Making sense of irrational numbers - Ganesh Pai

翻译人员: Nancy JIANG 校对人员: Runyu Hu

Like many heroes of Greek myths,

正如希腊神话中许多英雄一样

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

哲学家希帕索斯被传说要接受神的惩罚

But what was his crime?

但他错在哪儿了呢

Did he murder guests,

是他杀人了

or disrupt a sacred ritual?

还是他破坏了神圣的仪式

No, Hippasus's transgression was a mathematical proof:

都不是 希帕索斯的罪源于一个数学证明

the discovery of irrational numbers.

无理数的发现

Hippasus belonged to a group called the Pythagorean mathematicians

希帕索是毕达哥拉斯学派中的一员

who had a religious reverence for numbers.

他们对于数字有着宗教般的崇敬

Their dictum of, "All is number,"

他们的格言“万物皆数”

suggested that numbers were the building blocks of the Universe

暗示着他们认为数字是宇宙建立的基石

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

而且他们也相信任何事物 从宇宙研究到音乐发展

to music and morals followed eternal rules

从形而上学到道德观念

describable as ratios of numbers.

归根到底都是数字比例的问题

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

因此，任何数字都可以被写成一个比例（分数）

5 as 5/1,

5就是5/1

0.5 as 1/2

0.5就是1/2

and so on.

等等

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

甚至一个可以被无限延伸的十进制数字 也可以被准确表示成34/45

All of these are what we now call rational numbers.

这些数字都被称为有理数

But Hippasus found one number that violated this harmonious rule,

而希帕索斯却发现了一个背离这种和谐规律的数字

one that was not supposed to exist.

一个本不该存在的数字

The problem began with a simple shape,

这个问题起源于一个非常简单的图形

a square with each side measuring one unit.

一个四边长度均为单位1的正方形

According to Pythagoras Theorem,

根据毕达哥拉斯的理论

the diagonal length would be square root of two,

这个正方形的对角线长度应该为根号二

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

但是无论希帕索斯如何尝试 都不能将根号二变为两个整数的比例形式

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

他并没有选择放弃 而是决定证明这个数字确实无法被比例表示出来

Hippasus began by assuming that the Pythagorean worldview was true,

希帕索斯首先假设毕达哥拉斯的“万物皆数”的观点是正确的

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

根号二是可以被表示成两个整数的比例

He labeled these hypothetical integers p and q.

他假设这两个整数分别为p和q

Assuming the ratio was reduced to its simplest form,

假定这个比例已经被最简化

p and q could not have any common factors.

因此，p和q应该没有相同约数

To prove that root 2 was not rational,

要证明根号二并不是有理数

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

希帕索斯只需要证明p/q并不存在即可

So he multiplied both sides of the equation by q

他将等号两侧均乘以q

and squared both sides.

然后两侧均计算平方

which gave him this equation.

得到了这样一个等式

Multiplying any number by 2 results in an even number,

任何数字乘以2的结果都是偶数

so p^2 had to be even.

所以p的平方是偶数

That couldn't be true if p was odd

如果p是奇数，则p的平方不可能为偶数

because an odd number times itself is always odd,

因为奇数乘以本身，得到的还是奇数

so p was even as well.

所以p也应该是一个偶数

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

因此，p可以表示为2a 其中a也是一个整数

Substituting this into the equation and simplifying

把这个等式带入原来的方程，并简化

gave q^2 = 2a^2

得到：q^2 = 2a^2

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

再一次，任何数字乘以2得到的结果为偶数

so q^2 must have been even,

所以q的平方一定是偶数

and q must have been even as well,

那么q也一定是偶数

making both p and q even.

这就得到p和q都是偶数的结果

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

但如果这是正确的话 p和q就有一个共同的因子2

which contradicted the initial statement,

和最初的题设矛盾

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

至此，希帕索斯得以证明这样的比例是不存在的

That's called a proof by contradiction,

这被称为矛盾证明法

and according to the legend,

而根据传说

the gods did not appreciate being contradicted.

上帝并不喜欢矛盾的存在

Interestingly, even though we can't express irrational numbers

有趣的是，即便我们无法将无理数

as ratios of integers,

表示称为整数的比例

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

我们却可以将它准确表现在图形之中

Take root 2.

以根号二为例

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

我们需要做的就是准确的画出一个 两条直角边均为单位一的三角形

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

他的的斜边的长度就是单位根号二 这同时也可以被延伸下去

We can then form another right triangle

我们可以继续画另外一个直角三角形

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

其中一条边以刚才的斜边为基础，另一条边长度为单位一

and its hypotenuse would equal root three,

这个三角形的斜边程度就是单位根号三

which can be extended along the line, as well.

它同时还可以继续被延展下去

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

关键问题是 小数和分数都只是表现数字的方法之一

Root 2 simply is the hypotenuse of a right triangle

根号二只是一个边长为单位一的直角三角形的

with sides of a length one.

斜边长度罢了

Similarly, the famous irrational number pi

相似的，著名的无理数pi

is always equal to exactly what it represents,

也是与它描述的图形关系一样

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

代表者圆周长和半径的比例

Approximations like 22/7,

近似值 22/7 或者 355/133

or 355/113 will never precisely equal pi.

是永远无法准确的表达出pi值的

We'll never know what really happened to Hippasus,

我们永远也无法知道在希帕索斯身上到底发生过什么

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

但是我们知道他的发现带动了整个数学界的革命

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

所以无论神话里面怎么说 永远不要害怕去探索不可能