# Quod Erat Demonstrandum

## 2012/10/05

### Rationalization

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

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

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

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

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

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

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

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

1.答案要統一
2.表達要唯一

## 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 | 回應

• 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 | 回應

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 | 回應

• 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 | 回應

• 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 | 回應

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 | 回應