Pythagorean theorem in the context of "Right triangle"

Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during classical and late antiquity, mostly from the 5th century BC to the 6th century AD. Greek mathematicians lived in cities spread around the shores of the ancient Mediterranean, from Anatolia to Italy and North Africa, but were united by Greek culture and the Greek language. The development of mathematics as a theoretical discipline and the use of deductive reasoning in proofs is an important difference between Greek mathematics and those of preceding civilizations.

The early history of Greek mathematics is obscure, and traditional narratives of mathematical theorems found before the fifth century BC are regarded as later inventions. It is now generally accepted that treatises of deductive mathematics written in Greek began circulating around the mid-fifth century BC, but the earliest complete work on the subject is Euclid's Elements, written during the Hellenistic period. The works of renown mathematicians Archimedes and Apollonius, as well as of the astronomer Hipparchus, also belong to this period. In the Imperial Roman era, Ptolemy used trigonometry to determine the positions of stars in the sky, while Nicomachus and other ancient philosophers revived ancient number theory and harmonics. During late antiquity, Pappus of Alexandria wrote his Collection, summarizing the work of his predecessors, while Diophantus' Arithmetica dealt with the solution of arithmetic problems by way of pre-modern algebra. Later authors such as Theon of Alexandria, his daughter Hypatia, and Eutocius of Ascalon wrote commentaries on the authors making up the ancient Greek mathematical corpus.

In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.

In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as theorems only the most important results, and use the terms lemma, proposition and corollary for less important theorems.

The Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise written c. 300 BC by the Ancient Greek mathematician Euclid.

The Elements is the oldest extant large-scale deductive treatment of mathematics. Drawing on the works of earlier mathematicians such as Hippocrates of Chios, Eudoxus of Cnidus, and Theaetetus, the Elements is a collection in 13 books of definitions, postulates, geometric constructions, and theorems with their proofs that covers plane and solid Euclidean geometry, elementary number theory, and incommensurability. These include the Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the construction of regular polygons and polyhedra.

Samos (/ˈseɪmɒs/, also US: /ˈsæmoʊs, ˈsɑːmɔːs/; Greek: Σάμος, romanized: Sámos, Greek pronunciation: [ˈsa.mos]) is a Greek island in the eastern Aegean Sea, south of Chios, north of Patmos and the Dodecanese archipelago, and off the coast of western Turkey, from which it is separated by the 1.6-kilometre-wide (1.0 mi) Mycale Strait. It is also a separate regional unit of the North Aegean region.

In ancient times, Samos was an especially rich and powerful city-state, particularly known for its vineyards and wine production. It is home to Pythagoreion and the Heraion of Samos, a UNESCO World Heritage Site that includes the Eupalinian aqueduct, a marvel of ancient engineering. Samos is the birthplace of the Greek philosopher and mathematician Pythagoras, after whom the Pythagorean theorem is named, the philosophers Melissus of Samos and Epicurus, and the astronomer Aristarchus of Samos, the first known individual to propose that the Earth revolves around the Sun. Samian wine was well known in antiquity and is still produced on the island.

Pythagoras of Samos (Ancient Greek: Πυθαγόρας; c. 570 – c. 495 BC) was an ancient Ionian Greek philosopher, polymath, and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Modern scholars disagree regarding Pythagoras's education and influences, but most agree that he travelled to Croton in southern Italy around 530 BC, where he founded a school in which initiates were allegedly sworn to secrecy and lived a communal, ascetic lifestyle.

In antiquity, Pythagoras was credited with mathematical and scientific discoveries, such as the Pythagorean theorem, Pythagorean tuning, the five regular solids, the theory of proportions, the sphericity of the Earth, the identity of the morning and evening stars as the planet Venus, and the division of the globe into five climatic zones. He was reputedly the first man to call himself a philosopher ("lover of wisdom"). Historians debate whether Pythagoras made these discoveries and pronouncements, as some of the accomplishments credited to him likely originated earlier or were made by his colleagues or successors, such as Hippasus and Philolaus.

The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for taxation, commerce, trade, and in astronomy, to record time and formulate calendars.

The earliest mathematical texts available are from Mesopotamia and Egypt – Plimpton 322 (Babylonian c. 2000 – 1900 BC), the Rhind Mathematical Papyrus (Egyptian c. 1800 BC) and the Moscow Mathematical Papyrus (Egyptian c. 1890 BC). All these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development, after basic arithmetic and geometry.

In geometry, a hypotenuse is the side of a right triangle opposite to the right angle. It is the longest side of any such triangle; the two other shorter sides of such a triangle are called catheti or legs. Every rectangle can be divided into a pair of right triangles by cutting it along either diagonal; the diagonals are the hypotenuses of these triangles.

The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two legs. As an algebraic formula, this can be written as ${\displaystyle a^{2}+b^{2}=c^{2}}$, where ⁠${\displaystyle a}$⁠ is the length of one leg, ⁠${\displaystyle b}$⁠ is the length of the other leg, and ⁠${\displaystyle c}$⁠ is the length of the hypotenuse. For example, if the two legs of a right triangle have lengths 3 and 4, respectively, then the hypotenuse has length ⁠${\displaystyle 5}$⁠, because ⁠${\displaystyle \textstyle 3^{2}+4^{2}=25=5^{2}}$⁠.