Pythagoras theorem wikipedia

In mathematicsthe Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.

Such a triple is commonly written a , b , c , and a well-known example is 3, 4, 5. If a , b , c is a Pythagorean triple, then so is ka , kb , kc for any positive integer k. A primitive Pythagorean triple is one in which a , b and c are coprime that is, they have no common divisor larger than 1. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle and is a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. Pythagorean triples have been known since ancient times.

Pythagoras theorem wikipedia

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. Following Croton's decisive victory over Sybaris in around BC, Pythagoras's followers came into conflict with supporters of democracy , and Pythagorean meeting houses were burned. Pythagoras may have been killed during this persecution, or he may have escaped to Metapontum and died there. In antiquity, Pythagoras was credited with many mathematical and scientific discoveries, including the Pythagorean theorem , Pythagorean tuning , the five regular solids , the Theory of Proportions , the sphericity of the Earth , and the identity of the morning and evening stars as the planet Venus. It was said that he was the first man to call himself a philosopher "lover of wisdom" [c] and that he was the first to divide the globe into five climatic zones. 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. Some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important, but it is debated to what extent, if at all, he actually contributed to mathematics or natural philosophy. Pythagoras influenced Plato, whose dialogues , especially his Timaeus , exhibit Pythagorean teachings. Pythagorean ideas on mathematical perfection also impacted ancient Greek art.

Wong Archived at the Wayback MachineA generalized n-dimensional Pythagorean theorem, pythagoras theorem wikipedia. The triangle ABC is a right triangle, as shown in the upper part of the diagram, with BC the hypotenuse. O'Meara,Pythagoras Revived.

In geometry , the 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 Elements , the converse of the Pythagorean theorem, which states that if the square on one side 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 ,. The cruciform curve or cross curve is a quartic plane curve given by the equation. Substituting x with AC and y with BC gives.

In a right triangle The square of the hypotenuse is equal to the sum of the squares of the other two sides. The Khan Academy has video material relating to this topic which you may find easier to follow:. In a right triangle right-angled triangle the longest side is always furthest away from the right angle. It is called the hypotenuse. The length of the hypotenuse can be calculated from the lengths of the two other sides. In the diagram, c is the hypotenuse and we can calculate it from a and b.

Pythagoras theorem wikipedia

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.

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Drop a perpendicular from A to the side opposite the hypotenuse in the square on the hypotenuse. Then solving. These rectangles in their new position have now delineated two new squares, one having side length a is formed in the bottom-left corner, and another square of side length b formed in the top-right corner. 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. The article appeared in the peer-reviewed online journal Convergence published by the Mathematical Association of America. Communalism Metempsychosis Musica universalis. Tools Tools. Cryptographic applications of primitive Pythagorean triples. John Weale. In each right triangle, Pythagoras' theorem establishes the length of the hypotenuse in terms of this unit. Dear Wikiwand AI, let's keep it short by simply answering these key questions:. The dot product is called the standard inner product or the Euclidean inner product. Find the radius of the circle in the diagram. Representative see diagram. Aspastamba knew that the square on the diagonal of a rectangle is equal to the sum of the squares on the two adjacent sides.

The Pythagoras Theorem or the Pythagorean theorem, named after the Greek mathematician Pythagoras states that:.

Cyrene Mouseion of Alexandria Platonic Academy. Libeskind, Shlomo Wikimedia Commons has media related to Pythagorean theorem. Consequently, every primitive Pythagorean triple can be obtained in a unique way as a product of copies of the matrices U and L. Translated by Morrow, Glenn R. For the method of Plato argues from even numbers. This also follows from the converse to Thales' Theorem. OCLC A revival of Pythagorean teachings occurred in the first century BC [] when Middle Platonist philosophers such as Eudorus and Philo of Alexandria hailed the rise of a "new" Pythagoreanism in Alexandria. In Renger, Johannes ed. Leff