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众所周知,实数分为有理数和无理数,无理数又分为代数数和超越数。这是实数的一种划分法。实数集还可以分成代数数集和超越数集。如果一个实数是整系数的某个代数方程a_0x~n+a_1x~(n-1)+…+a(n-1)x+a~m=0的根,那么这个数叫做代数数。反之,不是任何整系数代数方程的根的实数称为超越数。因为全体有理数n/m是一次代数方程mx-n=0的根,所以有理数集是代数数集一个子数,因此超越数都是无理数。证明一个数a是无理数,统编高中《代数》课本用了反证法,但用反证法需要一定的技巧,学生往往不会使用。本文打算介绍证明代数数中无理数的一种一般方法、供教师们参考。这种方法要用到下列定理。这个定理在一般代数课本中都有、我们就不作证明了。定理:整系数代数方程a_0x~n+a_1~(n-1)+…+a(n-1)x+a_n=0有有理数根m/n(m、n互质)的必要条件是m是a_n的约数、r是a_0的约数。我们先举例说明如何用这个定理证明代数数中的无理数、然后总结这种方法的一般步骤。
As we all know, real numbers are divided into rational numbers and irrational numbers, and irrational numbers are divided into algebraic numbers and transcendental numbers. This is a division of real numbers. Real numbers can also be divided into algebraic data sets and transcendental data sets. If a real number is the root of an algebraic equation a_0x~n+a_1x~(n-1)+...+a(n-1)x+a~m=0, then the number is called an algebraic number. Conversely, a real number that is not the root of any algebraic equation is called the transcendental number. Since the whole rational number n/m is the root of an algebraic equation mx-n=0, the rational number set is a subnumber of algebraic numbers, so the transcendental numbers are irrational numbers. To prove that a number is an irrational number, the textbook of algebra in senior high school uses the anti-evidence method, but using anti-evidence requires a certain skill, and students often do not use it. This article intends to introduce a general method to prove irrational numbers in algebraic numbers for the reference of teachers. This method uses the following theorems. This theorem is in the general algebra textbook and we will not prove it. Theorem: The whole coefficient algebraic equation a_0x~n+a_1~(n-1)+...+a(n-1)x+a_n=0 The necessary condition for the rational root m/n (m, n prime) is m The divisor of a_n, r, is a divisor of a_0. We first illustrate how to use this theorem to prove irrational numbers in algebraic numbers and then summarize the general steps of this method.