PYTHAGORAS (569 BC 475 BC)
Born

569 BC in Samos, Ionia.

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}n^{2}.
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 2n4 right angles and sum of exterior angles
equal to four right angles.
Ø Pythagoras theorem–
for a rightangled 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. AB^{2}+BC^{2}=AC^{2}
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 algebrafor example they
solved equations such as a (ax)=x^{2 }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 sidebyside 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
ARE YOU STUPID? MENSURATION TURNED INTO MENSTRUATION?
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