The mathematical secrets of Pascal’s triangle - Wajdi Mohamed Ratemi

譯者: Kelly Liu 審譯者: Max Chern

This may look like a neatly arranged stack of numbers,

這看起來像是一堆整齊、 精心排列的數字

but it's actually a mathematical treasure trove.

其實是個數學百寶箱

Indian mathematicians called it the Staircase of Mount Meru.

印度數學家稱之為「須彌山之梯」

In Iran, it's the Khayyam Triangle.

在伊朗稱作「海亞姆三角形」

And in China, it's Yang Hui's Triangle.

在中國稱作「楊輝三角」

To much of the Western world, it's known as Pascal's Triangle

對多數西方世界來說， 它是「帕斯卡三角形」

after French mathematician Blaise Pascal,

由法國數學家 布萊茲·帕斯卡 而得名

which seems a bit unfair since he was clearly late to the party,

似乎有些不公平， 他的研究時間明顯較晚

but he still had a lot to contribute.

但他仍有許多貢獻

So what is it about this that has so intrigued mathematicians the world over?

究竟是什麼讓世界上的數學家 如此感興趣呢？

In short, it's full of patterns and secrets.

簡單來說，它充滿了許多型式和秘密

First and foremost, there's the pattern that generates it.

首先且最重要的， 有個產生三角形的型式

Start with one and imagine invisible zeros on either side of it.

從 1 開始，然後想像它的左右各有一個 0

Add them together in pairs, and you'll generate the next row.

將它們兩兩相加，便能得到下一列

Now, do that again and again.

然後不斷的重複

Keep going and you'll wind up with something like this,

繼續下去，你會得到像這樣的東西

though really Pascal's Triangle goes on infinitely.

按理來說，帕斯卡三角形是無限大的

Now, each row corresponds to what's called the coefficients of a binomial expansion

每一列對應到二項式 (x+y)^n 展開時的係數

of the form (x+y)^n,

where n is the number of the row,

n 代表列數

and we start counting from zero.

從 0 開始算起

So if you make n=2 and expand it,

所以，當 n=2 並將式子展開

you get (x^2) + 2xy + (y^2).

你會得到 (x^2) + 2xy + (y^2)

The coefficients, or numbers in front of the variables,

其係數，即在變數前的數字

are the same as the numbers in that row of Pascal's Triangle.

與帕斯卡三角形裡 對應列的數字完全吻合

You'll see the same thing with n=3, which expands to this.

同樣地，當 n=3 時 展開會得到這樣的係數

So the triangle is a quick and easy way to look up all of these coefficients.

所以，要查詢所有係數時， 這三角形是快又簡單的方式

But there's much more.

還不止這樣

For example, add up the numbers in each row,

譬如，個別把每列的數字加起來

and you'll get successive powers of two.

你會得到連續的 2 的次方

Or in a given row, treat each number as part of a decimal expansion.

或是將其中一列作十進位展開

In other words, row two is (1x1) + (2x10) + (1x100).

也就是說 第二列就變成 (1x1) + (2x10) + (1x100)

You get 121, which is 11^2.

會得到 121，也就是 11^2

And take a look at what happens when you do the same thing to row six.

看看如果對第六列也這樣做， 會發生什麼事

It adds up to 1,771,561, which is 11^6, and so on.

總和是 1,771,561, 也就是 11^6，以此類推

There are also geometric applications.

除此之外也有幾何的運用

Look at the diagonals.

看一下對角線

The first two aren't very interesting: all ones, and then the positive integers,

最前面兩個不怎麼有趣：全都是 1， 再來就是正整數

also known as natural numbers.

即是所謂的自然數

But the numbers in the next diagonal are called the triangular numbers

但下一個對角線數字就是三角形數

because if you take that many dots,

因為如果拿這些數目的點

you can stack them into equilateral triangles.

你可以把它們組成一個個正三角形

The next diagonal has the tetrahedral numbers

下一條對角線是四面體的數字

because similarly, you can stack that many spheres into tetrahedra.

因為同樣地， 你能用這數目的球堆出四面體

Or how about this: shade in all of the odd numbers.

或這樣，把奇數的部分上色

It doesn't look like much when the triangle's small,

當三角形還小時，看起來不怎麼樣

but if you add thousands of rows,

但若是加到好幾千列

you get a fractal known as Sierpinski's Triangle.

會得到一個碎形， 稱為「謝爾賓斯基三角形」

This triangle isn't just a mathematical work of art.

這三角形不只是個數學的藝術

It's also quite useful,

它也相當的實用

especially when it comes to probability and calculations

尤其在組合數學領域裡的 機率和計算

in the domain of combinatorics.

Say you want to have five children,

假設，你想要有 5 個小孩

and would like to know the probability

想知道理想中的家庭

of having your dream family of three girls and two boys.

有 3 個女孩和 2 個男孩的機率

In the binomial expansion,

在二項式展開中

that corresponds to girl plus boy to the fifth power.

相當於女加男的 5 次方

So we look at the row five,

所以我們看第五列

where the first number corresponds to five girls,

第一個數字 代表有 5 個女孩的可能性

and the last corresponds to five boys.

最後一個數字 代表有 5 個男孩的可能性

The third number is what we're looking for.

而第三個數字就是我們要找的

Ten out of the sum of all the possibilities in the row.

整列所有可能性總和 當中的 10 個可能性

so 10/32, or 31.25%.

因此機率為 10/32，也就是 31.25%

Or, if you're randomly picking a five-player basketball team

或是你隨機在 12 個朋友中

out of a group of twelve friends,

挑出 5 人組籃球隊

how many possible groups of five are there?

總共會有多少種五人組合呢？

In combinatoric terms, this problem would be phrased as twelve choose five,

在組合數學術語中， 這問題的用語表達是 12 取 5

and could be calculated with this formula,

可用此公式算出

or you could just look at the sixth element of row twelve on the triangle

或是你可查三角形第 12 列的第 6 個數字

and get your answer.

得到你要的答案

The patterns in Pascal's Triangle

帕斯卡三角形中的諸多型式

are a testament to the elegantly interwoven fabric of mathematics.

是由數學優雅交織而成的驗證

And it's still revealing fresh secrets to this day.

至今仍為我們揭開新的秘密

For example, mathematicians recently discovered a way to expand it

舉例來說， 數學家們最近找到一個方法來展開

to these kinds of polynomials.

像這樣的多項式

What might we find next?

接下來會有怎樣的發現呢？

Well, that's up to you.

就要看你囉！

翻譯：Kelly Liu