Pythagoras was a Greek philosopher, mathematician, and founder of the Pythagorean movement. He made important contributions to mathematics, discovering the Pythagorean theorem which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Pythagoras also studied properties of numbers, geometry, music, and astronomy. He believed that numbers underlie all things and that mathematical relations exist in music, discovering musical intervals based on whole number ratios between string lengths. Pythagoras made many advances in geometry as well, constructing regular polygons and the five regular solids.

1. (569-400 B.C)

2. ABOUT PYTHAGORAS
 Pythagoras was born on the island of samose in Greece.
 He is famous philosopher true mathematician and founder
of the religious movement called Pythagoranism.
 Father –Mbesarchus
 Mother-Parthenis
 School-Pythagoreanism
 Main subject-Mathematics, Ethics, Politics
 He has a teacher named Themistoclea , who introduced him
to the principles of ethics.
 Notable ideas-Musical Universalism, Golden ratio,
Pythagorean tuning, Pythagorean theorem
 He is often a great Mathematician, Mystic and Scientist.

3. CONTRIBUTIONS
 The Pythagorean theorom:In right angle triangle,
the sum of the squares of the two shorter sides (a & b)
equal to the squares of the hypotenuse(c).

5.  Pythagoras is remembered today for his above
Pythagoras theorem.
 He was also credited with devising the Tetractys,
the triangular figure of four rows,
which add up to the perfect number,
ten. As a mystical symbol, it was very
important to the worship of the
Pythagoreans.

6.  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
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1 1+2=3 1+2+3=6 1+2+3+4=10 1+2+3+4+5=
15

7. * **
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1 4 9 16 25
Square Numbers

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

9.  MUSIC: 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.
 Pythagoras made useful investigations into music and
began the theory of music by measuring the cords of
the Lyre, a musical instrument.
 Pythagorean Tuning : how the relationship between
musical notes can be expressed in mathematical
equations

10.  GEOMETRY
 Constructing figures of a given area and geometrical
algebra-for example they solved equations such as a (a-
x)=x2 by geometrical means.
 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.
 He construct a polygons equivalent
to one given polygon and similar to
another and could construct the five
regular polyhedron
 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.

11.  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.
 Many mathematical terms like parabola, lying side-by-side
ellipse etc. can be attributed to Pythagoras.
 The discovery of irrationals: this is certainly attributed to
the Pythagoras but it does seem unlikely to have been due to
Pythagoras himself.
 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.
 Pythagoras died at the age of about 75 years,in about 400 B.C