2007/04/08

0^0

標題不是表情符號 XD。今天又翻到,覺得值得摘錄下來 XD。Knuth 宣稱 00 應該定義為 1,理由在水泥數學 p.162:

Some textbooks leave the quantity 00 undefined, because the functions x0 and 0x have different limiting values when x decreases to 0. But this is a mistake. We must define

x0 = 1, for all x,

if the binomial theorem is to be valid when x = 0, y = 0, and/or x = -y. The theorem is too important to be arbitrarily restricted! By contrast, the function 0x is quite unimportant. (See [220] for further discussion.)

其中 [220] 是 Knuth 寫的〈Two Notes on Notation〉,可從 Knuth 的網頁下載 .tex 全文。這段敘述很有說服力 XD,同時也揭露所謂「定義」的些許本質。

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HTML 對程式碼(in particular C++ template code)和數學式實在不友善,尤其是後者!XD


看完 [220] 關於 00 的那一段(然後發現其實我以前看過 XD),就想到聰明引述 Laplace 的這句話:「做數學有一半工作是記號的戰爭。」XD

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