Quod Erat Demonstrandum

2012/10/05

Rationalization

Filed under: NSS,Teaching — johnmayhk @ 3:20 下午

何謂 inequality?

對,不平等。例如社經地位的不平等,性別上的不平等。但相信中學生和數學授課員多數想到:不等式。

何謂 rationalization?

對,合理化。比如問,某些行為是合理,是真正「合理」,還是「合理化」的結果?(有興趣看看 Weber 的論述吧。)但相信中學生和數學授課員多數想到諸如:

\frac{1}{\sqrt{2}}

=\frac{1}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}

=\frac{\sqrt{2}}{2}

嗯,近日忙到太無聊,在教這個「課題」時,自己問了一個問題。

教科書題目如下:

Rationalize \frac{1}{1+\sqrt{2}}.

應該大家也知道「做」甚麼,但

一個本來是無理(irrational)的數,怎能使它有理(rational)?怎能使無理數變成有理數?於是我會如此寫題目:

Rationalize the denominator of \frac{1}{1+\sqrt{2}}.

但是,這樣說似乎也有問題,先不談把分母乘以 (1-\sqrt{2}) 是否即是「把分母有理化」,

其實,(1+\sqrt{2}) 究竟是不是 \frac{1}{1+\sqrt{2}} 的分母?

因為我只知道分數(fraction)有分子和分母,

\frac{1}{1+\sqrt{2}} 是無理數,根本不是分數(分數即是有理數);

既然 \frac{1}{1+\sqrt{2}} 不是分數,何來有分母?

甚至那條橫線,可稱為「分線」嗎?

(和同事「吹水」時,他再問 \frac{1}{(2/3)} 中,\frac{2}{3} 是分母嗎?)

那麼,「Rationalize \frac{1}{1+\sqrt{2}}」的「正/精確」問法是甚麼?

哈哈,知吾無聊吧。

再和同事們進一步討論:做(所謂)rationalization 有何意義?

我又亂吹:這是歷史遺留的問題,以前計算工具不發達,若(比方說)只知 \sqrt{2}\simeq 1.1412...,那麼徒手計算 \frac{\sqrt{2}}{2} 比用長除法計算 \frac{1}{\sqrt{2}} (感覺上)方便。另外,處理 \frac{1}{\sqrt{2}}+\frac{1}{1+\sqrt{2}},先把「分母」轉成有理數,(感覺上)方便作進一步運算。

同事給的理由如下:

1.答案要統一
2.表達要唯一

應然命題?真相是甚麼?我無心查了。

8 則迴響 »

  1. 數學。妙
    宜家既題目唔係只係寫Rationalize xxx 嗎?
    路過問下

    迴響 由 Little Maths — 2012/10/11 @ 12:51 下午 | 回覆

    • 有些教科書也寫:Rationalize the denominator of xxx

      幾年前我也質疑過:

      Rationalize \frac{1}{1+\sqrt{x}}

      之類的教科書題目。比如,如果 x=4,那麼

      Rationalize \frac{1}{1+\sqrt{4}}

      好像有點奇怪吧。

      迴響 由 johnmayhk — 2012/10/11 @ 4:20 下午 | 回覆

  2. 1+\sqrt{x} is an “irrational function" of x, but 1+\sqrt{4} is a rational number.

    Rationalizing 1+\sqrt{x} is not the same as rationalizing 1+\sqrt{4}. So you need not mind what is considered strange.

    P.S. “irrational function" is termed in the sense that it does not take the form of a rational function, i.e. quotient of two polynomials.

    迴響 由 mm100100 — 2012/10/11 @ 7:34 下午 | 回覆

    • Thank you mm100100.

      The word ‘irrational function’ is new to me.

      Just a web search, it seems that there is no formal definition of ‘irrational function’ (?)

      Some may refer that

      f(x)\in \mathbb{R}\\mathbb{Q} for any x in the domain"

      If this is the case, then

      “rationalize the irrational function f(x)"

      is not that strange.

      迴響 由 johnmayhk — 2012/10/12 @ 1:41 下午 | 回覆

      • 新的 “rational" function 就不是本來的 irrational function 吧…

        迴響 由 Myst — 2012/10/25 @ 8:42 上午

  3. You may try “Get rid of the radicals in the denominator of……."

    reference from an article" http://arxiv.org/pdf/1110.1556v2.pdf

    迴響 由 Asathor — 2012/12/11 @ 6:47 下午 | 回覆

  4. I think the aim of rationalization is to make manipulation easier.
    Even for a computer program, I wonder if sqrt(2)/2 is easier to manipulate than 1/sqrt(2).
    I hope the experts in mathematical computing may share your opinion.

    迴響 由 lovely girl — 2014/08/29 @ 1:26 上午 | 回覆


RSS feed for comments on this post. TrackBack URI

發表迴響

在下方填入你的資料或按右方圖示以社群網站登入:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / 變更 )

Twitter picture

You are commenting using your Twitter account. Log Out / 變更 )

Facebook照片

You are commenting using your Facebook account. Log Out / 變更 )

Google+ photo

You are commenting using your Google+ account. Log Out / 變更 )

連結到 %s

在 WordPress.com 建立免費網站或網誌.

%d 位部落客按了讚: