MODULE 5 - PYTHAGORAS (569 BC -475 BC)

PYTHAGORAS (569 BC -475 BC)

FATHER OF NUMBERS

Born	569 BC in Samos, Ionia. (1707-04-15)
Died	475 BC
Residence	Egypt, Italy
Nationality	Samos
Fields	Mathematics, Music, Astronomy, Metaphysics, Ethics,
Institutions	Pythagoreanism
Philosophers	Pherekydes, Thalas, Alaximender

          Pythagoras was a Greek mathematician born on 569 BC in Samos, Ionia. His father was Mnesarchus, a merchant from Tyre and his mother Pythais a native of Samos. He is often described as the first pure mathematician who has contributed immensely towards the development of mathematics.

 

CONTRIBUTIONS OF PYTHAGORAS

PROPERTIES OF NUMBERS: 

Ø    Triangle and Square Numbers: Pythagoras studied properties of numbers, which would be familiar to mathematicians such as today, even, and odd numbers, triangular numbers etc. However to Pythagoras numbers had personalities, which we hardly recognize as mathematics today.

Triangle Numbers

*	* **	* ** ***	* ** *** ****	* ** *** **** *****
1	1+2=3	1+2+3=6	1+2+3+4=10	1+2+3+4+5=15

Square Numbers

*	** **	*** *** ***	**** **** **** ****	***** ***** ***** ***** *****
1	4	9	16	25

Ø    Personality of numbers: Each numbers had its own personality – masculine or feminine, perfect or incomplete, beautiful or ugly. This felling modern mathematics has deliberately eliminated, but we still find overtones of it in fiction and poetry.

Ø    Masculine or feminine: His followers said that odd number was masculine gender and even number was feminine gender. The number 1 was a mother of all numbers. The number 5 was the symbol of marriage ceremony.

Ø    Ten was the very best number: it contained in itself the first four integers one, two, three, and four [ 1 + 2 + 3 + 4 = 10] and these written in dot notation formed a perfect triangle.

Ø    Odd number: He discovered that any odd number (say 2n+1) can be expressed as the difference of two squares: 2n+1 = (n+1)²-n².

THEOREMS OF PYTHAGORAS:

The list of theorem attributed to Pythagoras, or rather more generally to the Pythagoreans are:

Ø    Properties of Triangle: The sum of the angles of a triangle is equal to two right angles.

Ø    Polygon Angles: Also the Pythagoras knew the generalization, which states that a polygon with n sides has sum of interior angles 2n-4 right angles and sum of exterior angles equal to four right angles.

Ø    Pythagoras theorem– for a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. We should note here that to Pythagoras the square on the hypotenuse would certainly not be thought of as a number multiplied by itself, but rather as a geometrical squares is equal to a third square meant that the two squares could be cut up and reassembled to form a square identical to the third square. AB²+BC²=AC²

MUSIC

Ø     Pythagoras believed that all relations could be reduced to number relations. Pythagoras noticed that vibrating strings produce harmonious tones when the ratios of the strings are whole numbers, and that theses ratio could be extended to other instruments. In facts Pythagoras made remarkable contribution to the mathematical theory of music.

Ø     Pythagoras made useful investigations into music and began the theory of music by measuring the cords of the Lyre, a musical instrument.

Ø     Pythagorean Tuning : how the relationship between musical notes can be expressed in mathematical equations

GEOMETRY

Ø    Constructing figures of a given area and geometrical algebra-for example they solved equations such as a (a-x)=x² by geometrical means.

Ø    The five regular solids: It is thought that Pythagoras himself knew how to construct the first three but it is unlikely that he would have known how to construct the other two.

Ø    He construct a polygons equivalent to one given polygon and similar to another and could construct the five regular polyhedron

Ø    Mensuration: Pythagoras studied the properties of area and volumes and he was the first to prove that the circle contains a greater area than any plane figure with the same perimeter while the sphere contains a greater volume than any other shape bounded by the same surface.

Ø    In solid geometry, Pythagoras called sphere the most perfect of all solids. He knew that were five regular solids which lie exactly in a sphere namely tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron.

Ø    Many mathematical terms like parabola, lying side-by-side ellipse etc. can be attributed to Pythagoras.

Ø    The discovery of irrationals: this is certainly attributed to the Pythagoras but it does seem unlikely to have been due to Pythagoras himself.

ASTRONOMY

Ø    In astronomy Pythagoras taught that the Earth was a sphere at the center of the Universe. He also recognized that the orbit of the moon was inclined to the equator of the earth and he was one of the first to realize that Venus as an evening star was the same planet as Venus as a morning star.