Making sense of irrational numbers - Ganesh Pai

譯者: Helen Lin 審譯者: Max Chern

Like many heroes of Greek myths,

如同許多希臘神話的英雄，

the philosopher Hippasus was rumored to have been mortally punished by the gods.

哲學家 希帕索斯 謠傳被眾神處死，

But what was his crime?

但他犯了什麼罪？

Did he murder guests,

他是謀殺了賓客

or disrupt a sacred ritual?

或妨礙神聖的宗教儀式？

No, Hippasus's transgression was a mathematical proof:

不是！希帕索斯的罪行 是一個數學的驗證：

the discovery of irrational numbers.

無理數的發現。

Hippasus belonged to a group called the Pythagorean mathematicians

希帕索斯屬於一群 稱為「畢達哥拉斯學派」的數學家，

who had a religious reverence for numbers.

他們對數字有著虔誠的敬畏。

Their dictum of, "All is number,"

他們的格言「萬物皆數」，

suggested that numbers were the building blocks of the Universe

認為數字是宇宙的主要組成部分，

and part of this belief was that everything from cosmology and metaphysics

這信仰的一部分是 萬物從宇宙學與形上學

to music and morals followed eternal rules

到音樂與倫理道德都遵循 可描寫成 ‘數字比例’ 的永恆規則 。

describable as ratios of numbers.

Thus, any number could be written as such a ratio.

因此，任何數字都能寫成這樣的比例，

5 as 5/1,

5 寫成 5/1，

0.5 as 1/2

0.5 寫成 1/2，

and so on.

等等。

Even an infinitely extending decimal like this could be expressed exactly as 34/45.

甚至像這個無限延伸的小數 也能以 34/45 來表示。

All of these are what we now call rational numbers.

現在我們稱這些為 有理數 (rational numbers)。

But Hippasus found one number that violated this harmonious rule,

但 希帕索斯 發現一個數字， 它違反這個和諧的規則，

one that was not supposed to exist.

那數字被認為不該存在。

The problem began with a simple shape,

這問題源自一個簡單的圖形，

a square with each side measuring one unit.

一個正方形其每邊長為 1 單位。

According to Pythagoras Theorem,

根據勾股定理（Pythagorean theorem），

the diagonal length would be square root of two,

對角線長度等於 √2 ，

but try as he might, Hippasus could not express this as a ratio of two integers.

不管怎樣努力，希帕索斯 無法用兩個整數的比例來表示 √2 ，

And instead of giving up, he decided to prove it couldn't be done.

他不放棄， 決定去證明它無法以比例表示。

Hippasus began by assuming that the Pythagorean worldview was true,

希帕索斯 首先假設 畢達哥拉斯的世界觀是正確的，

that root 2 could be expressed as a ratio of two integers.

就是 √2 可用兩個整數的比例來表示，

He labeled these hypothetical integers p and q.

他將這兩個假設的整數 命名為 p 及 q 。

Assuming the ratio was reduced to its simplest form,

假設這比例已被最簡化，

p and q could not have any common factors.

P 和 q 之間沒有任何共同因子，

To prove that root 2 was not rational,

欲證明 √2 不是有理數，

Hippasus just had to prove that p/q cannot exist.

希帕索斯 只要去證明 p/q 不可能存在。

So he multiplied both sides of the equation by q

所以他在方程式的兩邊都乘上 q，

and squared both sides.

然後兩邊平方，

which gave him this equation.

他得到這樣的方程式。

Multiplying any number by 2 results in an even number,

任何數字乘上 2 都會變成偶數，

so p^2 had to be even.

所以 p^2 必定是偶數，

That couldn't be true if p was odd

若 p 為奇數，那不可能是對的，

because an odd number times itself is always odd,

因為奇數自己相乘，永遠是奇數，

so p was even as well.

所以 p 也是偶數，

Thus, p could be expressed as 2a, where a is an integer.

因此，p 可以 2a 表示，a 是一個整數。

Substituting this into the equation and simplifying

將這帶入方程式並簡化，

gave q^2 = 2a^2

得到 q^2 = 2a^2

Once again, two times any number produces an even number,

再次，2 乘上任何數字會變成偶數，

so q^2 must have been even,

所以 q^2 一定是偶數，

and q must have been even as well,

而 q 一定也是偶數，

making both p and q even.

如此使得 p 和 q 都是偶數。

But if that was true, then they had a common factor of two,

但如果這是正確的， 那它們之間會有 2 的共同因子，

which contradicted the initial statement,

這與最初的假設相違背，

and that's how Hippasus concluded that no such ratio exists.

這就是希帕索斯推斷 這種比例不存在的方法，

That's called a proof by contradiction,

就是所謂的反證法 (Proof by contradiction)。

and according to the legend,

根據傳說

the gods did not appreciate being contradicted.

眾神並不樂見被反駁。

Interestingly, even though we can't express irrational numbers

有趣的是，雖然我們無法 用整數的比例來表示無理數，

as ratios of integers,

it is possible to precisely plot some of them on the number line.

但其中某些可正確地標定在數軸上。

Take root 2.

以 √2 為例，

All we need to do is form a right triangle with two sides each measuring one unit.

我們只須畫一個直角三角形， 其兩邊長各為 1 單位，

The hypotenuse has a length of root 2, which can be extended along the line.

其斜邊長則為 √2 ， 可延伸畫在數軸上，

We can then form another right triangle

然後，我們再畫另一個直角三角形，

with a base of that length and a one unit height,

底為那個斜邊長 √2，高為 1 單位，

and its hypotenuse would equal root three,

所成斜邊就等於 √3 ，

which can be extended along the line, as well.

也可延伸畫在數軸上。

The key here is that decimals and ratios are only ways to express numbers.

這裡的關鍵是 小數和比例是唯一表示數字的方法，

Root 2 simply is the hypotenuse of a right triangle

而 √2 只是兩邊長度為 1 的 直角三角形之斜邊。

with sides of a length one.

Similarly, the famous irrational number pi

同樣地，著名的無理數 π (pi)

is always equal to exactly what it represents,

總是正好等於

the ratio of a circle's circumference to its diameter.

一個圓的圓周與其半徑的比例。

Approximations like 22/7,

近似值例如 22/7

or 355/113 will never precisely equal pi.

或 355/113 都不能準確地等於 π 。

We'll never know what really happened to Hippasus,

我們永遠不會知道 希帕索斯到底發生什麼事，

but what we do know is that his discovery revolutionized mathematics.

但我們確實知道 他的發現徹底改變了數學。

So whatever the myths may say, don't be afraid to explore the impossible.

所以不管神話是怎麼說， 不要畏懼探索不可能的事物。

翻譯：Helen Lin