Absurd proof (proof

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Contradiction, also known as reduction to absurdity and paradox, is a way of argument. He first assumes that a proposition is not established (that is, the conclusion is not established under the condition of the original proposition), and then infers the obvious contradictory results, thus drawing the conclusion that the original hypothesis is not established and the original proposition is proved.

explain

The reduction to absurdity is an "indirect proof" and a proof method from a negative perspective, that is, affirming the topic and denying the conclusion, so as to draw contradictions. Hadamard, a French mathematician, summed up the essence of reduction to absurdity: "If we affirm the hypothesis of the theorem and deny its conclusion, it will lead to contradictions". Specifically, the reduction to absurdity is to start with the counter-proposition, take the negation of the proposition conclusion as the condition, make it contradict with the condition, affirm the proposition conclusion, and thus prove the proposition.

When using reduction to absurdity, we must use "anti-design", otherwise it is not reduction to absurdity. When using reduction to absurdity to prove a problem, if only one aspect of the proposition needs to be proved, then refute this situation, which is also called reduction to absurdity; If the conclusion is multifaceted, then all the negative situations must be refuted one by one in order to infer the original conclusion. This method of proof is also called "exhaustive method".

model

Proof: the root number two is irrational.

Assuming that the proposition is not true, then √2 is a rational number, and let √2=n/m, which is the simplest form of fraction.

Then n∧2/m∧2=2, 2m∧2=n∧2.

So n∧2 is an even number, then n is an even number, which can be expressed as 2x.

Then 2m∧2=4x∧2.

So m∧2=2x∧2.

M is also an even number.

So m and n have a common factor of 2, which contradicts the statement that n/m is simplest fraction.

So √2 is an irrational number!