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

譯者: Chen Chi-An 審譯者: Helen Chang

In the world of math,

在數學世界裡，

many strange results are possible when we change the rules.

改變規則可能會造成 很多奇怪的結果，

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

但有一個規則，多數人 都曾被告誡不要打破：

don’t divide by zero.

別除以 0 。

How can the simple combination of an everyday number

這個簡單的組合，

一個常用數字配上基本運算符號，

and a basic operation cause such problems?

何以造成問題呢？

Normally, dividing by smaller and smaller numbers

一般情況下，除以越來越小的數字

gives you bigger and bigger answers.

會得到越來越大的答案，

Ten divided by two is five,

10 除以 2 等於 5 ，

by one is ten,

除以 1 等於 10 ，

by one-millionth is 10 million,

除以百萬分之一等於一千萬，

and so on.

So it seems like if you divide by numbers

如此類推。

看起來，如果除以一個數字，

that keep shrinking all the way down to zero,

那數字趨近於 0 ，

the answer will grow to the largest thing possible.

答案會趨近無限大（ ∞ ）。

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

因此， 10 除以 0 的答案 不正是 ∞ 嗎？

That may sound plausible.

這聽起來很合理，

But all we really know is that if we divide 10

但我們知道，如果 10 除以 某個趨近於 0 的數字，

by a number that tends towards zero,

the answer tends towards infinity.

答案會趨近 ∞ ，

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

但這不等於表示，

10 除以 0 就等於 ∞ 。

is equal to infinity.

Why not?

為何不等於？

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

讓我們仔細看看除法的真正含義。

Ten divided by two could mean,

10 除以 2 可以代表

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

「我們要加幾次 2 才等於 10 ？」

or, “two times what equals 10?”

或「 2 乘以多少等於 10 ？」

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

除以某數本質上是乘以某數的相反，

in the following way:

以下解釋：

if we multiply any number by a given number x,

如果把任何數字乘以數字 X，

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

我們能問，是否有一個新數字，

乘以它會回到開始的數字。

to get back to where we started.

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

如果有，這新數字 就稱為 X 的「倒數」。

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

例如，將 3 乘以 2 得到 6 ，

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

接著再乘以 1/2 ，就能回到 3，

So the multiplicative inverse of two is one-half,

因此 2 的倒數是 1/2 ，

and the multiplicative inverse of 10 is one-tenth.

10 的倒數是 1/10 ，

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

你可能發現到，某數和其倒數的乘積

is always one.

永遠等於 1 。

If we want to divide by zero,

如果我們除以 0 ，

we need to find its multiplicative inverse,

我們要找到它的倒數，

which should be one over zero.

這應該是 1/0 ，

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

而 0 乘以它，應該要得到 1 。

But because anything multiplied by zero is still zero,

但因為任何數字 乘以 0 仍然是 0 ，

such a number is impossible,

這樣的數字是不存在的，

so zero has no multiplicative inverse.

因此， 0 沒有倒數。

Does that really settle things, though?

然而這能解釋事情嗎？

After all, mathematicians have broken rules before.

畢竟，數學家曾經打破規則，

For example, for a long time,

例如很長一段時間，

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

開根號內的數字不能是負數，

But then mathematicians defined the square root of negative one

但當數學家定義「-1」的開根號

as a new number called i,

是新數字「 i 」時，

opening up a whole new mathematical world of complex numbers.

這開啟一個全新的「複數」世界，

So if they can do that,

如果能這麼做，

couldn’t we just make up a new rule,

我們能不能訂定新規則說，

say, that the symbol infinity means one over zero,

∞ 代表 1/0，

and see what happens?

然後看看會發生什麼事？

Let's try it,

我們試試看，

imagining we don’t know anything about infinity already.

想像我們不知道 ∞ ，

Based on the definition of a multiplicative inverse,

根據倒數的定義，

zero times infinity must be equal to one.

0 乘以 ∞ 必等於 1，

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

這表示，0 乘以 ∞ 再加上 0 乘以 ∞ 等於 2，

Now, by the distributive property,

現在，根據分配率，

the left side of the equation can be rearranged

等式左邊的運算可以調整成

to zero plus zero times infinity.

0 加 0 ，然後乘以 ∞ 。

And since zero plus zero is definitely zero,

既然 0 加 0 一定等於 0，

that reduces down to zero times infinity.

則等式簡化成 0 乘以 ∞ ，

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

然而，我們已經定義這等於 1 ，

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

而等式另一端說這等於 2 ，

So, one equals two.

因此， 1 等於 2 。

Oddly enough, that's not necessarily wrong;

說來奇怪，這不見得是錯的；

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

只是在我們平常的 數字世界中錯誤而已。

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

仍有個方法能讓此在數學上成立：

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

假如 1 、 2 和任何數字都等於 0，

But having infinity equal to zero

但讓 ∞ 等於 0

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

對數學家或任何人並不實用，

There actually is something called the Riemann sphere

事實上，有個稱為 「黎曼球面」的概念，

that involves dividing by zero by a different method,

用不同的方法除以 0 ，

but that’s a story for another day.

但這故事改天再提。

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

同時，以最直覺的方式除以 0

doesn’t work out so great.

效果並不好，

But that shouldn’t stop us from living dangerously

但這不應該阻止我們冒點險

and experimenting with breaking mathematical rules

或嘗試打破數學規則，

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

看看我們是否能發明一個 有趣的新世界來探索。