi.Englishहिन्दी

Draw a square one unit on a side and draw its diagonal. That length cannot be written as one whole number over another — not hard, impossible.

The idea

The proof does not test fractions one by one — no amount of testing could settle an infinite list. It shows instead that the act of writing √2 in lowest terms destroys itself: assume a co-prime writing exists, and the algebra forces both numbers to be even, which is the one thing lowest terms forbids. The contradiction is not with arithmetic; it is with the assumption's own bookkeeping, and that is what makes one page of algebra decisive against every fraction at once. This is also where a geometrical length and an arithmetical impossibility meet: a square you can draw has a diagonal you cannot write down.

What you 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

Words to know

TermDefinition in one lineFirst introduced
Irrational Numbersnumbers on the line that no ratio of integers can expressprinted in bold in §3.5, p. 53
Proof by Contradictionassuming the opposite of a claim and deriving an impossibility from itprinted in bold in §3.5.1, p. 54
co-primesharing no common factor other than 1printed in §3.5.1, p. 54, and earlier at §3.4, p. 48
simplest formthe writing of a fraction in which the two parts are co-primeprinted in §3.5.1, p. 54
Baudhāyana–Pythagoras Theoremthe relation between the squares on a right triangle's three sidesprinted in §3.5, p. 53
ŚhulbasūtraBaudhāyana's manual for setting out geometric fire altarsprinted in italics in §3.5, p. 53
Baudhāyanathe author the chapter places at around 800 BCE, who met lengths fractions could not giveprinted in §3.5, p. 53
Hippasusthe member of the Pythagorean school credited with the first such proofprinted in §3.5.1, p. 53
Pythagorean schoolthe community Hippasus belonged to, placed at around 400 BCEprinted in §3.5.1, p. 53
even square lemmaan added name for the step that an even square forces an even rootan added term; the chapter states this fact inside Step 4 without naming or proving it
reductio ad absurduman added alternative name for the same proof patternan added term; the chapter uses only its English name

Where people slip up

  • "√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.

Worked answers to this chapter’s exercises · this video explains End-of-Chapter Exercises Q2

Transcript1,316 words

Around eight hundred years before the common era, someone laying out altars ran into a length they could not measure. The manual they left is called the Shulbasutra, and it is a builder's handbook. How to set out a square. How to raise a right angle. How to double an area. The tools are a cord and some pegs, and the cord is marked in units. Almost every length in that work divides cleanly into the units, or into some fraction of them.

One did not. And it was not that the fraction was awkward. It was that no marking of the cord, however fine, would ever land exactly on the end. Here is where that length comes from, and it takes one line. Draw a square whose side is exactly one unit. Draw the diagonal across it. The theorem about right angled triangles, the one that carries Baudhayana's name alongside Pythagoras's, says the square on the long side equals the squares on the other two added together. One plus one. Two.

So the diagonal squared is two, and the diagonal is the square root of two. That is the whole of the geometry here. From now on there is no square, no diagonal and no cord. There is a number, and a question about how it can be written. And the question is sharper than it looks, because of what people believed at the time. The belief was that every length you can draw is a ratio of two whole numbers. Not that the ratio is easy to find. Just that it is there, waiting.

Under that belief, the diagonal of a unit square is some fraction, and the only work left is to go and find it. What we are about to do is not find it. We are going to show that the search cannot succeed. Those are very different claims, and the difference between them is the whole point of what follows. So start by taking the belief seriously, and hunt. For every bottom number from one up to twenty thousand, work out what the top would have to be, and check whether it is a whole number. Twenty thousand bottoms. Nothing.

Some get close. Across two thousand of them the nearest miss was one, a single unit away from landing. But close is not equal. And here is why that search settles nothing at all. Run the identical hunt for four instead of two. It finds an answer at the very first bottom, two over one, and then at every one of the twenty thousand. So the search is perfectly capable of finding something. It simply did not find this. And running it longer never turns not yet into never.

To get from not yet to never you need a different shape of argument, and it has three parts. Assume the opposite of what you want to show. Reason forward carefully, leaving no gaps. Arrive at something impossible. If every step was sound and the end is impossible, then the beginning was impossible too. That is the whole method. It is not a trick and it is not a last resort. When you want to show that something does not exist, it is very often the only thing available, because you cannot check an endless list one entry at a time.

So assume the opposite. Suppose the square root of two is a fraction. Write it as a top over a bottom. And here is the part that matters: write it in lowest terms. We are allowed to do that, because any fraction can be brought to lowest terms. Divide both parts by whatever they share, and what is left shares nothing. So we may assume the top and the bottom have no factor in common at all.

Notice what that is. It is not decoration on the assumption. It is a second thing we have just promised, and it is the thing that is going to break. Now push it. Square both sides. The left becomes two. The right becomes the top squared over the bottom squared. Multiply both sides by the bottom squared, and the fractions disappear. Two times the bottom squared equals the top squared.

One equation, whole numbers only. Everything from here is arithmetic on that line. So read the left hand side. It is two times something. Which means the top squared is even. And now the step that almost always gets asserted and almost never gets shown. The top squared is even, so the top is even. Why? It looks obvious, and it costs one line. Suppose the top were odd. Then it is twice something, plus one. Square that, and you get four times something, plus four times something, plus one.

The first two pieces are even. The last one is not. So an odd number squared is always odd. A thousand and one of them were squared and checked, and not one came out even. Which means an even square cannot have come from an odd root. The top is even. So write the top as twice something, and put it back into the equation. Two times the bottom squared equals twice something, squared, which is four times that something squared.

Divide both sides by two. The bottom squared equals two times something squared. Look at the shape of that. It is the same line we had before, one level down. The bottom squared is two times a whole number, so the bottom squared is even. And by the step we have just proved, the bottom is even too. Stop and look at what is on the board. The top is even. The bottom is even. Both of them are divisible by two, which means they share a factor.

And in the very first line we promised they share no factor at all. That is the collision. Notice exactly what it collides with. Not with arithmetic. Not with the equation. With our own bookkeeping. Nine hundred pairs of even numbers were checked for sharing nothing, and not one of them managed it, because that is not a thing that can happen. So the assumption fails. There is no fraction here to find. Not one that nobody has found yet. One that cannot exist.

Now the trap, and it catches almost everybody who has just watched that. Try the same argument on eighty one. Its root is nine, and nine is nine over one, a perfectly ordinary fraction. So the argument has to fail somewhere. Where? Not at the deduction. Three divides eighty one, and three dividing the top squared really does force three to divide the top, exactly as before. That step is fine.

It fails at the ending. Hand the argument the actual pair, nine over one. The top is divisible by three. The bottom is one, and one is divisible by nothing. So the chain runs all the way to the last line and finds no collision waiting there. And an argument by contradiction with no contradiction in it proves nothing whatsoever. So what is this argument actually about? Not square roots. The number sitting under the root.

The one step that can genuinely break is the deduction. Does this divides the square force this divides the number? Scan every divisor up to forty and find out. It holds for exactly twenty five of them, and those twenty five are precisely the numbers with no repeated prime factor. Two, yes. Three, five and seven, yes. Ten, yes, and ten is not even prime. Four, no. And the smallest thing that breaks it for four is two itself, whose square is divisible by four while two is not.

Twelve, no, because twelve carries a factor of four inside it. So the argument runs straight through for three, for five, for seven. For twelve you have to pull the square factor out first.

Where this fits

Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.

Builds on

Comes up again in

Either side of this one

The book

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