FREE ESSAY ON PASCALS TRIANGLE

PASCALS TRIANGLE

Pascal's Triangle
Blase Pacal was born in France in 1623. He was a child prodigy and was
fascinated by mathematics. When Pascal was 19 he invented the first calculating
machine that actually worked. Many other people had tried to do the same but did not
succeed. One of the topics that deeply interested him was the likelihood of an event
happening (probability). This interest came to Pascal from a gambler who asked him
to help him make a better guess so he could make an educated guess. In the coarse of
his investigations he produced a triangular pattern that is named after him. The pattern
was known at least three hundred years before Pascal had discover it. The Chinese
were the first to discover it but it was fully developed by Pascal (Ladja , 2).
Pascal's triangle is a triangluar arrangement of rows. Each row except the first
row begins and ends with the number 1 written diagonally. The first row only has one
number which is 1. Beginning with the second row, each number is the sum of the
number written just above it to the right and the left. The numbers are placed midway
between the numbers of the row directly above it.
If you flip 1 coin the possibilities are 1 heads (H) or 1 tails (T). This
combination of 1 and 1 is the firs row of Pascal's Triangle. If you flip the coin twice
you will get a few different results as I will show below (Ladja, 3):
Let's say you have the polynomial x+1, and you want to raise it to some 
powers, like 1,2,3,4,5,.... If you make a chart of what you get when you 
do these power-raisins, you'll get something like this (Dr. Math, 3):
(x+1)^0 = 1
(x+1)^1 = 1 + x
(x+1)^2 = 1 + 2x + x^2
(x+1)^3 = 1 + 3x + 3x^2 + x^3
(x+1)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4
(x+1)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5 .....
If you just look at the coefficients of the polynomials that you get, you'll see
Pascal's Triangle! Because of this connection, the entries in Pascal's Triangle are
called
the binomial coefficients.There's a pretty simple formula for figuring out the binomial
coefficients (Dr. Math, 4):
n!
[n:k] = --------
k! (n-k)!
6 * 5 * 4 * 3 * 2 * 1
For example, [6:3] = ------------------------ = 20. 
3 * 2 * 1 * 3 * 2 * 1
The triangular numbers and the Fibonacci numbers can be found in 
Pascal's triangle. The triangular numbers are easier to find: starting with the third
one
on the left side go down to your right and you get 1, 3, 6, 10, etc (Swarthmore, 5) 
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
The Fibonacci numbers are harder to locate. To find them you need to go 
up at an angle: you're looking for 1, 1, 1+1, 1+2, 1+3+1, 1+4+3, 1+5+6+1 
(Dr. Math, 4). 
Another thing I found out is that if you multiply 11 x 11 you will get 121 which
is the 2nd line in Pascal's Triangle. If you multiply 121 x 11 you get 1331 which is the
3rd line in the triangle (Dr. Math, 4). 
If you then multiply 1331 x 11 you get 14641 which is the 4th line in Pascal's
Triangle, but if you then multiply 14641 x 11 you do not get the 5th line numbers. You
get 161051. But after the 5th line it doesn't work anymore (Dr. Math, 4). 
Another example of probability: Say there are four children Annie, Bob, 
Carlos, and Danny (A, B, C, D). The teacher wants to choose two of them to hand out
books; in how many ways can she choose a pair (ladja, 4)? 
1.A & B 
2.A & C 
3.A & D 
4.B & C 
5.B & D 
6.C & D 
There are six ways to make a choice of a pair. 
If the teacher wants to send three students: 
1.A, B, C 2.A, B, D 3.A, C, D 4.B, C, D 
If the teacher wants to send a group of K children where K may range 
from 0-4; in how many ways will she choose the children 
K=0 1 way (There is only one way to send no children) 
K=1 4 ways ( A; B; C; D) 
K=2 6 ways (like above with Annie, Bob, Carlos, Danny) 
K=3 4 ways (above with triplets) 
K=4 1 way (there is only one way to send a group of four) 
The above numbers (1 4 6 4 1) are the fourth row of numbers in Pascal 
Triangle (Ladja, 5). 
If we extend Pascal's triangle to infinitely many rows, and reduce the scale of
our picture in half each time that we double the number of rows, then the resulting
design is called self-similar -- that is, our picture can be reproduced by taking an
subtriangle and magnifying it, Granville notes.The pattern becomes more evident if
the numbers are put in cells and the cells colored according to whether the number is 1
or 0 (Peterson's, 5).Similar, though more complicated designs appear if one replaces
each number of the triangle with the remainder after dividing that number by 3. So, I
get: 
1
1 1
1 2 1
1 0 0 1
1 1 0 1 1
1 2 1 1 2 1
1 0 0 2 0 0 1
This time, one would need three different colors to reveal the patterns 
of triangles embedded in the array. One can also try other prime numbers 
as the divisor (or modulus), again writing down only the remainders in 
each position (Freedman, 5). Actually, there's a simpler way to try this out. With the
help of 
Jonathan Borwein of Simon Fraser University in Burnaby, British Columbia, and his
colleagues, Granville has created a Pascal's Triangle Interface on the web. One can
specify the number of rows (up to 100), the modulus (from 2 to 16), and the image size
to get a colorful rendering of the requested form.It's a neat way to explore the fractal
side of Pascal's triangle. Here's one example that I tried out, using 5 as the modulus
(Petetson's, 5).