Proof by contradiction, or reductio ad absurdum, is a powerful technique for proving a statement is true. Instead of building a direct case, you take the opposite approach: assume the statement is false and show that this assumption leads to a logical impossibility. If an assumption creates an absurd outcome, the assumption itself must be wrong.

Let's formalise this. Suppose you want to prove a conditional statement of the form 'If P, then Q' (symbolically, $P \implies Q$). The direct path would be to start with P and logically derive Q. The indirect path of contradiction starts differently. You assume the conclusion, Q, is false. So, you begin with the single assumption that both P is true AND 'not Q' is true.

Once you've made your initial assumption, your goal is to follow a chain of valid logical steps until you arrive at a contradiction. This isn't just any disagreement; it's a specific kind of logical impossibility where a statement and its exact opposite are both shown to be true. This violates the fundamental Law of Non-Contradiction, a cornerstone of classical logic which states that something cannot be both true and false at the same time in the same way.

The contradiction you are searching for will always take the form 'R and not R', for some statement R. The statement R can be anything, from a simple variable to a complex expression. The key is that you have derived both R and its negation, $\neg R$, from your initial assumption. This is the logical 'break' you need.

Since your logical steps were all valid, the only possible source of this absurd conclusion is your starting assumption. Therefore, the assumption ($P \land \neg Q$) must be false. If it's false that 'P and not Q' is true, then it must be true that 'If P, then Q'.

This method provides a clear template for constructing proofs. Let's look at the formal structure.

Let's apply this to a classic example: proving that $\sqrt{2}$ is irrational This can be framed as: If $x = \sqrt{2}$, then $x$ cannot be expressed as a fraction of two integers. We start by assuming the opposite: that $\sqrt{2}$ can be expressed as a fraction $a/b$ in its simplest form. Through a series of algebraic steps, we can show that this assumption implies that a must be even, and b must also be even. But if both are even, the fraction wasn't in its simplest form, which contradicts our starting condition. The assumption must be false, so $\sqrt{2}$ must be irrational.

Mastering this structure allows you to tackle proofs that are difficult or impossible to solve with direct methods. It's a fundamental tool for rigorous thinking in mathematics, computer science, and philosophy.