Proof by Contradiction — Method & Classic Example

Proof by contradiction 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.

This method is useful when the negation gives you something concrete to work with. A classic case is proving that $\sqrt{2}$ is irrational: once you assume it is rational, you can write it as a fraction and follow the consequences.

How proof by contradiction works

Suppose the statement you want to prove is $P$.

In a contradiction proof, you begin by assuming $P$ is false. Then you reason from that assumption until you reach a contradiction, such as:

$$1 = 0$$

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

At that point, the negation of $P$ cannot be correct, so $P$ must be true.

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.

When to use 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.

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.

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

A classic proof by contradiction shows that $\sqrt{2}$ is irrational.

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.

Why the contradiction is decisive

The contradiction is not just "something feels wrong." It 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.

Common mistakes in proof by contradiction

One common mistake is assuming the opposite too vaguely. You need the actual negation of the statement, not just a nearby statement that sounds similar.

Another mistake is reaching a contradiction because of an invalid algebra step. In that case, the contradiction proves only that the algebra was wrong, not that the original claim was true.

A third mistake is not naming the fact that is being contradicted. Good proofs make the conflict explicit: a parity rule, a definition, a minimality condition, or a previously proved theorem.

It is also easy to hide weak reasoning behind the phrase "this is a contradiction." If you cannot point to the exact fact that failed, the proof is probably incomplete.

A simple proof by contradiction template

For many beginner proofs, the structure looks like this:

1. Assume the statement is false.
2. Translate that assumption into a concrete form.
3. Push the consequences step by step.
4. Reach a contradiction with a known fact.
5. Conclude the original statement is true.

If step 2 is weak, the proof usually stays fuzzy. The strongest contradiction proofs often come from turning the negation into something very concrete.

Try a similar contradiction proof

Try the claim "there is no smallest positive rational number." Assume there is one, call it $r$, and ask what happens to $\frac{r}{2}$. If you want to check your reasoning step by step on a similar proof, try your own version in GPAI Solver and compare each claim with the contradiction you are aiming for.