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.