Let's say we're given the function f(x) = x^2 + 1.

有一個方程式，它長這樣：f(x) = x^2 + 1．

We can graph our function and get a nice parabola.

我們可以將這個方程式畫出來，得到一個漂亮的拋物綫

Now let's say we want to figure out where the equation equals zero

現在譬如我們想要知道這個方程式在哪裏會等於零

we want to find the roots.

我們其實是想要找到這個方程式的根

On our plot this should be where the function crosses the x-axis.

在這個圖上，就是圖形通過x軸的地方

As we can see, our parabola actually never crosses the x-axis,

我們可以看到，這個拋物線其實不會經過x軸

so according to our plot, there are no solutions to the equation x^2+1=0.

所以根據這圖型，我們的這個方程式x^2+1=0是無解的

But there's a small problem.

但是這有一個小問題

A little over 200 years ago a smart guy named Gauss

兩千多年以前，一個聰明的傢伙叫做高斯

proved that every polynomial equation of degree n

他證明了對一個最高n次方的多項式來說

has exactly n roots.

應該剛好有n個根

Our polynomial has a highest power, or degree, of two,

我們的這個多項式最高次方是2

so we should have two roots.

所以應當有兩個根才是

And Gauss' discovery is not just some random rule,

而高斯的發現並不是隨隨便便的規則

today we call it the FUNDAMENTAL THEOREM OF ALGEBRA.

今天這被稱之為 “代數的基本定理“

So our plot seems to disagree with something so important it's called the FUNDAMENTAL THEOREM OF ALGEBRA,

所以難道我們的這張圖難道違反了“代數的基本定理”?

which might be a problem.

這一定有一些小小的誤會

What Guass is telling us here, is that there are two perfectly good values of x

高斯所告訴我們的是我們應當可以找到兩個剛剛好的值

that we could plug into our function, and get zero out.

帶入方程式之後，消去之後可以得到0

Where could these 2 missing roots be?

那這兩個根到底躲到哪裡去了呢?

The short answer here is that we don't have enough numbers.

簡單地說就是我們沒有足夠的數字

We typically think of numbers existing on a 1 dimensional continuum - the number line.

一般來說我們把數字想像成存在於一條連續的線—-數線上

All our friends are here: 0, 1,

我們的數字小朋友們如 0 和 1

negative numbers, fractions, even irrational numbers like root 2 show up.

負數和分數，即便是無理數如根號2

--

都在這條線上站了一個小小的位置

But this system is incomplete.

但是這樣的系統並不完整

And our missing numbers are not just further left or right,

我們的那兩個數字小朋友到底躲到什麼地方去了呢？

-

他們並不是在更左邊或更右邊

they live in a whole new dimension.

他們是在一個全新的維度

Algebraically, this new dimension

依照代數學來說，這個全新的維度

has everything to do with a problem that was mathematically considered impossible for over two thousand years:

可以讓我們解答這個幾乎兩千年來數學上認為不可能的事

the square root of negative one.

-1的平方根

When we include this missing dimension in our analysis,

就讓我們加入這個消失的維度

our parabola gets way more interesting.

這樣子，我們的拋物線將會變得非常有趣．

Now that our input numbers are in their full two dimensional form,

既然我們的數字總共有了兩個維度

we see how our function x^2+1 really behaves.

我們來看看我們的方程式 x^2+1現在變成什麼樣子

And we can now see that our function does have exactly two roots!

現在我們的方程式的確通過 x軸

We were just looking in the wrong dimension.

我們之前只是從一個錯誤的維度看過去

So, why is this extra dimension that numbers possess not common knowledge?

所以，為什麼大家都不認識數字的這一個額外的維度呢?

Part of this reason is that it has been given a terrible, terrible name.

部分的原因是因為他們被取了一個很糟糕的名字

A name that suggest that these numbers aren't ever real!

讓它們看起來一點也不真實

In fact, Gauss himself had something to say about this naming convention.

事實上，高斯自己對於這件事可有點話要說

So yes, this missing dimension is comprised of numbers that have been given ridiculous name imaginary.

沒錯，消失的維度就是由這個叫作虛數的數字所構成的

Gauss proposed these numbers should instead be given the name lateral

高斯提議這些數字應當改名叫做側數

so from here on, let's let lateral mean imaginary.

所以現在我就用側數來稱呼虛數

To get a better handle on imaginary,

為了要對虛數更有感覺一點

I mean, lateral numbers,

喔，我的意思是側數

and really understand what's going on here,

同時對於接下的內容更容易理解

let's spend a little time thinking about numbers.

我們花一點時間來瞭解一下數字

Early humans really only had use for the natural numbers, that is 1, 2, 3, and so on.

