WEBVTT 00:00:07.603 --> 00:00:11.000 This may look like a neatly arranged stack of numbers, 00:00:11.000 --> 00:00:14.506 but it's actually a mathematical treasure trove. 00:00:14.506 --> 00:00:18.654 Indian mathematicians called it the Staircase of Mount Meru. 00:00:18.654 --> 00:00:21.131 In Iran, it's the Khayyam Triangle. 00:00:21.131 --> 00:00:23.738 And in China, it's Yang Hui's Triangle. 00:00:23.738 --> 00:00:28.033 To much of the Western world, it's known as Pascal's Triangle 00:00:28.033 --> 00:00:31.085 after French mathematician Blaise Pascal, 00:00:31.085 --> 00:00:35.234 which seems a bit unfair since he was clearly late to the party, 00:00:35.234 --> 00:00:37.476 but he still had a lot to contribute. 00:00:37.476 --> 00:00:42.270 So what is it about this that has so intrigued mathematicians the world over? 00:00:42.270 --> 00:00:46.124 In short, it's full of patterns and secrets. 00:00:46.124 --> 00:00:49.428 First and foremost, there's the pattern that generates it. 00:00:49.428 --> 00:00:54.477 Start with one and imagine invisible zeros on either side of it. 00:00:54.477 --> 00:00:58.592 Add them together in pairs, and you'll generate the next row. 00:00:58.592 --> 00:01:02.066 Now, do that again and again. 00:01:02.066 --> 00:01:05.784 Keep going and you'll wind up with something like this, 00:01:05.784 --> 00:01:09.325 though really Pascal's Triangle goes on infinitely. 00:01:09.325 --> 00:01:14.914 Now, each row corresponds to what's called the coefficients of a binomial expansion 00:01:14.914 --> 00:01:18.898 of the form (x+y)^n, 00:01:18.898 --> 00:01:21.307 where n is the number of the row, 00:01:21.307 --> 00:01:23.746 and we start counting from zero. 00:01:23.746 --> 00:01:26.552 So if you make n=2 and expand it, 00:01:26.552 --> 00:01:31.107 you get (x^2) + 2xy + (y^2). 00:01:31.107 --> 00:01:34.023 The coefficients, or numbers in front of the variables, 00:01:34.023 --> 00:01:38.397 are the same as the numbers in that row of Pascal's Triangle. 00:01:38.397 --> 00:01:43.256 You'll see the same thing with n=3, which expands to this. 00:01:43.256 --> 00:01:48.493 So the triangle is a quick and easy way to look up all of these coefficients. 00:01:48.493 --> 00:01:50.037 But there's much more. 00:01:50.037 --> 00:01:52.897 For example, add up the numbers in each row, 00:01:52.897 --> 00:01:56.039 and you'll get successive powers of two. 00:01:56.039 --> 00:02:01.221 Or in a given row, treat each number as part of a decimal expansion. 00:02:01.221 --> 00:02:07.835 In other words, row two is (1x1) + (2x10) + (1x100). 00:02:07.835 --> 00:02:12.111 You get 121, which is 11^2. 00:02:12.111 --> 00:02:15.872 And take a look at what happens when you do the same thing to row six. 00:02:15.872 --> 00:02:25.136 It adds up to 1,771,561, which is 11^6, and so on. 00:02:25.136 --> 00:02:27.890 There are also geometric applications. 00:02:27.890 --> 00:02:29.691 Look at the diagonals. 00:02:29.691 --> 00:02:34.117 The first two aren't very interesting: all ones, and then the positive integers, 00:02:34.117 --> 00:02:36.656 also known as natural numbers. 00:02:36.656 --> 00:02:40.707 But the numbers in the next diagonal are called the triangular numbers 00:02:40.707 --> 00:02:42.783 because if you take that many dots, 00:02:42.783 --> 00:02:46.389 you can stack them into equilateral triangles. 00:02:46.389 --> 00:02:49.307 The next diagonal has the tetrahedral numbers 00:02:49.307 --> 00:02:54.622 because similarly, you can stack that many spheres into tetrahedra. 00:02:54.622 --> 00:02:57.996 Or how about this: shade in all of the odd numbers. 00:02:57.996 --> 00:03:00.881 It doesn't look like much when the triangle's small, 00:03:00.881 --> 00:03:03.298 but if you add thousands of rows, 00:03:03.298 --> 00:03:07.439 you get a fractal known as Sierpinski's Triangle. 00:03:07.439 --> 00:03:10.756 This triangle isn't just a mathematical work of art. 00:03:10.756 --> 00:03:12.742 It's also quite useful, 00:03:12.742 --> 00:03:15.481 especially when it comes to probability and calculations 00:03:15.481 --> 00:03:18.566 in the domain of combinatorics. 00:03:18.566 --> 00:03:20.454 Say you want to have five children, 00:03:20.454 --> 00:03:22.270 and would like to know the probability 00:03:22.270 --> 00:03:26.590 of having your dream family of three girls and two boys. 00:03:26.590 --> 00:03:28.388 In the binomial expansion, 00:03:28.388 --> 00:03:32.116 that corresponds to girl plus boy to the fifth power. 00:03:32.116 --> 00:03:33.660 So we look at the row five, 00:03:33.660 --> 00:03:37.131 where the first number corresponds to five girls, 00:03:37.131 --> 00:03:39.929 and the last corresponds to five boys. 00:03:39.929 --> 00:03:42.692 The third number is what we're looking for. 00:03:42.692 --> 00:03:46.642 Ten out of the sum of all the possibilities in the row. 00:03:46.642 --> 00:03:51.490 so 10/32, or 31.25%. 00:03:51.490 --> 00:03:55.316 Or, if you're randomly picking a five-player basketball team 00:03:55.316 --> 00:03:57.084 out of a group of twelve friends, 00:03:57.084 --> 00:04:00.102 how many possible groups of five are there? 00:04:00.102 --> 00:04:05.062 In combinatoric terms, this problem would be phrased as twelve choose five, 00:04:05.062 --> 00:04:07.237 and could be calculated with this formula, 00:04:07.237 --> 00:04:11.708 or you could just look at the sixth element of row twelve on the triangle 00:04:11.708 --> 00:04:13.383 and get your answer. 00:04:13.383 --> 00:04:15.079 The patterns in Pascal's Triangle 00:04:15.079 --> 00:04:19.387 are a testament to the elegantly interwoven fabric of mathematics. 00:04:19.387 --> 00:04:23.271 And it's still revealing fresh secrets to this day. 00:04:23.271 --> 00:04:27.422 For example, mathematicians recently discovered a way to expand it 00:04:27.422 --> 00:04:30.019 to these kinds of polynomials. 00:04:30.019 --> 00:04:31.758 What might we find next? 00:04:31.758 --> 00:04:34.097 Well, that's up to you.