[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:07.60,0:00:11.00,Default,,0000,0000,0000,,This may look like a neatly arranged\Nstack of numbers,
Dialogue: 0,0:00:11.00,0:00:14.51,Default,,0000,0000,0000,,but it's actually\Na mathematical treasure trove.
Dialogue: 0,0:00:14.51,0:00:18.65,Default,,0000,0000,0000,,Indian mathematicians called it \Nthe Staircase of Mount Meru.
Dialogue: 0,0:00:18.65,0:00:21.13,Default,,0000,0000,0000,,In Iran, it's the Khayyam Triangle.
Dialogue: 0,0:00:21.13,0:00:23.74,Default,,0000,0000,0000,,And in China, it's Yang Hui's Triangle.
Dialogue: 0,0:00:23.74,0:00:28.03,Default,,0000,0000,0000,,To much of the Western world,\Nit's known as Pascal's Triangle
Dialogue: 0,0:00:28.03,0:00:31.08,Default,,0000,0000,0000,,after French mathematician Blaise Pascal,
Dialogue: 0,0:00:31.08,0:00:35.23,Default,,0000,0000,0000,,which seems a bit unfair\Nsince he was clearly late to the party,
Dialogue: 0,0:00:35.23,0:00:37.48,Default,,0000,0000,0000,,but he still had a lot to contribute.
Dialogue: 0,0:00:37.48,0:00:42.27,Default,,0000,0000,0000,,So what is it about this that has so \Nintrigued mathematicians the world over?
Dialogue: 0,0:00:42.27,0:00:46.12,Default,,0000,0000,0000,,In short, \Nit's full of patterns and secrets.
Dialogue: 0,0:00:46.12,0:00:49.43,Default,,0000,0000,0000,,First and foremost, there's the pattern\Nthat generates it.
Dialogue: 0,0:00:49.43,0:00:54.48,Default,,0000,0000,0000,,Start with one and imagine invisible\Nzeros on either side of it.
Dialogue: 0,0:00:54.48,0:00:58.59,Default,,0000,0000,0000,,Add them together in pairs, \Nand you'll generate the next row.
Dialogue: 0,0:00:58.59,0:01:02.07,Default,,0000,0000,0000,,Now, do that again and again.
Dialogue: 0,0:01:02.07,0:01:05.78,Default,,0000,0000,0000,,Keep going and you'll wind up \Nwith something like this,
Dialogue: 0,0:01:05.78,0:01:09.32,Default,,0000,0000,0000,,though really Pascal's Triangle \Ngoes on infinitely.
Dialogue: 0,0:01:09.32,0:01:14.91,Default,,0000,0000,0000,,Now, each row corresponds to what's called\Nthe coefficients of a binomial expansion
Dialogue: 0,0:01:14.91,0:01:18.90,Default,,0000,0000,0000,,of the form (x+y)^n,
Dialogue: 0,0:01:18.90,0:01:21.31,Default,,0000,0000,0000,,where n is the number of the row,
Dialogue: 0,0:01:21.31,0:01:23.75,Default,,0000,0000,0000,,and we start counting from zero.
Dialogue: 0,0:01:23.75,0:01:26.55,Default,,0000,0000,0000,,So if you make n=2 and expand it,
Dialogue: 0,0:01:26.55,0:01:31.11,Default,,0000,0000,0000,,you get (x^2) + 2xy + (y^2).
Dialogue: 0,0:01:31.11,0:01:34.02,Default,,0000,0000,0000,,The coefficients, \Nor numbers in front of the variables,
Dialogue: 0,0:01:34.02,0:01:38.40,Default,,0000,0000,0000,,are the same as the numbers in that row\Nof Pascal's Triangle.
Dialogue: 0,0:01:38.40,0:01:43.26,Default,,0000,0000,0000,,You'll see the same thing with n=3,\Nwhich expands to this.
Dialogue: 0,0:01:43.26,0:01:48.49,Default,,0000,0000,0000,,So the triangle is a quick and easy way\Nto look up all of these coefficients.
Dialogue: 0,0:01:48.49,0:01:50.04,Default,,0000,0000,0000,,But there's much more.
Dialogue: 0,0:01:50.04,0:01:52.90,Default,,0000,0000,0000,,For example, add up \Nthe numbers in each row,
Dialogue: 0,0:01:52.90,0:01:56.04,Default,,0000,0000,0000,,and you'll get successive powers of two.
Dialogue: 0,0:01:56.04,0:02:01.22,Default,,0000,0000,0000,,Or in a given row, treat each number\Nas part of a decimal expansion.
Dialogue: 0,0:02:01.22,0:02:07.84,Default,,0000,0000,0000,,In other words, row two is\N(1x1) + (2x10) + (1x100).
Dialogue: 0,0:02:07.84,0:02:12.11,Default,,0000,0000,0000,,You get 121, which is 11^2.
Dialogue: 0,0:02:12.11,0:02:15.87,Default,,0000,0000,0000,,And take a look at what happens\Nwhen you do the same thing to row six.
Dialogue: 0,0:02:15.87,0:02:25.14,Default,,0000,0000,0000,,It adds up to 1,771,561,\Nwhich is 11^6, and so on.
Dialogue: 0,0:02:25.14,0:02:27.89,Default,,0000,0000,0000,,There are also geometric applications.
Dialogue: 0,0:02:27.89,0:02:29.69,Default,,0000,0000,0000,,Look at the diagonals.
