作者armopen (八字-风水-姓名学)
看板tutor
标题Re: [解题] 高一数学 根号与有理数之问题
时间Sun Jul 14 22:15:53 2013
※ 引述《newgenius (枫叶)》之铭言：
: 1.年级：高一
: 2.科目：数学
: 3.章节：数与式
: 4.题目：证明√2 + √7 是有理数
: 5.想法：不知道如何做,我甚至用计算机来计算,都觉得不是有理数
: 是用反证法吗??
: 请各位指导一下...感谢
It is clear that both √2 and √7 are irrational numbers.
To show the irrationality of the sum of them, we assume the contray.
Suppose that √2 + √7 is a rational number, say r. Squaring both sides
of √2 + √7 = r, we have 9 + 2√14 = r^2. Hence √14 = (1/2)*(r^2 - 9) is
a rational number by the closeness of rational numbers under addition and
multiplication. This is a contraction since √14 is irrational.
Note. In fact, we can also prove the irrationality of √14 in detail.
Assume that √14 is a rational number, say m/n for m, n in Z and n is
non-zero. Squaring both sides, we obtain 14*n^2 = m^2. By the fundamental
Theorem of Arithmetic, any positive integers > 1 can be expressed as a
finite product of primes in only one way apart from the order of factors.
This is a cotradiction since the power of the prime factor 2 is odd on
the left side and even on the right side.
--
※ 发信站: 批踢踢实业坊(ptt.cc)
※ 编辑: armopen 来自: 114.37.171.55 (07/14 22:17)
1F：推 marra:我喜欢这个证法 :) 07/15 02:15
2F：推 bunjie:推这个方法 我也是用这方法:) 有点两段试证法的感觉 07/15 03:56
3F：推 nomorethings:高等微积分课本也都是酱证的^_^ 07/16 09:40