Scientific approach: an example in mathematics The history of humanity has had enough thinkers who changed the world by the genius of their work and their reflections. Some have been so influential that they have been awarded for discoveries that in reality were not invented by them. This is the case of Pythagoras of Samos (c. 570-480 BC), great thinker of antiquity, who is credited with the theorem says Pythagorean Theorem. S'ennonce and how this theorem and who are the proponents? What about its historical development?
I. Enunciation of the theorem
This theorem is a property of Euclidean geometry in arguing that a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
frome The best known of this theorem is: plane in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the sides of the right angle .
The length refers to the actual number on which the operation of squaring is perfectly defined.
The hypotenuse refers to the line segment, for which the geometric object squaring has meaningless.
Let the triangle ABC, the rectangle in C. Where AB is the hypotenuse AB = c, AC = b, BC = a, we shall have: BC ² = AB ² + AC ² or a ² + b ² = c ².
The Pythagorean theorem can, in fact, calculate the length of one side of the triangle, if one knows the other two. Thus:
if a = 3 and b = 4, the length c is: a ² + b ² = 3 ² + 4 ² = 25, hence c ² = 25, therefore, c = 5
All three integers (3, 4, 5) representing the lengths of three sides of a triangle is called Pythagorean triples.
II. Tenants and evolution of the theorem of Pythagoras
property of Pythagoras is known well before the historical period of Pythagoras.
The oldest depiction of pythoricien triplet (triangle whose sides are integers) is in Great Britain on megaliths (c. 2500 BC). There are also traces of Pythagorean triple on Babylonian tablets (tablets Plimptom 322, circa 1800 BC), this proves AUI More than a thousand years before Pythagoras, the surveyors were aware of Pythagorean triple.
It would certainly be the life of Pythagoras that his name would assoccié property. Legend has it that Pythagoras was so proud that he sacrificed to the gods a massacre, that is to say, one hundred (100) horse. The school of Pythagoras was, perhaps, the first to give a proof of the theorem. For, from the discovery of a property, its generalization and its proof, it often takes several centuries. Several developments have taken place on this theorem since antiquity until today.
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The first written record of the proof of this theorem is in Elements of Euclid in the following form: For triangles, the square of the side that supports the right corner of the square other two sides (Book I, proposition XLVII). With its inverse: If the square of one side of a triangle equals the square of the other two sides, the angle is supported by the right side (Book I, proposition XLVIII) .
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The property is also known in China. Found its mark in one of the oldest Chinese mathematical works: Zhoubi suajing . This book written between 220-206 BC. AD combines the computational techniques dating from the Zhou dynasty (tenth century BC. JC - 256AV. JC). The demonstrable theorem which bears the name of China theorem is contained in the Guge Jiuzhang suanshu (the nine chapters on mathematical art, 100 BC. BC - 50 AD.) Demonstration that nothing like that Euclid and shows the originality of the Chinese.
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In India, around 300 BC. AD, we find the trace of a numerical demonstration of the property; evidence performed on specific numbers but can easily be generalized. The demonstration of India, like that of Guge, leading to the property linking the square of the hypotenuse squared difference of the sides and the area of original triangle:
c ² = (ab) + ² 2ab.
On a geometrical property, the Pythagorean theorem takes développemnt arithmetic with the search for all triples of integers associated with the three sides of a triangle. This research opens the door to another: the quest for equality triple checking an + bn = cn.
There are still many demonstrations of this theorem:
Demonstration Leonardo da Vinci and even the American President James Garfield, he is also the theorem of Al-Kashi which gives for any triangle relationship.
The Pythagorean theorem was generalized to other figures and used in several areas. Already, it was announced by Euclid in his Elements (Proposition 31 of Book VI): "In the triangles, figure built on the side behind the right angle is equal to the similar and similarly described figures on the sides containing the right angle. " This property allows us to show that the area of triangle is equal to the sum of areas of crescents drawn on each side of the right angle.
The property is used in Cartesian coordinates in an orthonormal where she can express the distance between two points Plan ...
Today, this property is used in writing the vector in an inner product space and even in non-Euclidean geometry. The theory has inspired several demonstrations. Elisha Scott Leonis has gathered 370 in his book The Pythagorean proposal.
short, the theorem of Pythagoras is a palpable proof of the scientific process. For what, for thousands of years before our era, was in the form of observation and experimentation has developed from the Pythagorean theorem (VI century BC) and several demonstrations to inspire discovered until today. The track is still open to exploit this property or the theorem in other areas such as in physical space in the world of galaxies.
Sources:
1. Euclid's Elements , Book I, IV ° S.
2. Eliane Cousquer, Pythagoras's theorem, DUSQ, 1931.
3. Alexander Bogomolny, Proposal of 78 different demonstrations , 1938.
