The mathematical secrets of Pascal’s triangle  Wajdi Mohamed Ratemi

0:08  0:11This may look like a neatly arranged
stack of numbers, 
0:11  0:15but it's actually
a mathematical treasure trove. 
0:15  0:19Indian mathematicians called it
the Staircase of Mount Meru. 
0:19  0:21In Iran, it's the Khayyam Triangle.

0:21  0:24And in China, it's Yang Hui's Triangle.

0:24  0:28To much of the Western world,
it's known as Pascal's Triangle 
0:28  0:31after French mathematician Blaise Pascal,

0:31  0:35which seems a bit unfair
since he was clearly late to the party, 
0:35  0:37but he still had a lot to contribute.

0:37  0:42So what is it about this that has so
intrigued mathematicians the world over? 
0:42  0:46In short,
it's full of patterns and secrets. 
0:46  0:49First and foremost, there's the pattern
that generates it. 
0:49  0:54Start with one and imagine invisible
zeros on either side of it. 
0:54  0:59Add them together in pairs,
and you'll generate the next row. 
0:59  1:02Now, do that again and again.

1:02  1:06Keep going and you'll wind up
with something like this, 
1:06  1:09though really Pascal's Triangle
goes on infinitely. 
1:09  1:15Now, each row corresponds to what's called
the coefficients of a binomial expansion 
1:15  1:19of the form (x+y)^n,

1:19  1:21where n is the number of the row,

1:21  1:24and we start counting from zero.

1:24  1:27So if you make n=2 and expand it,

1:27  1:31you get (x^2) + 2xy + (y^2).

1:31  1:34The coefficients,
or numbers in front of the variables, 
1:34  1:38are the same as the numbers in that row
of Pascal's Triangle. 
1:38  1:43You'll see the same thing with n=3,
which expands to this. 
1:43  1:48So the triangle is a quick and easy way
to look up all of these coefficients. 
1:48  1:50But there's much more.

1:50  1:53For example, add up
the numbers in each row, 
1:53  1:56and you'll get successive powers of two.

1:56  2:01Or in a given row, treat each number
as part of a decimal expansion. 
2:01  2:08In other words, row two is
(1x1) + (2x10) + (1x100). 
2:08  2:12You get 121, which is 11^2.

2:12  2:16And take a look at what happens
when you do the same thing to row six. 
2:16  2:25It adds up to 1,771,561,
which is 11^6, and so on. 
2:25  2:28There are also geometric applications.

2:28  2:30Look at the diagonals.

2:30  2:34The first two aren't very interesting:
all ones, and then the positive integers, 
2:34  2:37also known as natural numbers.

2:37  2:41But the numbers in the next diagonal
are called the triangular numbers 
2:41  2:43because if you take that many dots,

2:43  2:46you can stack them
into equilateral triangles. 
2:46  2:49The next diagonal
has the tetrahedral numbers 
2:49  2:55because similarly, you can stack
that many spheres into tetrahedra. 
2:55  2:58Or how about this:
shade in all of the odd numbers. 
2:58  3:01It doesn't look like much
when the triangle's small, 
3:01  3:03but if you add thousands of rows,

3:03  3:07you get a fractal
known as Sierpinski's Triangle. 
3:07  3:11This triangle isn't just
a mathematical work of art. 
3:11  3:13It's also quite useful,

3:13  3:15especially when it comes
to probability and calculations 
3:15  3:19in the domain of combinatorics.

3:19  3:20Say you want to have five children,

3:20  3:22and would like to know the probability

3:22  3:27of having your dream family
of three girls and two boys. 
3:27  3:28In the binomial expansion,

3:28  3:32that corresponds
to girl plus boy to the fifth power. 
3:32  3:34So we look at the row five,

3:34  3:37where the first number
corresponds to five girls, 
3:37  3:40and the last corresponds to five boys.

3:40  3:43The third number
is what we're looking for. 
3:43  3:47Ten out of the sum
of all the possibilities in the row. 
3:47  3:51so 10/32, or 31.25%.

3:51  3:55Or, if you're randomly
picking a fiveplayer basketball team 
3:55  3:57out of a group of twelve friends,

3:57  4:00how many possible groups
of five are there? 
4:00  4:05In combinatoric terms, this problem would
be phrased as twelve choose five, 
4:05  4:07and could be calculated with this formula,

4:07  4:12or you could just look at the sixth
element of row twelve on the triangle 
4:12  4:13and get your answer.

4:13  4:15The patterns in Pascal's Triangle

4:15  4:19are a testament to the elegantly
interwoven fabric of mathematics. 
4:19  4:23And it's still revealing fresh secrets
to this day. 
4:23  4:27For example, mathematicians recently
discovered a way to expand it 
4:27  4:30to these kinds of polynomials.

4:30  4:32What might we find next?

4:32  4:34Well, that's up to you.
 Title:
 The mathematical secrets of Pascal’s triangle  Wajdi Mohamed Ratemi
 Speaker:
 Wajdi Mohamed Ratemi
 Description:

View full lesson: http://ed.ted.com/lessons/themathematicalsecretsofpascalstrianglewajdimohamedratemi
Pascal’s triangle, which at first may just look like a neatly arranged stack of numbers, is actually a mathematical treasure trove. But what about it has so intrigued mathematicians the world over? Wajdi Mohamed Ratemi shows how Pascal's triangle is full of patterns and secrets.
Lesson by Wajdi Mohamed Ratemi, animation by Henrik Malmgren.
 Video Language:
 English
 Team:
 closed TED
 Project:
 TEDEd
 Duration:
 04:50
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