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大家都知道2~(1/2),2~(1/3),…,是无理数,然而严格的証明可能并不是多数人所熟悉的,为此,我們在本文中列举出一些常見的无理数,并証明它們的无理性。 1.設N,n都是正整数,且N~(1/n)不是整数,則它必为无理数。用反証法証之。設N~(1/n)=p/q,其中p>0,q>1,且二者无公因数;将p,q分解成素因数的乘积: 由于p,q无公因数,故pi与qj中无相同者,又由于N~(1/n)=q/p, 由于pi与qj中无共同者,故上式是一不可約分式,从而p~n/q~n不可能为整数,这与假設矛盾。 2.設p,q是无公因数的正整数,且p~(1/N),q~(1/N)不同时为整数,則(p/q)~(1/N)是无理数。
Everyone knows that 2~(1/2), 2~(1/3),..., are irrational numbers, but strict proof may not be familiar to most people. For this reason, we list some common irrational numbers in this article. And prove their irrationality. 1. Let N, n be a positive integer, and N ~ (1/n) is not an integer, then it must be an irrational number. Use counter-proof to prove it. Let N ~ (1/n) = p/q, where p> 0, q> 1, and the two have no common factor; decompose p, q into a product of prime factors: Since p, q have no common factor, so pi There is no similarity in qj, and because N~(1/n)=q/p, since there is no common in pi and qj, the above equation is an irreducible fraction, so that p~n/q~n cannot be Integers, this contradicts the assumptions. 2. Let p, q be a positive integer without a common factor, and if p~(1/N) and q~(1/N) are different integers, then (p/q)~(1/N) is an irrational number.