Reduction to absurdity (also called 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. The reduction to absurdity is usually called Reductio ad absurdum, which means "transforming into impossible" in Latin and comes from Greek? ει? υυυυυυυυυυυυ▲ The reduction to absurdity is an "indirect proof" and a proof method from a negative perspective, that is, affirming the topic, denying the conclusion, and thus drawing 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". Reduction to absurdity is often used in mathematics. When the topic is not easy or can't be proved from the front, we should use the reduction to absurdity, which is called "if it is difficult, it will be reversed". Newton once said, "Reduction to absurdity is one of the most skilled weapons for mathematicians". Generally speaking, the reduction to absurdity is often used to prove that the positive proof is difficult, the situation is more or more complicated, and the negative proposition is relatively simple. This problem may be easy to solve. The proof of reduction to absurdity can be simply summarized as "negation → contradiction → negation". That is to say, from the negative conclusion, contradictions are drawn and new negation is drawn. We can think that the basic idea of reduction to absurdity is dialectical negation of negation. The purpose of applying reduction to absurdity is to prove that "if p is q" is a true proposition and lead to contradictions from opposite conclusions. Therefore, the reduction to absurdity mainly uses the conclusion that a proposition and its negative proposition are true and false. Why? This conclusion can be proved by exhaustive method: Proposition A: If A is B, this proposition has four situations: 1. When A is true and B is true, A→B is true and B → A is true; 2. When A is true and B is false, A→B is false and B → A is false; 3. When A is false and B is true, A→B is true and B → A is true; 4. When A is false and B is false, A→B is true and B → A is true; The truth of a proposition and its negation, that is, reduction to absurdity, is correct. Equivalent to "If A is B first" is its negative proposition. If "b" is "a" and "b" is assumed and "a" is deduced, it means that the negative proposition is true, and so is the original proposition. However, in the actual deduction process, it is quite difficult to deduce "A", so it is transformed into "deduction sum". (2) Starting from this proposition, the contradiction is proved by reasoning. (3) Judging that the hypothesis is not established through contradiction, thus affirming that the conclusion of the proposition is correct. The reduction to absurdity is suitable for simple logic: (1) uniqueness proposition (2) negation proposition (3) "at most" and "at least" proposition. Two examples of reduction to absurdity prove that there are infinitely many prime numbers. This ancient proposition was originally given by the ancient Greek mathematician Euclid (living in Alexandria, about 330 BC ~ about 275 BC, the most famous mathematician in ancient Greece) in his immortal book "Elements of Geometry": If the proposition is not true, then there are only a limited number of prime numbers, and all prime numbers are 2 = A 1