PYTHAGORAS

Born: about 569 BC in Samos (today's Turkey)
Died: about 475 BC

" Everything is number "

 


" The eternal silence of these infinite spaces frightens me "
Blaise Pascal, French Mathematician and Philosopher (1623-1662)

Philosopher, a word created by Pythagoras, is a person who seeks to discover the secrets of  Nature.The word comes from philo (love) and sophia (wisdom or knowledge). Philosophers and mathematicians were usually the same people. For the Pythagorean school, numbers (whole and rational ) are the principle of everything : they govern the harmony of the universe. His mystic interpretation of numbers led Pythagoras to state : "everything is number." (Numbers mysticism  can be found in Judaism and Islam. In the latter the number 19 is an important number). Pythagoras said  that                    the relationship between him and his friend is the same as the relationship between 220 and 284.  What does he mean ? 

The table suggests the answer :

220 284


all factors of 220 are, except 220 itself :

1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110


all factors of 284 are, except 284 itself :

1, 2,  4, 71, 142

 


sum of the factors of 220 = 284

sum of the factors of 284 = 220


220 and 284 are called amicable numbers or friendly numbers. Can you find another pair of friendly numbers ?

In the Bible (Genesis) : angry at his brother (Esau) who "stole" the blessing their father (Isaac) intended for him, Jacob was determined to kill his brother. To defuse his brother's homicidal mood, Esau offered him  " 200 female goats and 20 male goats, 200 ewes and 20 rams..." (Gen 32 : 14). 

What amicable numbers have to do with  9th century Baghdad ? To find out click here > http://martin.winkler.net/friendly.html

Pythagoras ' work includes triangular numbers as described below : 

                                    Two interesting properties of triangular numbers :

1) The sum of two consecutive triangular numbers is a perfect square ( i.e. 4, 9,16,etc. What are the fifth and sixth triangular numbers? Check that their sum is a perfect square). 

2) Any whole number is the sum of three triangular numbers.

The Pythagorean school was thrown into a profound crisis when they could not find  a rational number whose square equals 2.

 

The Pythagorean theorem is the most renowned work of  the philosopher's contribution to mathematics. The theorem states that 

a2 + b2 = c2 where a, b, and c measure the sides of a rectangle triangle, with c being the hypotenuse. 

Conversely,  a triangle whose sides satisfy a2+b2=c2 is right angled (example :  if a triangle sides are 3,4, and 5, then it is right angled).

                                                                                        

Geometrically, the theorem claims that the sum of  areas of the smaller squares equals the area of the larger one. 

 

Links to Learn More About Pythagoras :

 Pythagorean Theorem Made Simple : http://www-istp.gsfc.nasa.gov/stargaze/Spyth.htm

What the concept of ratio has to do with music ? http://www.aboutscotland.com/harmony/prop.html

A wealth of information on Pythagoras and Pythagoreans : http://www.math.tamu.edu/~don.allen/history/pythag/pythag.html