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我们知道,大凡一个数学命题“若A则B”,一般都能用反证法给予证明。但在以下几种情况下用反证法,显得特别方便和有效。一、当结论的反面B,比B本身的形式较为具体,或情况较为简单的时候。例1．若a、b、c为奇数(可以是负数),则方程ax~2+bx+c= 0没有有理根。这个命题中的结论B——方程无有理根(即只有无理根);(?)——方程有有理根。相比之下,因有理数可写成互质整数之比,故(?)形式具体。
We know that generally a mathematical proposition “If A then B” can generally be proved by anti-evidence. However, using the counter-evidence method in the following situations is particularly convenient and effective. 1. When the negative B of the conclusion is more specific than the form of B itself, or when the situation is relatively simple. example 1. If a, b, and c are odd (can be negative), then the equation ax~2+bx+c=0 does not have a rational root. The conclusion B in this proposition is that the equation has no rational roots (that is, only irrational roots); (?) The equation has a rational root. In contrast, because the rational number can be written as the ratio of coprime integers, the (?) form is concrete.