Pure · Proof
Chapter 1 · 3
The idea
Proof by contradiction
Assuming the opposite of what you want to prove, deriving a logical contradiction, and concluding the original must be true. Negating a statement correctly, the classic results (√2 is irrational), and why reaching an actual contradiction is the whole point. (A-level / Year 2.)
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Pure · Proof
Proof by contradiction
Assuming the opposite of what you want to prove, deriving a logical contradiction, and concluding the original must be true. Negating a statement correctly, the classic results (√2 is irrational), and why reaching an actual contradiction is the whole point. (A-level / Year 2.)
Why it works
Assume the opposite, break something
How do you prove that is irrational — that among the infinitely many fractions, not one of them squares to ? You cannot check them all, and no direct construction reaches a statement about what doesn't exist. The way in is a detective's move: suppose it did. In a proof by contradiction you assume the opposite of what you want to prove, then reason until you hit something impossible. Since every reasoning step was valid, the only thing that can be wrong is the assumption — so the assumption is false and the original statement is true.Keep reading — free
The rest of the explanation, plus 3 worked examples you step through move by move.
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