Proof by contradiction works by assuming the opposite of what you want to prove, then showing that this assumption leads to something impossible. Once you’ve reached that impossibility, the assumption must be false, which means your original statement must be true. It’s one of the most powerful and widely used proof techniques in mathematics, and once you see the pattern, you can apply it to a huge range of problems.
Why Contradiction Proofs Work
The technique rests on a simple logical principle: every statement is either true or false. There’s no middle ground. This is called the law of excluded middle, and it means that if you can rule out one possibility, the other one must hold. So if you assume a statement is false and that assumption blows up into nonsense, the statement has to be true.
There’s a second principle doing work here: the law of non-contradiction, which says a statement can’t be both true and false at the same time. When your assumption forces you into claiming something that’s simultaneously true and false, you’ve hit a wall. That wall is the contradiction, and it’s what makes the whole proof click.
The Three Core Steps
Every proof by contradiction follows the same skeleton:
- Assume the opposite of the statement you want to prove. If you’re trying to prove “X is true,” you begin by supposing “X is false.”
- Reason forward from that assumption using definitions, algebra, known theorems, or any valid logical steps. Keep going until you reach a conclusion that contradicts something you know to be true, or contradicts the assumption itself.
- Conclude the assumption is false. Since the assumption led to an impossibility, it can’t hold. Therefore, the original statement must be true.
The hard part is always the middle step. You need to find the right chain of reasoning that produces the contradiction, and that takes practice. But the framework never changes.
What “Assume the Opposite” Actually Means
For a simple statement like “the square root of 2 is irrational,” the opposite is straightforward: assume the square root of 2 is rational. But many theorems take the form “if P, then Q.” In that case, assuming the opposite means assuming P is true and Q is false at the same time. You keep the premise and negate the conclusion.
For example, if you want to prove “if n² is even, then n is even,” you’d assume n² is even (keep the premise) and n is odd (negate the conclusion). Then you work from there until something breaks.
Getting the negation right is critical. If you negate the wrong thing, you’ll either prove something different from what you intended or chase reasoning that goes nowhere.
A Classic Example: Infinitely Many Primes
One of the oldest and most elegant contradiction proofs comes from Euclid. The claim is that there are infinitely many prime numbers.
Start by assuming the opposite: there are only finitely many primes. That means you could write them all down in a complete list: 2, 3, 5, 7, 11, and so on, up to some largest prime. Now multiply every prime in that list together and add 1. Call the result N.
N is bigger than any prime on your list, so if your list is truly complete, N itself isn’t prime. That means some prime on the list must divide N evenly. But here’s the problem: every prime on the list divides the product you started with (since N was built by multiplying them all together), and N is that product plus 1. If a number divides both the product and N, it would have to divide the difference between them, which is 1. No prime divides 1.
That’s the contradiction. You assumed the list of primes was complete, but then constructed a number that no prime on the list can divide. The assumption of finitely many primes is impossible, so there must be infinitely many.
A Simpler Example: Square Root of 2
This is often the first contradiction proof students encounter. The claim is that the square root of 2 is irrational, meaning it can’t be written as a fraction of two whole numbers.
Assume the opposite: the square root of 2 is rational. Then you can write it as a fraction a/b, where a and b are whole numbers with no common factors (the fraction is fully reduced). Squaring both sides gives you 2 = a²/b², which rearranges to a² = 2b². This tells you a² is even, and since the square of an odd number is always odd, a itself must be even. Write a as 2k for some whole number k.
Substituting back: (2k)² = 2b², so 4k² = 2b², so b² = 2k². Now b² is even, which means b is also even. But if both a and b are even, they share a factor of 2. That contradicts the assumption that a/b was fully reduced. The square root of 2 can’t be written as a fraction, so it’s irrational.
Contradiction vs. Contrapositive
These two techniques look similar and are easy to confuse, but they work differently. A contrapositive proof applies only to “if P, then Q” statements. Instead of proving P implies Q, you prove that “not Q” implies “not P,” which is logically equivalent. You never reach a contradiction. You just prove a different (but equivalent) statement directly.
Proof by contradiction is more general. You can use it on any statement, not just conditional ones. And instead of proving an equivalent statement, you assume the negation and chase it until it collapses. If you’re proving “if P, then Q” by contradiction, you assume P is true and Q is false, then derive an impossibility. With contrapositive, you’d assume Q is false and directly show P must also be false, no impossibility needed.
When you have a choice, contrapositive proofs are often considered cleaner because they don’t require you to hunt for a contradiction. But many results, like the irrationality of the square root of 2 or the infinitude of primes, don’t lend themselves to contrapositive reasoning. Contradiction is the natural tool.
Tips for Writing Your Own
Start every contradiction proof by clearly stating your assumption. A phrase like “Suppose, for the sake of contradiction, that…” signals to your reader (and to yourself) that you’re entering the contradiction framework. This is not just a formality. It keeps your logic organized and makes the final step, where you identify the contradiction, land cleanly.
Next, look for what “known fact” your assumption might conflict with. Sometimes the contradiction will be with a basic definition (a prime can’t equal 1). Sometimes it contradicts your own assumption (the fraction was supposed to be fully reduced). Before you start writing the proof, it helps to have at least a rough idea of where the contradiction will come from. Work backward from that target.
Keep your logical chain tight. Every line should follow from the previous one, and each step should be justified by a definition, an axiom, or a previously proven result. Loose reasoning is the most common mistake in contradiction proofs because you’re working in a hypothetical world where your assumption is true, and it’s easy to accidentally use the thing you’re trying to prove.
Finally, close the proof explicitly. State what the contradiction is, name both sides of it (“this means b is both even and odd” or “this requires a prime to divide 1”), and conclude that the original assumption was false. A reader should never have to guess where the contradiction happened.
When To Reach for Contradiction
Contradiction is especially useful when a direct proof would require you to show something doesn’t exist or can’t happen. Proving a number is irrational, proving a set is infinite, or proving that no object with certain properties can exist are all natural fits. In each case, directly constructing the proof is awkward, but assuming the opposite gives you something concrete to work with: a fraction to manipulate, a finite list to examine, an object whose properties you can push until they break.
It’s also a good fallback when you’ve tried a direct approach and gotten stuck. Flipping the assumption often opens up new avenues of reasoning that weren’t available before, because you suddenly have an extra piece of information (the negation) to combine with everything else you know.