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.

因此，任何數字都能寫成這樣的比例，

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

5 寫成 5/1，

5 as 5/1,

0.5 寫成 1/2，

0.5 as 1/2

等等。

and so on.

甚至像這個無限延伸的小數 也能以 34/45 來表示。

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

現在我們稱這些為 有理數 (rational numbers)。

All of these are what we now call rational numbers.

但 希帕索斯 發現一個數字， 它違反這個和諧的規則，

But Hippasus found one number that violated this harmonious rule,

那數字被認為不該存在。

one that was not supposed to exist.

這問題源自一個簡單的圖形，

The problem began with a simple shape,

一個正方形其每邊長為 1 單位。

a square with each side measuring one unit.

根據勾股定理（Pythagorean theorem），

According to Pythagoras Theorem,

對角線長度等於 √2 ，

the diagonal length would be square root of two,

不管怎樣努力，希帕索斯 無法用兩個整數的比例來表示 √2 ，

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

他不放棄， 決定去證明它無法以比例表示。

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

希帕索斯 首先假設 畢達哥拉斯的世界觀是正確的，

Hippasus began by assuming that the Pythagorean worldview was true,

就是 √2 可用兩個整數的比例來表示，

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

他將這兩個假設的整數 命名為 p 及 q 。

He labeled these hypothetical integers p and q.

假設這比例已被最簡化，

Assuming the ratio was reduced to its simplest form,

P 和 q 之間沒有任何共同因子，

p and q could not have any common factors.

欲證明 √2 不是有理數，

To prove that root 2 was not rational,

希帕索斯 只要去證明 p/q 不可能存在。

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

所以他在方程式的兩邊都乘上 q，

So he multiplied both sides of the equation by q

然後兩邊平方，

and squared both sides.

他得到這樣的方程式。

which gave him this equation.

任何數字乘上 2 都會變成偶數，

Multiplying any number by 2 results in an even number,

所以 p^2 必定是偶數，

so p^2 had to be even.

若 p 為奇數，那不可能是對的，

That couldn't be true if p was odd

因為奇數自己相乘，永遠是奇數，

because an odd number times itself is always odd,

所以 p 也是偶數，

so p was even as well.

因此，p 可以 2a 表示，a 是一個整數。

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

將這帶入方程式並簡化，

Substituting this into the equation and simplifying

得到 q^2 = 2a^2

gave q^2 = 2a^2

再次，2 乘上任何數字會變成偶數，

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

所以 q^2 一定是偶數，

so q^2 must have been even,

而 q 一定也是偶數，

and q must have been even as well,

如此使得 p 和 q 都是偶數。

making both p and q even.

但如果這是正確的， 那它們之間會有 2 的共同因子，

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

這與最初的假設相違背，

which contradicted the initial statement,

這就是希帕索斯推斷 這種比例不存在的方法，

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

就是所謂的反證法 (Proof by contradiction)。

That's called a proof by contradiction,

根據傳說

and according to the legend,

眾神並不樂見被反駁。

the gods did not appreciate being contradicted.

有趣的是，雖然我們無法 用整數的比例來表示無理數，

Interestingly, even though we can't express irrational numbers

但其中某些可正確地標定在數軸上。

as ratios of integers,

以 √2 為例，

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

我們只須畫一個直角三角形， 其兩邊長各為 1 單位，

Take root 2.

其斜邊長則為 √2 ， 可延伸畫在數軸上，

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

然後，我們再畫另一個直角三角形，

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

底為那個斜邊長 √2，高為 1 單位，

We can then form another right triangle

所成斜邊就等於 √3 ，

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

也可延伸畫在數軸上。

and its hypotenuse would equal root three,

這裡的關鍵是 小數和比例是唯一表示數字的方法，

which can be extended along the line, as well.

而 √2 只是兩邊長度為 1 的 直角三角形之斜邊。

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

同樣地，著名的無理數 π (pi)

Root 2 simply is the hypotenuse of a right triangle

總是正好等於

with sides of a length one.

一個圓的圓周與其半徑的比例。

Similarly, the famous irrational number pi

近似值例如 22/7

is always equal to exactly what it represents,

或 355/113 都不能準確地等於 π 。

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

我們永遠不會知道 希帕索斯到底發生什麼事，

Approximations like 22/7,

但我們確實知道 他的發現徹底改變了數學。

or 355/113 will never precisely equal pi.

所以不管神話是怎麼說， 不要畏懼探索不可能的事物。

We'll never know what really happened to Hippasus,

翻譯：Helen Lin

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

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