Proof by Contradiction — Method & Classic Example

Why can no fraction in lowest terms ever square to give exactly $2$? Questions like this are where proof by contradiction earns its place. The method proves a claim by assuming the opposite and showing that the assumption leads to an impossibility. If the logic is valid and the contradiction is genuine, the original claim must be true.

When to reach for proof by contradiction

Proof by contradiction works best when the opposite of a claim has a rigid structure. That happens often in number theory, impossibility proofs, and some existence questions.

For example, if you want to prove that a number is irrational, assuming it is rational gives you a fraction form to work with. That extra structure often makes the contradiction easier to find than a direct proof. The negation gives you something concrete to push on.

It is less useful when a short direct proof already exists. Contradiction is a valid method, but it should clarify the argument rather than make it feel longer than necessary.

The step-by-step structure

Suppose the statement you want to prove is $P$. A contradiction proof moves through these steps:

1. Assume the statement is false. Replace $P$ with its negation and treat that as a temporary assumption.
2. Translate that assumption into a concrete form you can manipulate.
3. Push the consequences step by step, using definitions, algebra, and known theorems.
4. Reach a contradiction with a known fact, such as:

$$1 = 0$$

or a conflict with a definition, or a statement that cannot hold at the same time as an earlier step.

5. Conclude the original statement is true, because the negation cannot be correct.

The key condition is that the contradiction must come from valid reasoning and accepted facts. A result that merely looks strange is not enough. It has to be impossible under the definitions or theorems you are using. If step 2 is weak, the proof usually stays fuzzy. The strongest contradiction proofs come from turning the negation into something very concrete.

Full example: why $\sqrt{2}$ is irrational

This is the classic case, worked end to end.

Start by assuming the opposite:

$$\sqrt{2} \text{ is rational.}$$

If that were true, then $\sqrt{2}$ could be written as

$$\sqrt{2} = \frac{a}{b}$$

where $a$ and $b$ are integers, $b \neq 0$, and the fraction is in lowest terms.

Now square both sides:

$$2 = \frac{a^2}{b^2}$$

so

$$a^2 = 2b^2.$$

This tells us that $a^2$ is even. That forces $a$ to be even too, because the square of an odd integer is odd. So write

$$a = 2k$$

for some integer $k$.

Substitute that into $a^2 = 2b^2$:

$$(2k)^2 = 2b^2$$ $$4k^2 = 2b^2$$ $$2k^2 = b^2.$$

Now $b^2$ is even, so $b$ is even as well.

But now both $a$ and $b$ are even. That means both are divisible by $2$, which contradicts the assumption that $\frac{a}{b}$ was already in lowest terms.

So the original assumption was false. Therefore $\sqrt{2}$ is irrational.

Where the proof can stall, and how to check yourself

Each step has a place where beginners get stuck. Use these self-checks.

At the assumption step, the danger is assuming the opposite too vaguely. You need the actual negation of the statement, not a nearby statement that sounds similar. Self-check: state the negation precisely before continuing.

At the reasoning step, the danger is an invalid algebra step. If you reach a contradiction because of bad algebra, the contradiction proves only that the algebra was wrong, not that the original claim was true. Self-check: every line should follow from accepted rules.

At the contradiction step, the danger is not naming the fact that is being contradicted. The conflict in the $\sqrt{2}$ proof is specific:

1. The assumption says $\sqrt{2} = \frac{a}{b}$ in lowest terms.
2. The algebra shows both $a$ and $b$ must be even.
3. A fraction cannot be both in lowest terms and have numerator and denominator divisible by $2$.

That direct clash is the contradiction. Self-check: point to the exact rule that failed, whether a parity rule, a definition, a minimality condition, or a previously proved theorem. If you cannot name it, the proof is probably incomplete. It is easy to hide weak reasoning behind the phrase "this is a contradiction."

Build the skill on a new claim

A good next claim to prove this way: "there is no smallest positive rational number." Assume there is one, call it $r$, and ask what happens to $\frac{r}{2}$. Translate the assumption into concrete form, push the consequences, and watch for the exact fact that breaks. Naming that fact is what turns a vague feeling into a finished proof.