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给你讲清楚什么是无理数 – 译学馆
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给你讲清楚什么是无理数

Making sense of irrational numbers - Ganesh Pai

正如希腊神话中许多英雄一样
Like many heroes of Greek myths,
哲学家希帕索斯被传说要接受神的惩罚
the philosopher Hippasus was rumored to have been mortally punished by the gods.
但他错在哪儿了呢
But what was his crime?
是他杀人了
Did he murder guests,
还是他破坏了神圣的仪式都不是
or disrupt a sacred ritual?
希帕索斯的罪源于一个数学证明
No, Hippasus’s transgression was a mathematical proof:
无理数的发现
the discovery of irrational numbers.
希帕索是毕达哥拉斯学派中的一员
Hippasus belonged to a group called the Pythagorean mathematicians
他们对于数字有着宗教般的崇敬
who had a religious reverence for numbers.
他们的格言“万物皆数”
Their dictum of, “All is number,”
暗示着他们认为数字是宇宙建立的基石
suggested that numbers were the building blocks of the Universe
而且他们也相信任何事物从宇宙研究到音乐发展
and part of this belief was that everything from cosmology and metaphysics
从形而上学到道德观念
to music and morals followed eternal rules
归根到底都是数字比例的问题因此
describable as ratios of numbers.
任何数字都可以被写成一个比例(分数)
Thus, any number could be written as such a ratio.
5就是5/1
5 as 5/1,
0.5就是1/2
0.5 as 1/2
等等
and so on.
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
Even an infinitely extending decimal like this could be expressed exactly as 34/45.
这些数字都被称为有理数
All of these are what we now call rational numbers.
而希帕索斯却发现了一个背离这种和谐规律的数字
But Hippasus found one number that violated this harmonious rule,
一个本不该存在的数字
one that was not supposed to exist.
这个问题起源于一个非常简单的图形
The problem began with a simple shape,
一个四边长度均为单位1的正方形
a square with each side measuring one unit.
根据毕达哥拉斯的理论
According to Pythagoras Theorem,
这个正方形的对角线长度应该为根号二
the diagonal length would be square root of two,
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
but try as he might, Hippasus could not express this as a ratio of two integers.
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
And instead of giving up, he decided to prove it couldn’t be done.
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
Hippasus began by assuming that the Pythagorean worldview was true,
根号二是可以被表示成两个整数的比例
that root 2 could be expressed as a ratio of two integers.
他假设这两个整数分别为p和q
He labeled these hypothetical integers p and q.
假定这个比例已经被最简化因此
Assuming the ratio was reduced to its simplest form,
p和q应该没有相同约数
p and q could not have any common factors.
要证明根号二并不是有理数
To prove that root 2 was not rational,
希帕索斯只需要证明p/q并不存在即可
Hippasus just had to prove that p/q cannot exist.
他将等号两侧均乘以q
So he multiplied both sides of the equation by q
然后两侧均计算平方
and squared both sides.
得到了这样一个等式
which gave him this equation.
任何数字乘以2的结果都是偶数
Multiplying any number by 2 results in an even number,
所以p的平方是偶数
so p^2 had to be even.
如果p是奇数 则p的平方不可能为偶数
That couldn’t be true if p was odd
因为奇数乘以本身 得到的还是奇数
because an odd number times itself is always odd,
所以p也应该是一个偶数因此
so p was even as well.
p可以表示为2a其中a也是一个整数
Thus, p could be expressed as 2a, where a is an integer.
把这个等式带入原来的方程
Substituting this into the equation and simplifying
并简化得到:q^2 =
gave q^2 = 2a^2
2a^2再一次 任何数字乘以2得到的结果为偶数
Once again, two times any number produces an even number,
所以q的平方一定是偶数
so q^2 must have been even,
那么q也一定是偶数
and q must have been even as well,
这就得到p和q都是偶数的结果
making both p and q even.
但如果这是正确的话p和q就有一个共同的因子2
But if that was true, then they had a common factor of two,
和最初的题设矛盾至此
which contradicted the initial statement,
希帕索斯得以证明这样的比例是不存在的
and that’s how Hippasus concluded that no such ratio exists.
这被称为矛盾证明法
That’s called a proof by contradiction,
而根据传说
and according to the legend,
上帝并不喜欢矛盾的存在
the gods did not appreciate being contradicted.
有趣的是 即便我们无法将无理数
Interestingly, even though we can’t express irrational numbers
表示称为整数的比例
as ratios of integers,
我们却可以将它准确表现在图形之中
it is possible to precisely plot some of them on the number line.
以根号二为例
Take root 2.
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
All we need to do is form a right triangle with two sides each measuring one unit.
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
The hypotenuse has a length of root 2, which can be extended along the line.
我们可以继续画另外一个直角三角形
We can then form another right triangle
其中一条边以刚才的斜边为基础 另一条边长度为单位一
with a base of that length and a one unit height,
这个三角形的斜边程度就是单位根号三
and its hypotenuse would equal root three,
它同时还可以继续被延展下去
which can be extended along the line, as well.
关键问题是小数和分数都只是表现数字的方法之一
The key here is that decimals and ratios are only ways to express numbers.
根号二只是一个边长为单位一的直角三角形的
Root 2 simply is the hypotenuse of a right triangle
斜边长度罢了相似的
with sides of a length one.
著名的无理数pi
Similarly, the famous irrational number pi
也是与它描述的图形关系一样
is always equal to exactly what it represents,
代表者圆周长和半径的比例近似值
the ratio of a circle’s circumference to its diameter.
22/7 或者 355/133
Approximations like 22/7,
是永远无法准确的表达出pi值的
or 355/113 will never precisely equal pi.
我们永远也无法知道在希帕索斯身上到底发生过什么
We’ll never know what really happened to Hippasus,
但是我们知道他的发现带动了整个数学界的革命
but what we do know is that his discovery revolutionized mathematics.
所以无论神话里面怎么说永远不要害怕去探索不可能
So whatever the myths may say, don’t be afraid to explore the impossible.

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