Pythagorean Theorem is one of the oldest theorems known to humans and also is one of the most widely used. Its discovery has been attributed to Pythagoras of Greece but various historical documents show that this theorem has been known to humans from earlier times. Various references have been found to this theorem in ancient Mesopotamia, Egypt, India, Babylon and China. The statement that “The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides” seems so simple that it almost veils its immense impact.
This theorem has its roots in geometry but it can also be algebraically stated as a^2+b^2=c^2. Using this theorem, the sides of a right angled triangle can be solved easily. This theorem is the foundation stone of two-dimensional rectangular co-ordinate geometry in which the coordinates to a point are the perpendicular distances from the origin. Using this theorem, the distance between two points can be found. The theorem has been extended to be used for three-dimensional solid geometry and for inner product spaces.
Pythagorean Theorem is the first theorem that gave the idea of existence of irrational numbers. An irrational number is a number that cannot be expressed as the fraction involving two integers. This led to the idea of distance that could not be measured exactly. The root obtained as a solution for side in Pythagorean theorem is an irrational number. It is said that Hippasus of Metaponum was first to find this and was drowned in the sea by his fellow mathematicians because it was sort of a blasphemy to mathematics.
An important consequence of the theorem is the existence of Pythagorean triples. Three co-prime numbers (all of the numbers are prime) are called Pythagorean triples if they satisfy the Pythagorean Theorem, that is, square of one of them is the sum of the squares of others. Some examples are (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97). There are many more and can be found by running simple codes on common programming languages like C++ or FORTRAN. Megalithic structures (structures from time before recorded history) with sides as Pythagorean triplets have been found in Circa of Egypt and Northern Europe.
This theorem is also said to have the greatest number of proofs. Proofs include geometric proofs with axioms and statements, geometric proof with rearrangement, algebraic proof and proof from calculus to name a few. Many more proofs can be found at this site. http://www.cut-the-knot.org/pythagoras/
There, however, is a catch to this theorem. This theorem works with Euclidian mathematics in Euclidian Space. The space we live in is supposed to be Riemannian space and it follows Riemannian geometry. So, Pythagorean Theorem does not work in the real world; though it may provide greatest approximation using simple mathematics. In fact, non-Euclidean geometry specifically requires the three sides of a right angled triangle not to follow the Pythagorean Theorem. Still, Pythagorean theorem is one of the oldest and one of the most important theorems of mathematics. This theorem will remain used because the alternatives that provide more accurate results are too complicated and for general purposes, Pythagorean Theorem is most suitable.
