JOSSICA - jossica.com

无理数是指无限不循环小数，任何无理数都不能写成两个整数相除的分数形式。 那么我们应该如何来证明√2是无理数呢？ 如果从正面思考这个问题，就需要证明√2是无限不循环小数。 很显然，无论你把√2精确计算到多少位，都不能说明其小数部分是无限的，更不能说明是不循环的。 也就是说，根本就无法做到从正面来说明√2是无理数。 所以我们必须转换思维，在实数范围内，除了无理数就是有理数，如果我们能够说明√2不是有理数，那就一定能够说明√2是无理数。 这就是反证法的思想。 求证：√2是无理数. 证明：假设√2是有理数. 显然√2＞0，则√2=m/n. m、n∈N*，且 (m，n)=1. m= (√2)n，m^2= [ (√2)n]^2=2 (n^2) m^2为偶数，则m为偶数. 设m=2p.

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