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

Translator: Winnie Ling Reviewer: Alan Watson

This may look like a neatly arranged stack of numbers,

呢啲睇落可能只係一堆排列整齊既數字

但事實上佢係數學嘅寶藏

but it's actually a mathematical treasure trove.

Indian mathematicians called it the Staircase of Mount Meru.

印度數學家稱為「梅魯火山之梯」

In Iran, it's the Khayyam Triangle.

喺伊朗，佢係「海亞姆三角」

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

而喺中國，佢係「楊輝三角」

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

喺大部份嘅西方國家

佢係「帕斯卡三角」

以法國數學家布萊茲 ‧ 帕斯卡嚟命名

after French mathematician Blaise Pascal,

咁嘅名命睇落有啲唔公平

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

因為帕斯卡係後期嘅人 去研究呢款三角形

但佢嘅貢獻都唔少

but he still had a lot to contribute.

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

咁到底係咩

令到世界嘅數學家都咁著迷呢？

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

簡單啲嚟講

係因為佢充滿咗唔同嘅規律同秘密

首先講下畫呢個三角形嘅方法

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

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

由 1 開始

想像兩邊各有一個見唔到嘅 0

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

將佢哋兩個兩個咁相加

你就會得到下一行

Now, do that again and again.

重覆咁做

繼續做，你就會得到呢個三角形

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

though really Pascal's Triangle goes on infinitely.

但其實，帕斯卡三角係無限延伸

而家，每一行嘅數字

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

就係喺二項式 (x+y)^n 展開嘅系數

of the form (x+y)^n,

where n is the number of the row,

而 n 就係行數

and we start counting from zero.

由 0 開始數

So if you make n=2 and expand it,

如果 n=2 ，你代入佢

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

你會得到 x^2 + 2xy + y^2

系數，即係變數前嘅數字

The coefficients, or numbers in front of the variables,

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

同帕斯卡三角嗰行嘅數字一樣

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

當 n=3

展開之後，你會見到相同嘅情況

所以呢個三角形係一個

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

快捷而且簡單嘅方法去搵呢啲系數

But there's much more.

不過，秘密仲有好多

For example, add up the numbers in each row,

例如，將同一行嘅數字加起嚟

and you'll get successive powers of two.

你會得到 2 嘅 n 次方

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

或者喺指定嘅一行

當每個數字都係十進制展開嘅一部份

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

即係話

第三行係 (1x1) + (2x10) + (1x100)

You get 121, which is 11^2.

等於 121，即係 11^2

睇下如果喺第六行做相同嘅嘢會點？

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

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

一共係 1,771,561，亦即係 11^6

之後嘅都係咁

呢三角形仲有唔同嘅幾何應用

There are also geometric applications.

Look at the diagonals.

睇下啲對角線

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

第一同第二條對角線並唔係好有趣

全部都係 1 ，同埋正整數

also known as natural numbers.

亦即係自然數

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

而喺下一條對角數嘅數字

我哋稱為三角數

because if you take that many dots,

因為當你將咁多點排列

you can stack them into equilateral triangles.

你可以排出一個等邊三角形

The next diagonal has the tetrahedral numbers

喺跟住落嚟嘅對角線上嘅係三角錐體數

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

同樣，你可以將呢啲數目砌成三角錐體

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

或者咁，遮住所有單數

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

當個三角形仲細嘅時候

你睇唔出係啲咩

but if you add thousands of rows,

但當你加上成千上萬咁多行之後

you get a fractal known as Sierpinski's Triangle.

你就會得到一個碎形

亦即係謝爾賓斯三角形

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

呢個三角形唔單只係數學嘅藝術

It's also quite useful,

佢都幾有用

especially when it comes to probability and calculations

特別係計概率同埋組合數學

in the domain of combinatorics.

Say you want to have five children,

例如你想要 5 個小朋友

and would like to know the probability

而且想知道

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

有 3 個女仔同 2 個男仔 呢個理想家庭嘅概率

In the binomial expansion,

喺二項式入面

that corresponds to girl plus boy to the fifth power.

呢個即係女仔加男仔嘅 5 次方

So we look at the row five,

咁我哋睇下第五行

where the first number corresponds to five girls,

第一個數字代表 5 個女仔

and the last corresponds to five boys.

而最尾嗰個代表 5 個男仔

The third number is what we're looking for.

第三個數字就係我哋搵緊嗰個

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

呢一行所有可能嘅總和分之 10

so 10/32, or 31.25%.

即係 10/32 ，或者 31.25%

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

或者，你隨機喺 12 個朋友入面

out of a group of twelve friends,

揀出一隊 5 人籃球隊

how many possible groups of five are there?

可以有幾多種組合呢？

喺組合數學嚟講

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

呢個問題可以睇成 12 揀 5

而且可以用呢條式去計

and could be calculated with this formula,

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

或者你可以喺呢個三角形入面

搵第十二行第六個數字，就會得到答案

and get your answer.

The patterns in Pascal's Triangle

帕斯卡三角形嘅規律

展現數學優雅交織嘅一面

are a testament to the elegantly interwoven fabric of mathematics.

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

我哋至今仍然繼續發現佢新嘅秘密

For example, mathematicians recently discovered a way to expand it

例如

數學家最近發現咗 展開呢種多項式嘅方法

to these kinds of polynomials.

What might we find next?

跟住落嚟我哋會發現啲咩？

Well, that's up to you.

咁就睇你啦