### Pascal`s Triangle

```Pascal's Triangle
Andy Hopkins
A Brief History…
 Pascal was not the first to have discovered this triangle.
 Al-Karaji knew of the triangle during his lifetime(953-1029).
 Also was discussed by Omar Khayyam. (It is known as
Khayyam’s triangle in Iran)
 Yang Hui discovered this triangle in China during his
lifetime(1238-1298). (Known as Yang Hui’s triangle in
China).
 Known as Tartaglia’s triangle in Italy (Tartaglia lived from
1500-1577).
 Pascal published his work in 1655.
http://en.wikipedia.org/wiki/Pascal_triangle#
History
http://milan.milanovic.org/math/english/fibo
/fibo0.html
Pascal’s work
 Blaise Pascal published his paper with the help of Pierre de
Fermat.
 In the paper he explained everything that he had discovered
 It starts with how to construct the triangle and then went on
to list the 19 corollaries he found.
http://milan.milanovic.org/math/english/fibo
/fibo0.html
Smith, David E. A Source Book in Mathematics.
First ed. New York: McGraw-Hill Book
Company, Inc., 1929.
How Pascal Starts the Triangle
 What Pascal did was took a point (which he called G) and drew a




line to the right of it and down from it.
Then he drew lines from one line to the other to create 10
different triangles.
He then proceeded to draw lines parallel to each line that was
constructed from point G to create cells.
After this he started with a number that he calls the generator.
This, of course, is the number one. However, it can be a different
number than one if you want to construct a different triangle.
The final thing that he states is that you can find the other cells
based on this rule: The numbers of each cell are equal to the sum
of the previous cells to the left and above the cell you are trying to
find.
Smith, David E. A Source Book in
Mathematics. First ed. New York:
McGraw-Hill Book Company, Inc.,
1929.
Pascal’s Corollaries
 To do all of Pascal’s corollaries in a paper or a presentation
would take a lot of time. I decided to do them on corollaries
1-3, 5-9, and 11.
 The first one just states that the number one is in each cell of
the first row and column.
 The next two are similar in saying that if you take consecutive
numbers in a row or column and add them up your result
will be in the next row or column. We call this the Hockey
Stick pattern.
Smith, David E. A Source Book in Mathematics. First ed. New
York: McGraw-Hill Book Company, Inc., 1929.
http://ptri1.tripod.com/
Corollaries con’t
 There are also corollaries that use previous corollaries to help
solve the current one.
 Corollary 7 says that the sum of a row divided by 2 is equal
to the sum of the previous row.
 Corollary 8 says that the sum of each row is a number in the
geometric progression (1,2,4,8,16,32…). This is true if we
add up the numbers in each row and divide by 2 (corollary
7).
Smith, David E. A Source Book in Mathematics.
First ed. New York: McGraw-Hill Book Company,
Inc., 1929.
Examples of previous triangles.
This is a pre-Pascal
triangle in Japanese
http://cas.umkc.edu/math/ExtraCap
tion.htm
This is a pre-Pascal Chinese
triangle.
http://milan.milanovic.org/math/english/fib
o/fibo0.html
This was published in a 1407 editon of Jordanus' de
Arithmetica.
http://milan.milanovic.org/math/english/fibo/fibo0.html
This was published in
1544 by Michael Stifel
of Germany
http://milan.milanovic.org/mat
h/english/fibo/fibo0.html
This table was
published in 1636 by
Father Marin
Mersenne.
http://milan.milanovic.org/mat
h/english/fibo/fibo0.html
This is Pascal’s
original
triangle.
http://threesixty36
0.wordpress.com/2
007/11/
```