Proof by Contradiction

Understanding the Concept

A proof by contradiction (also known as reductio ad absurdum) is a mathematical method used to show that a statement is true by assuming its negation is true and then reaching a contradiction.
This type of proof involves several steps, starting with assuming the opposite of what you’re trying to prove, then showing that this assumption leads to an illogical conclusion.

The first step in a proof by contradiction is to assume the negation of the statement you are trying to prove.
Next, using logical and mathematical reasoning, derive a result that contradicts a known fact or axiom. This shows that the initial assumption (the negation of the statement) must be false.
Consequently, the original statement whose negation we had taken must be true.
The contradiction may also be something that contradicts the assumption made, proving the assumption incorrect.

To demonstrate how this method works, consider the proof that the square root of 2 is irrational.
The negation of this statement would be that the square root of 2 is rational, meaning it can be expressed as a fraction of two integers.
If we square both sides of the assumed fraction (let’s say root 2 is ‘a/b’ where ‘a’ and ‘b’ have no common factor other than 1), we get 2=(a^2 )/(b^2 ). This leads to a contradiction because both ‘a’ and ‘b’ become even, hence having at least 2 as a common factor thereby contradicting the original assumption.
Therefore, the statement that square root of 2 is rational leads to a contradiction. Hence, the square root of 2 must be irrational.

Proving by contradiction is crucial because it allows mathematicians to establish the truth of a statement by showing that its opposite leads to an impossible or illogical situation.
It has been employed in many mathematical proofs including famous ones like the proof of infinite primes, and several statements in algebra, calculus, and topology.