Dialogue: 0,0:02:29.69,0:02:34.12,Default,,0000,0000,0000,,The first two aren't very interesting:\Nall ones, and then the positive integers,
Dialogue: 0,0:02:34.12,0:02:36.66,Default,,0000,0000,0000,,also known as natural numbers.
Dialogue: 0,0:02:36.66,0:02:40.71,Default,,0000,0000,0000,,But the numbers in the next diagonal\Nare called the triangular numbers
Dialogue: 0,0:02:40.71,0:02:42.78,Default,,0000,0000,0000,,because if you take that many dots,
Dialogue: 0,0:02:42.78,0:02:46.39,Default,,0000,0000,0000,,you can stack them \Ninto equilateral triangles.
Dialogue: 0,0:02:46.39,0:02:49.31,Default,,0000,0000,0000,,The next diagonal \Nhas the tetrahedral numbers
Dialogue: 0,0:02:49.31,0:02:54.62,Default,,0000,0000,0000,,because similarly, you can stack\Nthat many spheres into tetrahedra.
Dialogue: 0,0:02:54.62,0:02:57.100,Default,,0000,0000,0000,,Or how about this:\Nshade in all of the odd numbers.
Dialogue: 0,0:02:57.100,0:03:00.88,Default,,0000,0000,0000,,It doesn't look like much\Nwhen the triangle's small,
Dialogue: 0,0:03:00.88,0:03:03.30,Default,,0000,0000,0000,,but if you add thousands of rows,
Dialogue: 0,0:03:03.30,0:03:07.44,Default,,0000,0000,0000,,you get a fractal \Nknown as Sierpinski's Triangle.
Dialogue: 0,0:03:07.44,0:03:10.76,Default,,0000,0000,0000,,This triangle isn't just \Na mathematical work of art.
Dialogue: 0,0:03:10.76,0:03:12.74,Default,,0000,0000,0000,,It's also quite useful,
Dialogue: 0,0:03:12.74,0:03:15.48,Default,,0000,0000,0000,,especially when it comes \Nto probability and calculations
Dialogue: 0,0:03:15.48,0:03:18.57,Default,,0000,0000,0000,,in the domain of combinatorics.
Dialogue: 0,0:03:18.57,0:03:20.45,Default,,0000,0000,0000,,Say you want to have five children,
Dialogue: 0,0:03:20.45,0:03:22.27,Default,,0000,0000,0000,,and would like to know the probability
Dialogue: 0,0:03:22.27,0:03:26.59,Default,,0000,0000,0000,,of having your dream family \Nof three girls and two boys.
Dialogue: 0,0:03:26.59,0:03:28.39,Default,,0000,0000,0000,,In the binomial expansion,
Dialogue: 0,0:03:28.39,0:03:32.12,Default,,0000,0000,0000,,that corresponds \Nto girl plus boy to the fifth power.
Dialogue: 0,0:03:32.12,0:03:33.66,Default,,0000,0000,0000,,So we look at the row five,
Dialogue: 0,0:03:33.66,0:03:37.13,Default,,0000,0000,0000,,where the first number \Ncorresponds to five girls,
Dialogue: 0,0:03:37.13,0:03:39.93,Default,,0000,0000,0000,,and the last corresponds to five boys.
Dialogue: 0,0:03:39.93,0:03:42.69,Default,,0000,0000,0000,,The third number \Nis what we're looking for.
Dialogue: 0,0:03:42.69,0:03:46.64,Default,,0000,0000,0000,,Ten out of the sum \Nof all the possibilities in the row.
Dialogue: 0,0:03:46.64,0:03:51.49,Default,,0000,0000,0000,,so 10/32, or 31.25%.
Dialogue: 0,0:03:51.49,0:03:55.32,Default,,0000,0000,0000,,Or, if you're randomly \Npicking a five-player basketball team
Dialogue: 0,0:03:55.32,0:03:57.08,Default,,0000,0000,0000,,out of a group of twelve friends,
Dialogue: 0,0:03:57.08,0:04:00.10,Default,,0000,0000,0000,,how many possible groups \Nof five are there?
Dialogue: 0,0:04:00.10,0:04:05.06,Default,,0000,0000,0000,,In combinatoric terms, this problem would\Nbe phrased as twelve choose five,
Dialogue: 0,0:04:05.06,0:04:07.24,Default,,0000,0000,0000,,and could be calculated with this formula,
Dialogue: 0,0:04:07.24,0:04:11.71,Default,,0000,0000,0000,,or you could just look at the sixth \Nelement of row twelve on the triangle
Dialogue: 0,0:04:11.71,0:04:13.38,Default,,0000,0000,0000,,and get your answer.
Dialogue: 0,0:04:13.38,0:04:15.08,Default,,0000,0000,0000,,The patterns in Pascal's Triangle
Dialogue: 0,0:04:15.08,0:04:19.39,Default,,0000,0000,0000,,are a testament to the elegantly \Ninterwoven fabric of mathematics.
Dialogue: 0,0:04:19.39,0:04:23.27,Default,,0000,0000,0000,,And it's still revealing fresh secrets\Nto this day.
Dialogue: 0,0:04:23.27,0:04:27.42,Default,,0000,0000,0000,,For example, mathematicians recently \Ndiscovered a way to expand it
Dialogue: 0,0:04:27.42,0:04:30.02,Default,,0000,0000,0000,,to these kinds of polynomials.
Dialogue: 0,0:04:30.02,0:04:31.76,Default,,0000,0000,0000,,What might we find next?
Dialogue: 0,0:04:31.76,0:04:34.10,Default,,0000,0000,0000,,Well, that's up to you.