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.
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 印度數學家稱為「梅魯火山之梯」 喺伊朗,佢係「海亞姆三角」 而喺中國,佢係「楊輝三角」 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 咁嘅名命睇落有啲唔公平 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 咁到底係咩 令到世界嘅數學家都咁著迷呢? 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 由 1 開始 想像兩邊各有一個見唔到嘅 0 將佢哋兩個兩個咁相加 你就會得到下一行 重覆咁做 繼續做,你就會得到呢個三角形 但其實,帕斯卡三角係無限延伸 而家,每一行嘅數字 就係喺二項式 (x+y)^n 展開嘅系數 而 n 就係行數 由 0 開始數 如果 n=2 ,你代入佢 你會得到 x^2 + 2xy + y^2 系數,即係變數前嘅數字 同帕斯卡三角嗰行嘅數字一樣 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 快捷而且簡單嘅方法去搵呢啲系數 不過,秘密仲有好多 例如,將同一行嘅數字加起嚟 你會得到 2 嘅 n 次方 或者喺指定嘅一行 當每個數字都係十進制展開嘅一部份 即係話 第三行係 (1x1) + (2x10) + (1x100) 等於 121,即係 11^2 睇下如果喺第六行做相同嘅嘢會點? 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 睇下啲對角線 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 亦即係自然數 而喺下一條對角數嘅數字 我哋稱為三角數 因為當你將咁多點排列 你可以排出一個等邊三角形 喺跟住落嚟嘅對角線上嘅係三角錐體數 同樣,你可以將呢啲數目砌成三角錐體 或者咁,遮住所有單數 當個三角形仲細嘅時候 你睇唔出係啲咩 但當你加上成千上萬咁多行之後 你就會得到一個碎形 亦即係謝爾賓斯三角形 呢個三角形唔單只係數學嘅藝術 佢都幾有用 特別係計概率同埋組合數學 例如你想要 5 個小朋友 而且想知道 有 3 個女仔同 2 個男仔 呢個理想家庭嘅概率 喺二項式入面 呢個即係女仔加男仔嘅 5 次方 咁我哋睇下第五行 第一個數字代表 5 個女仔 而最尾嗰個代表 5 個男仔 第三個數字就係我哋搵緊嗰個 呢一行所有可能嘅總和分之 10 即係 10/32 ,或者 31.25% 或者,你隨機喺 12 個朋友入面 揀出一隊 5 人籃球隊 可以有幾多種組合呢? 喺組合數學嚟講 呢個問題可以睇成 12 揀 5 而且可以用呢條式去計 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 帕斯卡三角形嘅規律 展現數學優雅交織嘅一面 我哋至今仍然繼續發現佢新嘅秘密 例如 數學家最近發現咗 展開呢種多項式嘅方法 跟住落嚟我哋會發現啲咩? 咁就睇你啦