This may look like a neatly arranged stack of numbers, but it's actually a mathematical treasure trove. Indian mathematicians called it the Staircase of Mount Meru. In Iran, it's the Khayyam Triangle. And in China, it's Yang Hui's Triangle. To much of the Western world, it's known as Pascal's Triangle after French mathematician Blaise Pascal, which seems a bit unfair since he was clearly late to the party, but he still had a lot to contribute. So what is it about this that has so intrigued mathematicians the world over? In short, it's full of patterns and secrets. First and foremost, there's the pattern that generates it. Start with one and imagine invisible zeros on either side of it. Add them together in pairs, and you'll generate the next row. Now, do that again and again. Keep going and you'll wind up with something like this, though really Pascal's Triangle goes on infinitely. Now, each row corresponds to what's called the coefficients of a binomial expansion of the form (x+y)^n, where n is the number of the row, and we start counting from zero. So if you make n=2 and expand it, you get (x^2) + 2xy + (y^2). The coefficients, or numbers in front of the variables, are the same as the numbers in that row of Pascal's Triangle. You'll see the same thing with n=3, which expands to this. So the triangle is a quick and easy way to look up all of these coefficients. But there's much more. For example, add up the numbers in each row, and you'll get successive powers of two. Or in a given row, treat each number as part of a decimal expansion. In other words, row two is (1x1) + (2x10) + (1x100). You get 121, which is 11^2. 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. There are also geometric applications. Look at the diagonals. The first two aren't very interesting: all ones, and then the positive integers, also known as natural numbers. But the numbers in the next diagonal are called the triangular numbers because if you take that many dots, you can stack them into equilateral triangles. The next diagonal has the tetrahedral numbers because similarly, you can stack that many spheres into tetrahedra. Or how about this: shade in all of the odd numbers. It doesn't look like much when the triangle's small, but if you add thousands of rows, you get a fractal known as Sierpinski's Triangle. This triangle isn't just a mathematical work of art. It's also quite useful, especially when it comes to probability and calculations in the domain of combinatorics. Say you want to have five children, and would like to know the probability of having your dream family of three girls and two boys. In the binomial expansion, that corresponds to girl plus boy to the fifth power. So we look at the row five, where the first number corresponds to five girls, and the last corresponds to five boys. The third number is what we're looking for. Ten out of the sum of all the possibilities in the row. so 10/32, or 31.25%. Or, if you're randomly picking a five-player basketball team out of a group of twelve friends, how many possible groups of five are there? In combinatoric terms, this problem would be phrased as twelve choose five, and could be calculated with this formula, or you could just look at the sixth element of row twelve on the triangle and get your answer. The patterns in Pascal's Triangle are a testament to the elegantly interwoven fabric of mathematics. And it's still revealing fresh secrets to this day. For example, mathematicians recently discovered a way to expand it to these kinds of polynomials. What might we find next? Well, that's up to you.
這看起來像是一堆整齊、 精心排列的數字 其實是個數學百寶箱 印度數學家稱之為「須彌山之梯」 在伊朗稱作「海亞姆三角形」 在中國稱作「楊輝三角」 對多數西方世界來說, 它是「帕斯卡三角形」 由法國數學家 布萊茲·帕斯卡 而得名 似乎有些不公平, 他的研究時間明顯較晚 但他仍有許多貢獻 究竟是什麼讓世界上的數學家 如此感興趣呢? 簡單來說,它充滿了許多型式和秘密 首先且最重要的, 有個產生三角形的型式 從 1 開始,然後想像它的左右各有一個 0 將它們兩兩相加,便能得到下一列 然後不斷的重複 繼續下去,你會得到像這樣的東西 按理來說,帕斯卡三角形是無限大的 每一列對應到二項式 (x+y)^n 展開時的係數 n 代表列數 從 0 開始算起 所以,當 n=2 並將式子展開 你會得到 (x^2) + 2xy + (y^2) 其係數,即在變數前的數字 與帕斯卡三角形裡 對應列的數字完全吻合 同樣地,當 n=3 時 展開會得到這樣的係數 所以,要查詢所有係數時, 這三角形是快又簡單的方式 還不止這樣 譬如,個別把每列的數字加起來 你會得到連續的 2 的次方 或是將其中一列作十進位展開 也就是說 第二列就變成 (1x1) + (2x10) + (1x100) 會得到 121,也就是 11^2 看看如果對第六列也這樣做, 會發生什麼事 總和是 1,771,561, 也就是 11^6,以此類推 除此之外也有幾何的運用 看一下對角線 最前面兩個不怎麼有趣:全都是 1, 再來就是正整數 即是所謂的自然數 但下一個對角線數字就是三角形數 因為如果拿這些數目的點 你可以把它們組成一個個正三角形 下一條對角線是四面體的數字 因為同樣地, 你能用這數目的球堆出四面體 或這樣,把奇數的部分上色 當三角形還小時,看起來不怎麼樣 但若是加到好幾千列 會得到一個碎形, 稱為「謝爾賓斯基三角形」 這三角形不只是個數學的藝術 它也相當的實用 尤其在組合數學領域裡的 機率和計算 假設,你想要有 5 個小孩 想知道理想中的家庭 有 3 個女孩和 2 個男孩的機率 在二項式展開中 相當於女加男的 5 次方 所以我們看第五列 第一個數字 代表有 5 個女孩的可能性 最後一個數字 代表有 5 個男孩的可能性 而第三個數字就是我們要找的 整列所有可能性總和 當中的 10 個可能性 因此機率為 10/32,也就是 31.25% 或是你隨機在 12 個朋友中 挑出 5 人組籃球隊 總共會有多少種五人組合呢? 在組合數學術語中, 這問題的用語表達是 12 取 5 可用此公式算出 或是你可查三角形第 12 列的第 6 個數字 得到你要的答案 帕斯卡三角形中的諸多型式 是由數學優雅交織而成的驗證 至今仍為我們揭開新的秘密 舉例來說, 數學家們最近找到一個方法來展開 像這樣的多項式 接下來會有怎樣的發現呢? 就要看你囉!