Why can't you divide by zero? - TED-Ed

翻译人员: Varlum Wei 校对人员: Lipeng Chen

In the world of math,

在数学世界里，

many strange results are possible when we change the rules.

当我们改变规则时， 有可能产生许多奇怪的结果。

But there’s one rule that most of us have been warned not to break:

但其中有一条规则，我们很多人 都被告诫说不要去打破它：

don’t divide by zero.

不要把零当除数去除。

How can the simple combination of an everyday number

为什么将这一日常数字

and a basic operation cause such problems?

与基本运算结合起来会导致问题呢？

Normally, dividing by smaller and smaller numbers

通常，随着除数的变小，

gives you bigger and bigger answers.

得到的结果就会变大。

Ten divided by two is five,

10 除以 2 等于 5，

by one is ten,

10 除以 1 等于 10，

by one-millionth is 10 million,

10 除以百万分之一等于一千万，

and so on.

So it seems like if you divide by numbers

以此类推。

所以看起来似乎是如果你除以

that keep shrinking all the way down to zero,

一个小至趋于 0 的数字，

the answer will grow to the largest thing possible.

得到的结果可能就会无限大。

Then, isn’t the answer to 10 divided by zero actually infinity?

那么 10 除以 0 结果是不是无限大呢？

That may sound plausible.

这听起来似乎很合理。

But all we really know is that if we divide 10

但我们所知道的是，如果我们用 10

by a number that tends towards zero,

除以一个趋于 0 的数字，

the answer tends towards infinity.

结果会趋于无穷大。

And that’s not the same thing as saying that 10 divided by zero

这与 10 除以 0

is equal to infinity.

等于无穷大是不同的。

Why not?

为什么不同呢？

Well, let’s take a closer look at what division really means.

我们仔细来看看除法的真正含义吧。

Ten divided by two could mean,

10 除以 2 可以理解成，

"How many times must we add two together to make 10,”

“有多少个 2 相加等于 10，”

or, “two times what equals 10?”

或者说，“2 乘以什么等于 10？”

Dividing by a number is essentially the reverse of multiplying by it,

除以某个数其实就是乘以这个数的倒数，

in the following way:

如下面这些例子：

if we multiply any number by a given number x,

如果我们用任何一个数乘以已知数 X，

we can ask if there’s a new number we can multiply by afterwards

我们可能会问，是否可以乘以一个新的数

to get back to where we started.

让我们得到开始时的数字。

If there is, the new number is called the multiplicative inverse of x.

如果有的话，这个新的数字就叫 X 的倒数。

For example, if you multiply three by two to get six,

例如，如果你用 3 乘以 2，得到 6，

you can then multiply by one-half to get back to three.

然后你可以用 6 乘以 1/2 得回原来的数 3。

So the multiplicative inverse of two is one-half,

所以，2 的倒数是 1/2，

and the multiplicative inverse of 10 is one-tenth.

10 的倒数是 1/10。

As you might notice, the product of any number and its multiplicative inverse

你可能会注意到，任何一个数与其倒数相乘

is always one.

结果总是 1。

If we want to divide by zero,

如果我们想除以 0 的话，

we need to find its multiplicative inverse,

我们需要找到它的倒数，

which should be one over zero.

那应该是 1/0。

This would have to be such a number that multiplying it by zero would give one.

这个数乘以 0 的话会等于1。

But because anything multiplied by zero is still zero,

但是因为所有数字乘以 0 结果仍然是 0，

such a number is impossible,

那 1/0 这样的数字是不可能的，

so zero has no multiplicative inverse.

所以 0 没有倒数。

Does that really settle things, though?

然而这样就解决问题了吗？

After all, mathematicians have broken rules before.

毕竟，数学家们以前还是破例了。

For example, for a long time,

例如，长期以来，

there was no such thing as taking the square root of negative numbers.

负数是不能取平方根的。

But then mathematicians defined the square root of negative one

但后来数学家们取 -1 的平方根

as a new number called i,

为一个叫 i 的新数字，

opening up a whole new mathematical world of complex numbers.

在数学领域里，这为复杂数字打开了全新的世界。

So if they can do that,

所以，如果他们可以那样做的话，

couldn’t we just make up a new rule,

我们不能创建一个新的规则吗，

say, that the symbol infinity means one over zero,

即，无穷大等于 1/0，

and see what happens?

看看会怎样？

Let's try it,

我们试试，

imagining we don’t know anything about infinity already.

假设我们对无穷大一无所知。

Based on the definition of a multiplicative inverse,

基于倒数的定义，

zero times infinity must be equal to one.

0 乘以无穷大一定等于 1。

That means zero times infinity plus zero times infinity should equal two.

那意味着 0 乘以无穷大再加上 0 乘以无穷大应该等于2。

Now, by the distributive property,

现在，根据乘法分配律，

the left side of the equation can be rearranged

可以将等式左边的式子整理成

to zero plus zero times infinity.

(0 + 0) 乘以 1/0。

And since zero plus zero is definitely zero,

既然 0 + 0 一定是等于 0，

that reduces down to zero times infinity.

那可以缩简成 0 乘以 1/0。

Unfortunately, we’ve already defined this as equal to one,

很遗憾，我们一开始已经得到答案这等于 1 了，

while the other side of the equation is still telling us it’s equal to two.

然而等式另一边答案仍然是 2。

So, one equals two.

所以，1 等于 2。

Oddly enough, that's not necessarily wrong;

太奇怪了，这也不一定错；

it's just not true in our normal world of numbers.

只是在我们正常的数字世界里，这不对。

There’s still a way it could be mathematically valid,

在数学上，还是有方法可以证明其是合理的，

if one, two, and every other number were equal to zero.

如果 1, 2 或其它任何一个数字都等于 0 的话。

But having infinity equal to zero

但是无穷大等于 0

is ultimately not all that useful to mathematicians, or anyone else.

对于数学家或其他任何人来说 最终并不那么有用。

There actually is something called the Riemann sphere

事实上，有个叫黎曼球面的东西

that involves dividing by zero by a different method,

它涉及到通过不同的方法来除以 0，

but that’s a story for another day.

但今天我们且不谈这个。

In the meantime, dividing by zero in the most obvious way

同时，很显然，直接除以 0

doesn’t work out so great.

并没有什么意义。

But that shouldn’t stop us from living dangerously

但那不应该阻止我们在生活中去冒险

and experimenting with breaking mathematical rules

及打破数学规则去进行实验，

to see if we can invent fun, new worlds to explore.

以看看我们是否能创造 新的有趣的世界去探索。