Re: [解題] 高一數學 根號與有理數之問題
※ 引述《newgenius (楓葉)》之銘言：
: 1.年級：高一
: 2.科目：數學
: 3.章節：數與式
: 4.題目：證明√2 + √7 是有理數
: 5.想法：不知道如何做,我甚至用計算機來計算,都覺得不是有理數
: 是用反證法嗎??
: 請各位指導一下...感謝
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.
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※ 編輯: armopen 來自: 114.37.171.55 (07/14 22:17)
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