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The mathematical secrets of Pascal’s triangle - Wajdi Mohamed Ratemi

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    This may look like a neatly arranged
    stack of numbers,
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    but it's actually
    a mathematical treasure trove.
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    Indian mathematicians called it
    the Staircase of Mount Meru.
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    In Iran, it's the Khayyam Triangle.
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    And in China, it's Yang Hui's Triangle.
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    To much of the Western world,
    it's known as Pascal's Triangle
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    after French mathematician Blaise Pascal,
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    which seems a bit unfair
    since he was clearly late to the party,
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    but he still had a lot to contribute.
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    So what is it about this that has so
    intrigued mathematicians the world over?
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    In short,
    it's full of patterns and secrets.
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    First and foremost, there's the pattern
    that generates it.
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    Start with one and imagine invisible
    zeros on either side of it.
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    Add them together in pairs,
    and you'll generate the next row.
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    Now, do that again and again.
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    Keep going and you'll wind up
    with something like this,
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    though really Pascal's Triangle
    goes on infinitely.
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    Now, each row corresponds to what's called
    the coefficients of a binomial expansion
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    of the form (x+y)^n,
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    where n is the number of the row,
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    and we start counting from zero.
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    So if you make n=2 and expand it,
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    you get (x^2) + 2xy + (y^2).
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    The coefficients,
    or numbers in front of the variables,
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    are the same as the numbers in that row
    of Pascal's Triangle.
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    You'll see the same thing with n=3,
    which expands to this.
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    So the triangle is a quick and easy way
    to look up all of these coefficients.
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    But there's much more.
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    For example, add up
    the numbers in each row,
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    and you'll get successive powers of two.
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    Or in a given row, treat each number
    as part of a decimal expansion.
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    In other words, row two is
    (1x1) + (2x10) + (1x100).
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    You get 121, which is 11^2.
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    And take a look at what happens
    when you do the same thing to row six.
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    It adds up to 1,771,561,
    which is 11^6, and so on.
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    There are also geometric applications.
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    Look at the diagonals.
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    The first two aren't very interesting:
    all ones, and then the positive integers,
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    also known as natural numbers.
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    But the numbers in the next diagonal
    are called the triangular numbers
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    because if you take that many dots,
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    you can stack them
    into equilateral triangles.
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    The next diagonal
    has the tetrahedral numbers
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    because similarly, you can stack
    that many spheres into tetrahedra.
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    Or how about this:
    shade in all of the odd numbers.
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    It doesn't look like much
    when the triangle's small,
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    but if you add thousands of rows,
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    you get a fractal
    known as Sierpinski's Triangle.
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    This triangle isn't just
    a mathematical work of art.
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    It's also quite useful,
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    especially when it comes
    to probability and calculations
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    in the domain of combinatorics.
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    Say you want to have five children,
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    and would like to know the probability
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    of having your dream family
    of three girls and two boys.
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    In the binomial expansion,
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    that corresponds
    to girl plus boy to the fifth power.
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    So we look at the row five,
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    where the first number
    corresponds to five girls,
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    and the last corresponds to five boys.
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    The third number
    is what we're looking for.
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    Ten out of the sum
    of all the possibilities in the row.
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    so 10/32, or 31.25%.
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    Or, if you're randomly
    picking a five-player basketball team
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    out of a group of twelve friends,
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    how many possible groups
    of five are there?
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    In combinatoric terms, this problem would
    be phrased as twelve choose five,
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    and could be calculated with this formula,
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    or you could just look at the sixth
    element of row twelve on the triangle
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    and get your answer.
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    The patterns in Pascal's Triangle
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    are a testament to the elegantly
    interwoven fabric of mathematics.
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    And it's still revealing fresh secrets
    to this day.
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    For example, mathematicians recently
    discovered a way to expand it
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    to these kinds of polynomials.
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    What might we find next?
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    Well, 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/the-mathematical-secrets-of-pascal-s-triangle-wajdi-mohamed-ratemi

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.

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Video Language:
English
Team:
closed TED
Project:
TED-Ed
Duration:
04:50

English subtitles

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