PASCAL S TRIANGLE AND FIBONACCI

ΔΙΑΘΕΜΑΤΙΚΗ ΔΡΑΣΗ: ΑΓΓΛΙΚΑ +ΜΑΘΗΜΑΤΙΚΑ

ΤΜΗΜΑ Γ3

Π.Π. ΓΥΜΝΑΣΙΟ ΙΩΝΙΔΕΙΟΥ ΣΧΟΛΗΣ ΠΕΙΡΑΙΑ 2014 2015

Ιωάννα Λύρα

Βίκτωρ Μαρτίνος

Μυρτώ Φιλιππάκη

Βασίλης Μιχαλίτσης

Σακκάς Γιώργος

Μαρίνης Στέλιος

HISTORY

Pascal s triangle is perhaps the most famous of all number patterns. It is very ancient standing, being probably more than a thousand year old. Its hidden properties have been revealed more and more as mathematics has developed through the centuries.

The Chinese Knew About It

This drawing is entitled "The Old Method Chart of the Seven Multiplying Squares".

It is from the front of Chu Shi-Chieh's book "Ssu Yuan Yü Chien" (Precious Mirror of the Four Elements), written in AD 1303 (over 700 years ago, and more than 300 years before Pascal!), and in the book it says the triangle was known about more than two centuries before that.

CONSTRUCTION

To build the triangle, start with "1" at the top,

then continue placing numbers below it in a

triangular pattern.

Each number is the two numbers above it added

together (except for the edges, which are all "1").

THEORY

Pascal’s Triangle, developed by the French

Mathematician Blaise Pascal, is formed by starting with

an apex of 1. Every number below in the triangle is the

sum of the two numbers diagonally above it to the left

and the right, with positions outside the triangle counting

as zero.

The numbers on diagonals of the triangle add to the

Fibonacci series, as shown below:

PATTERNS WITHIN THE TRIANGLE

Diagonals

The first diagonal is, of course, just "1"s, and the

next diagonal has the Counting numbers(1,2,3, etc).

The third diagonal has the triangular numbers

(The fourth diagonal, not highlighted, has the

tetrahedral numbers.)

FIBONACCI SEQUENCE

The Fibonacci

Sequence

starts "0, 1"

and then

continues by

adding the

two previous

numbers, for

example

3+5=8, then

5+8=13, etc

PROPERTIES Horizontal rows add to powers of 2 (i.e., 1, 2, 4, 8, 16,

etc.)

The horizontal rows represent powers of 11 (1, 11, 121, 1331, 14641) for the first 5 rows, in which the numbers have only a single digit.

Adding any two successive numbers in the diagonal 1-3-6-10-15-21-28… results in a perfect square (1, 4, 9, 16, etc.)

It can be used to find combinations in probability problems (if, for instance, you pick any two of five items, the number of possible combinations is 10, found by looking in the second place of the fifth row. Do not count the 1’s.)

When the first number to the right of the 1 in any row is a prime number, all numbers in that row are divisible by that prime number

Pascal's Triangle can also show you the

coefficients in binomial expansion

SOURCES

http://www.mathsisfun.com/pascals-triangle.html

http://www.goldennumber.net/pascals-triangle/

H.E. HUNTLEY, The divine proportion,

Dover Publications, N.York 1970

Α.ΑΛΙΜΠΙΝΙΣΗΣ, Σ.ΓΡΗΓΟΡΙΑΔΗΣ,

Ε.ΕΥΣΤΑΘΟΠΟΥΛΟΣ, Ν.ΚΛΑΟΥΔΑΤΟΣ,

Σ.ΠΑΠΑΣΤΑΥΡΙΔΗΣ, Α.ΣΒΕΡΚΟΣ ,

ΜΑΘΗΜΑΤΙΚΑ Γ΄ ΓΥΜΝΑΣΙΟΥ . ΟΕΔΒ,ΑΘΗΝΑ

2004