古代人類只會使用自然的數字，1 2 3 等

This makes sense because of how numbers were used.

這很說得通，因為數字就是這樣用的

So to early humans, the number line would have just been a series of dots.

所以早期人類的數線上只有一些點而已

As civilizations advanced,

當文明越來越進步

people needed answers to more sophisticated math questions –

人們越來越需要數學來回答一些複雜的問題

ike when to plant seeds,

例如什麼時候要播種了

how to divide land, and how to keep track of financial transactions.

土地該怎麼分配，還有有金錢交易要如何記帳

The natural numbers just weren’t cutting it anymore,

自然數漸漸的就不夠用了

so the Egyptians innovated

因此古埃及人發揮他們的創意

and developed a new, high tech solution: fractions.

發展了一個全新的高科技：分數

Fractions filled in the gaps in our number line,

分數幫我們填滿了數線上的空隙

and were basically cutting edge technology for a couple thousand years.

而且基本上，在幾千年間，他是最先進的高科技

The next big innovations to hit the number line were the number zero and negative numbers,

下一個數線上的大發明是數字零和負數

but it took some time to get everyone on board.

但是我們可是花了好多時間讓大夥都上了道

Since it’s not obvious what these numbers mean

因為這些數字代表的意義可不是那麼明顯

or how they fit into the real world,

他們代表的意義比較抽象一點

zero and negative numbers were met with skepticism,

零和負數讓人很懷疑

and largely avoided or ignored .

而且很多人避免使用他，忽視它們

Some cultures were more suspicious than others,

一些文明對於他們比較猜疑

depending largely on how people viewed the connection between mathematics and reality.

一大部分是取決於人們如何看待數學和真實世界的連結

And this is not all ancient history -

這可不是發生在真正的古代

just a few centuries ago,

只不過是幾世紀前

mathematicians would intentionally move terms around to avoid having negatives show up in equations.

數學家會故意用移項來避免他們的等式裡出現負數

Suspicion of zero and negative numbers did eventually fade -

最終對於零和負數的猜疑逐漸地淡去了

partially because negatives are useful for expressing concepts like debt,

部分原因是因為負數對於代表負債時特別有用

but mostly because negatives just kept sneaking into mathematics.

但是絕大多數是因為負數已經漸漸地深入到很多數學中

It turns out there’s just a whole lot of math you just can’t do

結果就是如果你不讓負數參一腳的話

if you don’t allow negative numbers to play.

很多數學運算你都沒辦法做

Without negatives, simple algebra problems like x + 3 = 2 have no answer.

如果沒有負數，像x + 3 = 2 的簡單代數就無解了

Before negatives were accepted,

如果你不接納負數的存在

this problem would have no solution,

這個式子就沒有解

just like we thought our original problem had no solution.

就像我們原來的式子一樣

The thing is, it’s not crazy or weird to think problems like this have no solutions –

其實是這樣的，認為這樣的問題沒有解答並不瘋狂

in words, this algebra problem basically says:

這個代數問題其實可以這樣詮釋：

“if I have 2 things and I take away 3, how many things do I have left?”

如果我有兩個東西，我拿走了三個，那還剩下幾個呢？

It’s not surprising that most of the people who have lived on our planet would be suspicious of questions like this.

一般地球人大概都會很懷疑怎麼會有這種問題

These problems don’t make any sense.

因為這一點意義也沒有

Even brilliant mathematicians of the 18th century, such as Leonard Euler,

即便是十八世紀最聰明的數學家--李奧納多 歐拉

didn’t really know what to do with negatives –

也不知道要怎麼處理負數

he at one point wrote that negatives were greater than infinity.

他一度還把負數當作是比無限大還要大

So it’s fair to say that

我們可以這麼說

negative and imaginary numbers raise a lot of very good, very valid questions.

負數和虛數的確給我們帶來了很多很好的問題

-

讓我們思考

Like why do we require students to understand and work with numbers

例如，這些問題已被最偉大的數學家逃避了幾千年

Like why do we require students to understand and work with numbers

為什麼我們還要求學生們去了解它們

-

用它們來運算

Why did we even come accept negative and imaginary numbers in the first place,

既然它們和現實世界幾乎毫不相關

when they don’t really seem connected to anything in the real world?

為什麼我們卻先接受它們呢？

And how do these extra numbers help explain the missing solutions to our problem?

這些缺少的數字如何能夠合理解釋我們的問題呢？

Next time, we’ll begin to address these questions

下一次我們會來談談這些問題

by going way back to the discovery of complex numbers.

我們會回到以前來看看複數是怎麼發現的

-

[結束-中文翻譯：游國仁]