最近看到几个有趣的数学谬证,想写下来与大家分享;结果写到这个又想到那个,一写就写个没完,于是想到干脆做一篇谬证大全,收集各种荒谬的证明。
如果你有什么更棒的“证明”,欢迎来信与我分享,我会更新到这篇日志中。我的邮箱是 matrix67 at tom.com ,或者 gs.matrix67 at gmail.com 。
1=2?史上最经典的“证明”
设 a = b ,则 a·b = a^2 ,等号两边同时减去 b^2 就有 a·b – b^2 = a^2 – b^2 。注意,这个等式的左边可以提出一个 b ,右边是一个平方差,于是有 b·(a – b) = (a + b)(a – b) 。约掉 (a – b) 有 b = a + b 。然而 a = b ,因此 b = b + b ,也即 b = 2b 。约掉 b ,得 1 = 2 。
这可能是有史以来最经典的谬证了。 Ted Chiang 在他的短篇科幻小说 Division by Zero 中写到:
There is a well-known “proof” that demonstrates that one equals two. It begins with some definitions: “Let a = 1; let b = 1.” It ends with the conclusion “a = 2a,” that is, one equals two. Hidden inconspicuously in the middle is a division by zero, and at that point the proof has stepped off the brink, making all rules null and void. Permitting division by zero allows one to prove not only that one and two are equal, but that any two numbers at all—real or imaginary, rational or irrational—are equal.
这个证明的问题所在想必大家都已经很清楚了:等号两边是不能同时除以 a – b 的,因为我们假设了 a = b ,也就是说 a – b 是等于 0 的。
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