Indian mathematicsIndian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, Bhaskara II, and Varāhamihira. The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra.
Cube rootIn mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 23 = 8, while the other cube roots of 8 are and .
Square rootIn mathematics, a square root of a number x is a number y such that ; in other words, a number y whose square (the result of multiplying the number by itself, or ) is x. For example, 4 and −4 are square roots of 16 because . Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by where the symbol "" is called the radical sign or radix. For example, to express the fact that the principal square root of 9 is 3, we write .
The Compendious Book on Calculation by Completion and BalancingThe Compendious Book on Calculation by Completion and Balancing (الكتاب المختصر في حساب الجبر والمقابلة, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah; Liber Algebræ et Almucabola), also known as al-Jabr (Arabic: الجبر), is an Arabic mathematical treatise on algebra written in Baghdad around 820 CE by the Persian polymath Muḥammad ibn Mūsā al-Khwārizmī. It was a landmark work in the history of mathematics, establishing algebra as an independent discipline.
TranslationTranslation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between translating (a written text) and interpreting (oral or signed communication between users of different languages); under this distinction, translation can begin only after the appearance of writing within a language community.
Tensor algebraIn mathematics, the tensor algebra of a vector space V, denoted T(V) or T^•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of T(V).
Algebraic curveIn mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation h(x, y, t) = 0 can be restricted to the affine algebraic plane curve of equation h(x, y, 1) = 0.
Chinese mathematicsMathematics in China emerged independently by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system (base 2 and base 10), algebra, geometry, number theory and trigonometry. Since the Han dynasty, as diophantine approximation being a prominent numerical method, the Chinese made substantial progress on polynomial evaluation. Algorithms like regula falsi and expressions like continued fractions are widely used and have been well-documented ever since.
History of mathematicsThe 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 the 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 purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.
Machine translationMachine translation is use of either rule-based or probabilistic (i.e. statistical and, most recently, neural network-based) machine learning approaches to translation of text or speech from one language to another, including the contextual, idiomatic and pragmatic nuances of both languages. History of machine translation The origins of machine translation can be traced back to the work of Al-Kindi, a ninth-century Arabic cryptographer who developed techniques for systemic language translation, including cryptanalysis, frequency analysis, and probability and statistics, which are used in modern machine translation.
Heron's formulaIn geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths a, b, c. If is the semiperimeter of the triangle, the area A is, It is named after first-century engineer Heron of Alexandria (or Hero) who proved it in his work Metrica, though it was probably known centuries earlier. Let △ABC be the triangle with sides a = 4, b = 13 and c = 15. This triangle's semiperimeter is and so the area is In this example, the side lengths and area are integers, making it a Heronian triangle.
Computer-assisted translationComputer-aided translation (CAT), also referred to as computer-assisted translation or computer-aided human translation (CAHT), is the use of software to assist a human translator in the translation process. The translation is created by a human, and certain aspects of the process are facilitated by software; this is in contrast with machine translation (MT), in which the translation is created by a computer, optionally with some human intervention (e.g. pre-editing and post-editing).
