ROHAN KARMAKAR

PYTHAGORAS

Pythagoras was an Ionian Greek philosopher, mathematician. Pythagoras made influential contributions to philosophy and religion in the late 6th century BC. He is often revered as a great mathematician and scientist and is best known for the Pythagorean theorem which

bears his name.

WHO WAS PYTHAGORAS ?

Pythagoras of Samos was a Greek mathematician and philosopher (570 – 495 BC). He is known for the proof of the important Pythagorean theorem.He had a great impact on mathematics, theory of music and astronomy. He was an influence for Plato.

PYTHAGORAS DEPICTING HIS TETRACTYS

EARLY LIFE Pythagoras spent most of

his early childhood at Samos. At first, he studied from the scholars of Syria.

His meeting with Thales elicited in him an interest in science, mathematics and astronomy.

He then went to Babylon and began to study mathematics.

PYTHAGORAS DURING HIS EARLY LIFE

LATER LIFE The Pythagoreans, as the

followers of Pythagoras were called. They lived and worked at the school.

Like most geniuses, Pythagoras too created many enemies. One of them instigated the mob against the Pythagoreans and set fire to the building where they were staying. However, Pythagoras was able to escape. He then went to Metapontum and starved himself to death

PYTHAGORAS IN HIS SCHOOL

BELIEFS OF PYTHAGORAS

He believed that everything could be reduced to numbers and these numbers had their own characteristics.

To him 10 was the most complete number because it was made up of the first 4 digits and when written in dot notation, they formed a triangle.

He also believed geometry to be the highest form of mathematical studies through which one can explain the physical world.

WORKS OF PYTHAGORAS

PYTHAGOREAN THEOREM

It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

i.e.

where a and b are adjacent sides opposite to the hypotenuse which is c.

PROOF OF PYTHAGOREAN THEOREM

The two large squares shown in the figure each

contain four identical triangles, and the only difference between the

two large squares is that the triangles are arranged differently. Therefore, the white space within each of

the two large squares must have equal area.

Equating the area of the white space yields the Pythagorean theorem.

THIS CAN BE MORE EASILY UNDERSTOOD BY THIS…:

PROOF OF PYTHAGOREAN THEOREM

USES OF PYTHAGOREAN THEOREM

The Pythagorean theorem occurs in the following:

In Engineering Calculations In Oceanography (determining speed of

sound in water and related uses) It is also used in aerospace industry. It is also used in meteorology and geology.

PYTHAGOREAN TRIPLETSPythagorean triple consists of 3 positive

integers a, b, and c, such that . Such a triple is commonly written (a, b, c). Pythagorean triple is

one in which and are coprime.

Some examples are :

THIS CAN BE MORE EASILY UNDERSTOOD BY THIS…:PYTHAGOREAN TRIPLETS

GENERATING A TRIPLETThough the idea of triplet was primarily given by Pythagoras, the idea and mathematical generation of triplets was given by Euclid.

Euclid's formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of integers m and n with . The formula states that the integers.

form a Pythagorean triple. The triple generated by Euclid's formula is primitive if and only if m and n are coprime and not both odd.

PYTHAGOREAN TETRACTYSThe tetractys is a triangular figure consisting of

ten points arranged in four rows which is the geometrical representation of the fourth

triangular number.

AN EXAMPLE OF PYTHAGORAS TETRACTYS HAVING 10 DOT

STRUCTURE

USES OF PYTHAGOREAN TETRACTYS

The tetractys occurs (generally coincidentally) in the following:

The baryon decuplet An archbishop's coat of arms The arrangement of pins in ten-pin bowling A Chinese checkers board The "Christmas Tree" formation in

association football

ALL GOOD THINGSMUST COME TO AN END !

THANKS

© Rohan Karmakar (original)