In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
The Pythagorean theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof, although it is often argued that knowledge of the theorem predates him. There is evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they used it in a mathematical framework.
The theorem has numerous proofs, possibly the most of any mathematical theorem. These are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps and cartoons abound.
Other articles related to "pythagorean theorem, theorem, pythagorean":
... "rules of inference" to generate new strings from given ones), one can prove that the Pythagorean theorem holds (that is, you can generate the string corresponding to the Pythagorean theorem) ... In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one if you assign meaning to the strings in such a way that the rules of the game become true (i.e ... to the axioms and the rules of inference are truth-preserving), then you have to accept the theorem, or, rather, the interpretation you have given it must be a true ...
... is possible to give a more geometric proof than using the Pythagorean theorem alone ... Algebraic manipulations (in particular the binomial theorem) are avoided ... Apply the Pythagorean theorem to obtain Then use the tangent secant theorem (Euclid's Elements Book 3, Proposition 36), which says that the square on the tangent through a point B ...
... The Pythagorean theorem has arisen in popular culture in a variety of ways ... and Sullivan comic opera The Pirates of Penzance, "About binomial theorem I'm teeming with a lot o' news, With many cheerful facts about the square of the hypotenuse", makes an oblique reference to the theorem ... in the film The Wizard of Oz makes a more specific reference to the theorem ...
... earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians." They contain lists of Pythagorean triples, which are particular cases of ... Sulba Sutra, which contains examples of simple Pythagorean triples, such as, , and as well as a statement of the Pythagorean theorem for the sides of a square "The rope which is. 1850 BCE "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple, indicating, in particular ...
... The Pythagorean theorem can be proven without words as shown in the second diagram on right ... The two different methods for determining the area of the large square give the relation between the sides ...
Famous quotes containing the words theorem and/or pythagorean:
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (19131960)
“Come now, let us go and be dumb. Let us sit with our hands on our mouths, a long, austere, Pythagorean lustrum. Let us live in corners, and do chores, and suffer, and weep, and drudge, with eyes and hearts that love the Lord. Silence, seclusion, austerity, may pierce deep into the grandeur and secret of our being, and so diving, bring up out of secular darkness, the sublimities of the moral constitution.”
—Ralph Waldo Emerson (18031882)