Pythagoras theorem wikipedia

In geometrythe inverse Pythagorean theorem also known as the reciprocal Pythagorean theorem [1] or the upside down Pythagorean theorem [2] is as follows: [3]. This theorem should not be confused with proposition 48 in book 1 of Euclid 's Elementsthe converse of the Pythagorean theorem, pythagoras theorem wikipedia, which states that if the square on one pythagoras theorem wikipedia of a triangle is equal to the sum of the squares on the other two sides then the other two sides contain a right angle. Using the Pythagorean theorem .

His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato , Aristotle , and, through them, the West in general. Knowledge of his life is clouded by legend. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around BC, he travelled to Croton in southern Italy, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included aspects of vegetarianism. The teaching most securely identified with Pythagoras is metempsychosis , or the "transmigration of souls", which holds that every soul is immortal and, upon death, enters into a new body. He may have also devised the doctrine of musica universalis , which holds that the planets move according to mathematical equations and thus resonate to produce an inaudible symphony of music. Scholars debate whether Pythagoras developed the numerological and musical teachings attributed to him, or if those teachings were developed by his later followers, particularly Philolaus of Croton.

Pythagoras theorem wikipedia

In mathematics , the Pythagorean theorem or Pythagoras's theorem is a statement about the sides of a right triangle. One of the angles of a right triangle is always equal to 90 degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite to the right angle, and it is always the longest side. The Pythagorean theorem says that the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs. In this picture, the area of the blue square added to the area of the red square makes the area of the purple square. It was named after the Greek mathematician Pythagoras :. One proof of the Pythagorean theorem was found by a Greek mathematician, Eudoxus of Cnidus. We can get another proof of the Pythagorean theorem by using similar triangles. The triangle with sides of 3, 4, and 5 is a well known example. The three-four-five triangle works for all multiples of 3, 4, and 5. In other words, numbers such as 6, 8, 10 or 30, 40 and 50 are also Pythagorean triples. A Pythagorean triple that is not a multiple of other triples is called a primitive Pythagorean triple.

Classical historians debate whether Pythagoras made these discoveries, and many of the accomplishments credited to him likely originated earlier or were made by his colleagues or successors.

Pythagoras of Samos was a famous Greek mathematician and philosopher c. He started a group of mathematicians, called the Pythagoreans, who worshiped numbers and lived like monks. He had an influence on Plato. He had a great impact on mathematics, theory of music and astronomy. His theories are still used in mathematics today.

Garfield's proof of the Pythagorean theorem is an original proof the Pythagorean theorem invented by James A. Garfield November 19, — September 19, , the 20th president of the United States. He assumed the office of President on March 4, , and served in that position only for a brief period up to September 19, The proof is nontrivial and, according to the historian of mathematics, William Dunham , "Garfield's is really a very clever proof. The theorem is proved by computing the area of this trapezoid in two different ways. Contents move to sidebar hide. Article Talk. Read Edit View history.

Pythagoras theorem wikipedia

In mathematics , the Pythagorean theorem or Pythagoras's theorem is a statement about the sides of a right triangle. One of the angles of a right triangle is always equal to 90 degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite to the right angle, and it is always the longest side. The Pythagorean theorem says that the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs. In this picture, the area of the blue square added to the area of the red square makes the area of the purple square.

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Aristotle, de Anima , i. Ancient Greece. For such a triple, either a or b is even, and the other is odd; from this, it follows that c is also odd. Each square is first sheared into a parallelogram, and then into a rectangle which can be translated onto one section of the square on the hypotenuse. The Pythagorean school dealt with proportions by comparison of integer multiples of a common subunit. Babylonian knowledge of proof of the Pythagorean theorem is discussed by J. Beginning functional analysis. These two triangles are shown to be congruent , proving this square has the same area as the left rectangle. This can be generalised to find the distance between two points, z 1 and z 2 say. Methods of applied mathematics Reprint of Prentice-Hall 2nd ed. The "hypotenuse" is the base of the tetrahedron at the back of the figure, and the "legs" are the three sides emanating from the vertex in the foreground. Therefore, the ratios of their sides must be the same, that is:. The smallest known counterexamples are [45] [46] [16]. Atiyah Gromov.

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The underlying question is why Euclid did not use this proof, but invented another. Knowledge of his life is clouded by legend. Assen , pp. Pythagoras may have been killed during this persecution, or he may have escaped to Metapontum and died there. Therefore, the angle between the side of lengths a and b in the original triangle is a right angle. A History of Greek Mathematics 2 Vols. Choosing m and n from certain integer sequences gives interesting results. Then two rectangles are formed with sides a and b by moving the triangles. University Press of America, Lanham , p. Running Press. The basic idea behind this generalization is that the area of a plane figure is proportional to the square of any linear dimension, and in particular is proportional to the square of the length of any side. Mathematical Association of America. Pythagoras of Samos was a famous Greek mathematician and philosopher c. Outline History Timeline.