What Is a Proof by Contradiction?
A proof by contradiction is a method of mathematical proof where you begin by assuming that the statement you want to prove is false. You then use logical steps to show that this assumption leads to a logical contradiction or an impossible result. Because the assumption leads to an impossibility, the original assumption must be false, which means the statement you wanted to prove must be true.
Section 2: The Core Steps
This method, also known as 'reductio ad absurdum,' follows three main steps. First, assume the opposite (the negation) of the proposition you aim to prove. Second, using this assumption, logically derive a consequence that is impossible or contradicts a known fact or a previous assumption. Third, conclude that since your initial assumption led to a contradiction, it must be false, thereby proving the original proposition is true.
Section 3: A Classic Example
A famous example is proving that the square root of 2 is an irrational number. The proof starts by assuming the opposite: that √2 is rational. This means √2 can be written as a fraction a/b in its simplest form. Squaring both sides gives 2 = a²/b², or 2b² = a². This implies a² is even, so 'a' must also be even. If 'a' is even, we can write it as 2k. Substituting this back gives 2b² = (2k)², which simplifies to b² = 2k². This shows b² is even, meaning 'b' is also even. The contradiction is that if both 'a' and 'b' are even, the fraction a/b was not in its simplest form, which contradicts our initial setup.
Section 4: Importance in Logic and Math
Proof by contradiction is a powerful tool because it allows for the proof of statements that are difficult to prove directly. It is particularly useful for proving the non-existence of something or demonstrating properties like irrationality. It is a fundamental technique for building rigorous logical arguments and is essential in fields from number theory to computer science.