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• This may look like a neatly arranged stack of numbers,

這看起來可能只是一堆排列整齊的數字

• but it's actually a mathematical treasure trove.

但這其實是數學史上的瑰寶

• Indian mathematicians called it the Staircase of Mount Meru.

印度數學家們稱它為須彌山之梯

• In Iran, it's the Khayyam Triangle.

在伊朗，這被稱為開儼三角形

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

而在中國，稱之為楊輝三角形

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

對大半部分的西方世界而言，這就是所謂的帕斯卡三角

• after French mathematician Blaise Pascal,

得名於法國數學家布萊茲˙帕斯卡

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

這聽起來實在有點不公平，他明明比前面所述的人都還要晚出生

• but he still had a lot to contribute.

儘管如此，他還是貢獻良多

言歸正傳，帕斯卡三角到底是什麼玩意兒，能讓全世界的數學家為之著迷

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

簡言之，它蘊藏了許多形式與秘密

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

首先，帕斯卡三角有一個組合模式

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

從1開始，左右兩側各有一個假想的0

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

將左右數字兩兩相加，就能得出下一列

• Now, do that again and again.

現在，重複這個步驟

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

一直做下去，你就會得出像這樣的結果

• though really Pascal's Triangle goes on infinitely.

雖然帕斯卡三角是無限延伸的

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

每一列都對應到二項式定理(x+y)^n中

• of the form (x+y)^n,

所謂的係數

• where n is the number of the row,

n對應到的就是列數

• and we start counting from zero.

我們從0開始數

• So if you make n=2 and expand it,

如果令n=2再將其展開

• you get (x^2) + 2xy + (y^2).

會得到 (x^2) + 2xy + (y^2)

• The coefficients, or numbers in front of the variables,

這些係數，也就是變數前的數字

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

都和帕斯卡三角中每一列的數字相同

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

當我們令n=3時也會發生同樣的事，展開後會變成這樣

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

所以這個三角形是找尋這些係數的簡單捷徑

• But there's much more.

但還不只這些呢

• For example, add up the numbers in each row,

像是把每一列的數字加總

• and you'll get successive powers of two.

你就能得到連續的2的次方

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

或是在選定的一列中，把每個數字視為十進位展開

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

換言之，第二列就變成 (1x1) + (2x10) + (1x100)

• You get 121, which is 11^2.

會得到121，也就是11的平方

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

再來看看當你對第六列做同樣計算，會發生什麼事吧

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

加總起來為1,771,561，也就是11的六次方，再繼續往下做便會無限延伸

• There are also geometric applications.

帕斯卡三角中也有幾何應用

• Look at the diagonals.

注意對角線

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

最外面兩條對角線沒什麼意思：全都是1，接下來是正整數

• also known as natural numbers.

也就是自然數

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

但再下一層對角線的數字稱為三角形數

• because if you take that many dots,

因為只要你取一定數量的點

• you can stack them into equilateral triangles.

就能將它們排成等邊三角形

• The next diagonal has the tetrahedral numbers

下一條對角線的數字為四面體數

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

因為同樣地，你可以把一定數量的球體堆疊成四面體

那麼看看這個：把所有奇數遮住

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

當三角形很小的時候還看不太出有什麼變化

• but if you add thousands of rows,

不過如果你加至數千列

• you get a fractal known as Sierpinski's Triangle.

就會得到一個碎形，我們稱之為謝爾賓斯基三角形

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

這個三角形不只是數學上的藝術品

• It's also quite useful,

它的用途也非常廣

• especially when it comes to probability and calculations

尤其是談到組合學中的

• in the domain of combinatorics.

機率及其計算

• Say you want to have five children,

假設你想要有五個小孩

• and would like to know the probability

而你又想知道

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

你夢寐以求的三女二男的機率為何

• In the binomial expansion,

從二項式定理來看

• that corresponds to girl plus boy to the fifth power.

可以想成(女+男)^5

• So we look at the row five,

所以我們來看一下第五列

• where the first number corresponds to five girls,

第一個數字對應到五個女生

• and the last corresponds to five boys.

而最後一個數字對應到五個男生

• The third number is what we're looking for.

我們要找的是第三個數字

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

以十為分子，除以該列所有可能性的加總

• so 10/32, or 31.25%.

得到了10/32，也就是31.25%

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

或者，如果你想要隨機從十二位朋友中

• out of a group of twelve friends,

挑選五位組成籃球隊

• how many possible groups of five are there?

會有幾種可能呢？

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

在組合學名詞中，這種類型的問題我們稱為12取5

• and could be calculated with this formula,

而用這條公式就能算出答案

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

或者你其實可以去找帕斯卡三角中第十二列的第六個數字

馬上得到答案

• The patterns in Pascal's Triangle

帕斯卡三角中的種種模式

• are a testament to the elegantly interwoven fabric of mathematics.

是數學結構錯綜複雜的絕佳證明

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

直至今日它也仍不斷地在綻放全新的秘密

• For example, mathematicians recently discovered a way to expand it

舉例來說，數學家們最近找出能將帕斯卡三角延伸至

• to these kinds of polynomials.

像這類型的多項式的方法了

• What might we find next?

接下來還會有什麼新發現？

• Well, that's up to you.

嗯，就交給你囉。

This may look like a neatly arranged stack of numbers,

B1 中級 中文 美國腔 TED-Ed 數字 三角形 對角線 數學家 係數

# 【Ted-Ed】聽過「帕斯卡三角」嗎? 來一窺數學原理背後的祕密吧！(The mathematical secrets of Pascal’s triangle - Wajdi Mohamed Ratemi)

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