0:00:07.603,0:00:11.000
This may look like a neatly arranged[br]stack of numbers,
0:00:11.000,0:00:14.506
but it's actually[br]a mathematical treasure trove.
0:00:14.506,0:00:18.654
Indian mathematicians called it [br]the Staircase of Mount Meru.
0:00:18.654,0:00:21.131
In Iran, it's the Khayyam Triangle.
0:00:21.131,0:00:23.738
And in China, it's Yang Hui's Triangle.
0:00:23.738,0:00:28.033
To much of the Western world,[br]it's known as Pascal's Triangle
0:00:28.033,0:00:31.085
after French mathematician Blaise Pascal,
0:00:31.085,0:00:35.234
which seems a bit unfair[br]since he was clearly late to the party,
0:00:35.234,0:00:37.476
but he still had a lot to contribute.
0:00:37.476,0:00:42.270
So what is it about this that has so [br]intrigued mathematicians the world over?
0:00:42.270,0:00:46.124
In short, [br]it's full of patterns and secrets.
0:00:46.124,0:00:49.428
First and foremost, there's the pattern[br]that generates it.
0:00:49.428,0:00:54.477
Start with one and imagine invisible[br]zeros on either side of it.
0:00:54.477,0:00:58.592
Add them together in pairs, [br]and you'll generate the next row.
0:00:58.592,0:01:02.066
Now, do that again and again.
0:01:02.066,0:01:05.784
Keep going and you'll wind up [br]with something like this,
0:01:05.784,0:01:09.325
though really Pascal's Triangle [br]goes on infinitely.
0:01:09.325,0:01:14.914
Now, each row corresponds to what's called[br]the coefficients of a binomial expansion
0:01:14.914,0:01:18.898
of the form (x+y)^n,
0:01:18.898,0:01:21.307
where n is the number of the row,
0:01:21.307,0:01:23.746
and we start counting from zero.
0:01:23.746,0:01:26.552
So if you make n=2 and expand it,
0:01:26.552,0:01:31.107
you get (x^2) + 2xy + (y^2).
0:01:31.107,0:01:34.023
The coefficients, [br]or numbers in front of the variables,
0:01:34.023,0:01:38.397
are the same as the numbers in that row[br]of Pascal's Triangle.
0:01:38.397,0:01:43.256
You'll see the same thing with n=3,[br]which expands to this.
0:01:43.256,0:01:48.493
So the triangle is a quick and easy way[br]to look up all of these coefficients.
0:01:48.493,0:01:50.037
But there's much more.
0:01:50.037,0:01:52.897
For example, add up [br]the numbers in each row,
0:01:52.897,0:01:56.039
and you'll get successive powers of two.
0:01:56.039,0:02:01.221
Or in a given row, treat each number[br]as part of a decimal expansion.
0:02:01.221,0:02:07.835
In other words, row two is[br](1x1) + (2x10) + (1x100).
0:02:07.835,0:02:12.111
You get 121, which is 11^2.
0:02:12.111,0:02:15.872
And take a look at what happens[br]when you do the same thing to row six.
0:02:15.872,0:02:25.136
It adds up to 1,771,561,[br]which is 11^6, and so on.
0:02:25.136,0:02:27.890
There are also geometric applications.
0:02:27.890,0:02:29.691
Look at the diagonals.
0:02:29.691,0:02:34.117
The first two aren't very interesting:[br]all ones, and then the positive integers,
0:02:34.117,0:02:36.656
also known as natural numbers.
0:02:36.656,0:02:40.707
But the numbers in the next diagonal[br]are called the triangular numbers
0:02:40.707,0:02:42.783
because if you take that many dots,
0:02:42.783,0:02:46.389
you can stack them [br]into equilateral triangles.
0:02:46.389,0:02:49.307
The next diagonal [br]has the tetrahedral numbers
0:02:49.307,0:02:54.622
because similarly, you can stack[br]that many spheres into tetrahedra.
0:02:54.622,0:02:57.996
Or how about this:[br]shade in all of the odd numbers.
0:02:57.996,0:03:00.881
It doesn't look like much[br]when the triangle's small,
0:03:00.881,0:03:03.298
but if you add thousands of rows,
0:03:03.298,0:03:07.439
you get a fractal [br]known as Sierpinski's Triangle.
0:03:07.439,0:03:10.756
This triangle isn't just [br]a mathematical work of art.
0:03:10.756,0:03:12.742
It's also quite useful,
0:03:12.742,0:03:15.481
especially when it comes [br]to probability and calculations
0:03:15.481,0:03:18.566
in the domain of combinatorics.
0:03:18.566,0:03:20.454
Say you want to have five children,
0:03:20.454,0:03:22.270
and would like to know the probability
0:03:22.270,0:03:26.590
of having your dream family [br]of three girls and two boys.
0:03:26.590,0:03:28.388
In the binomial expansion,
0:03:28.388,0:03:32.116
that corresponds [br]to girl plus boy to the fifth power.
0:03:32.116,0:03:33.660
So we look at the row five,
0:03:33.660,0:03:37.131
where the first number [br]corresponds to five girls,
0:03:37.131,0:03:39.929
and the last corresponds to five boys.
0:03:39.929,0:03:42.692
The third number [br]is what we're looking for.
0:03:42.692,0:03:46.642
Ten out of the sum [br]of all the possibilities in the row.
0:03:46.642,0:03:51.490
so 10/32, or 31.25%.
0:03:51.490,0:03:55.316
Or, if you're randomly [br]picking a five-player basketball team
0:03:55.316,0:03:57.084
out of a group of twelve friends,
0:03:57.084,0:04:00.102
how many possible groups [br]of five are there?
0:04:00.102,0:04:05.062
In combinatoric terms, this problem would[br]be phrased as twelve choose five,
0:04:05.062,0:04:07.237
and could be calculated with this formula,
0:04:07.237,0:04:11.708
or you could just look at the sixth [br]element of row twelve on the triangle
0:04:11.708,0:04:13.383
and get your answer.
0:04:13.383,0:04:15.079
The patterns in Pascal's Triangle
0:04:15.079,0:04:19.387
are a testament to the elegantly [br]interwoven fabric of mathematics.
0:04:19.387,0:04:23.271
And it's still revealing fresh secrets[br]to this day.
0:04:23.271,0:04:27.422
For example, mathematicians recently [br]discovered a way to expand it
0:04:27.422,0:04:30.019
to these kinds of polynomials.
0:04:30.019,0:04:31.758
What might we find next?
0:04:31.758,0:04:34.097
Well, that's up to you.