Keep a Running List of Assumed Facts — The Contradiction Must Clash With One of Them

Partway through a proof by contradiction, the algebra has produced something that looks suspicious. Is it the contradiction? Which earlier line is it fighting with? If you cannot answer that instantly, you have been proving blindly — running the algebra without tracking the targets it might collide with.

A simple habit fixes this. Before you start derivations, list every fact you have assumed in this proof and every known theorem you intend to cite. Write the list at the top of your scratch paper. Then when a suspicious line appears, your eyes go straight to the list and ask: which of these does this clash with? If none, keep going. If one, you are done.

The habit

Separate your proof into three regions on the page:

The assumption box. Every statement you have assumed in this proof — the main assumption-for-contradiction, plus any supporting ones (e.g., \gcd(p, q) = 1 for a "lowest terms" proof).
The known-facts box. Theorems you can cite — fundamental theorem of arithmetic, Euclid's division lemma, whatever is in play for this problem.
The derivation column. The actual algebraic work, one line per step.

Every time you produce a new line in the derivation, scan the two boxes above and ask: does this line deny any of them? If yes, flag it and close the proof. If no, continue.

Why this is worth the overhead: the whole point of contradiction is to collide a derived fact with something you have previously asserted. If you do not track what you have asserted, you are forced to scan the whole proof every time a suspicious line appears — or worse, you miss the collision entirely and keep deriving after the proof should have ended. The list makes the collision a pattern-match instead of a search.

What goes in the assumption box

Every single line that starts with "Let ...", "Suppose ...", "Assume ...", or "We may take ... without loss of generality." Also implicit assumptions that follow from how you set up the proof. In the \sqrt{2} proof the box contains:

\sqrt{2} = p/q (the contradiction-for assumption).
p, q are positive integers.
q \neq 0.
\gcd(p, q) = 1 (the "lowest terms" addendum).

In Euclid's infinite-primes proof the box contains:

There are finitely many primes: p_1, p_2, \ldots, p_n (the contradiction-for assumption).
Every integer greater than 1 has a prime factor (a supporting fact, often justified by well-ordering).

That is it. Every proof has a short, finite list. Writing it out takes thirty seconds and saves you ten minutes of scanning later.

What goes in the known-facts box

Theorems you are allowed to use without re-proving. For most number-theoretic contradiction proofs:

Euclid's lemma: if p is prime and p \mid ab, then p \mid a or p \mid b.
Fundamental theorem of arithmetic.
Every positive integer has a unique prime factorisation.
\gcd(a, b) = 1 means no prime divides both.

For an \sqrt{n} irrationality proof at school level, you would write these in the box at the top: "Prime p divides a^2 iff p divides a." Then when the algebra produces a new divisibility, you instantly know whether it collides with the lowest-terms assumption.

The scan pattern

When a new line of algebra appears, run this scan:

Step 1. Read the new line. What is its claim? ("p is even," "2 \mid p^2," "\gcd(p, q) \geq 2.")

Step 2. Look at the assumption box, top-to-bottom. Does the new line deny any entry?

Step 3. If yes, stop. That is your contradiction, and the pair is the new line paired with the assumption it denied.

Step 4. If no, look at the known-facts box. Does the new line deny a known theorem? (Rare, but happens — e.g., 2^p = 3^q denies unique factorisation for positive p, q.)

Step 5. If no, the line is a derived fact, not a contradiction. Add it to a running list of derived facts (a fourth column, if you want) and continue.

This scan is a few seconds per line once you have the habit. It also forces you to be honest: if you cannot point to which entry a line contradicts, the line is not a contradiction yet — see When Is It Really a Contradiction (vs Just Something Weird)?.

A side-effect: clean proofs

The box habit also improves the write-up of your proof. When you finish and write the proof in paragraph form, you can reference the assumption box explicitly: "This contradicts the assumption that \gcd(p, q) = 1." That is a clean, precise flag. A reader knows exactly which earlier line is being denied because you wrote it at the top in the first place.

Students who skip the box habit often have vague closing lines: "which is impossible." Vague flags leave graders (and themselves, months later) confused about which exact statement was being denied.

The box method in action

Consider proving that \sqrt{3} is irrational. Set up the proof with visible boxes.

Assumption box.

\sqrt{3} = p/q for integers p, q with q > 0.
\gcd(p, q) = 1.

Known-facts box.

A prime p divides a^2 iff p divides a.

Derivation column.

Line 1: Square both sides: p^2 = 3q^2.
Line 2: 3 \mid p^2 (since p^2 = 3q^2). Scan the boxes. Does 3 \mid p^2 deny any entry? No. Continue.
Line 3: 3 \mid p (by the known fact). Scan. No collision with boxes. Continue.
Line 4: Write p = 3k, then 9k^2 = 3q^2, so q^2 = 3k^2. Scan. Derived. Continue.
Line 5: 3 \mid q^2, so 3 \mid q. Scan. Does 3 \mid q deny anything? Combined with line 3 (3 \mid p), we have 3 \mid \gcd(p, q), so \gcd(p, q) \geq 3 > 1. That denies assumption box entry 2 directly.

Flag: "We have derived 3 \mid \gcd(p, q), which contradicts the assumption \gcd(p, q) = 1. Therefore \sqrt{3} is irrational." \blacksquare

At every step, the collision check is a quick look at a short list. The proof is impossible to get lost inside, because there are only two boxes to scan and neither changes during the proof.

When the list grows

Some proofs involve many supporting assumptions — a case split here, a "without loss of generality" there. Keep the box tidy: number each entry, and cite by number when you derive a conclusion. "Combining line 7 with assumption box entry 3 gives a contradiction." That precision is what graders reward and what you need six months later when you re-read the proof.

If the box exceeds ten entries, your proof is probably overcomplicated — look for a cleaner formulation. Contradiction proofs typically need only two or three assumption-box entries plus a handful of known facts.

The takeaway

Proof by contradiction is not algebra followed by "hope a wall appears." It is a directed argument: you are steering the algebra toward a collision with a pre-announced target. Writing the target down — the assumption box and the known-facts box — is the small discipline that turns hopeful wandering into deliberate argument. Thirty seconds at the top of your page. Every time.