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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

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

    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

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

    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

  • and get your answer.

    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,

這看起來可能只是一堆排列整齊的數字

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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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    SylviaQQ 發佈於 2016 年 06 月 30 日
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