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Euler's Identity
In mathematics, Euler's identity (also known as Euler's equation) is the Equality (mathematics), equality e^ + 1 = 0 where :e is E (mathematical constant), Euler's number, the base of natural logarithms, :i is the imaginary unit, which by definition satisfies i^2 = -1, and :\pi is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula e^ = \cos x + i\sin x when evaluated for x = \pi. Euler's identity is considered an exemplar of mathematical beauty, as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in Lindemann–Weierstrass theorem#Transcendence of e and π, a proof that is Transcendental number, transcendental, which implies the impossibility of squaring the circle. Mathematical beauty Euler's identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), division. The addition of two Natural number, whole numbers results in the total or ''summation, sum'' of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as , which is read as "three plus two Equality (mathematics), equals five". Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers, and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as Euclidean vector, vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Nahin
Paul J. Nahin (born November 26, 1940) is an American electrical engineer, author, and former college professor. He has written over 20 books on topics in physics and mathematics. Biography Born in California, Nahin graduated from Brea Olinda High School in 1958, and thereafter received a B.S. from Stanford University in 1962, an M.S. from the California Institute of Technology in 1963, and a Ph.D. from the University of California, Irvine, in 1972, all in electrical engineering.Electrical and computer engineering expert offers annual Sampson Lecture '' News'' (March 23, 2011). Nahin thereafter taught at |
Sonnet
A sonnet is a fixed poetic form with a structure traditionally consisting of fourteen lines adhering to a set Rhyme scheme, rhyming scheme. The term derives from the Italian word ''sonetto'' (, from the Latin word ''sonus'', ). Originating in 13th-century Sicily, the sonnet was in time taken up in many European-language areas, mainly to express romantic love at first, although eventually any subject was considered acceptable. Many formal variations were also introduced, including abandonment of the quatorzain limit – and even of rhyme altogether in modern times. Romance languages Sicilian Giacomo da Lentini is credited with the sonnet's invention at the Court of Frederick II, Holy Roman Emperor, Frederick II in the Sicilian city of Palermo. The Sicilian School of poets who surrounded Lentini then spread the form to the mainland. Those earliest sonnets no longer survive in the original Sicilian language, however, but only after being translated into Tuscan dialect. The form c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Keith Devlin
Keith James Devlin (born 16 March 1947) is a British mathematician and popular science writer. Since 1987 he has lived in the United States. He has dual British-American citizenship.Curriculum vitae Profkeithdevlin.com, accessed 3 February 2014. Education He was born and grew up in England, in Kingston upon Hull, where he attended Greatfield Estate, Kingston upon Hull#Schools, Greatfield High School. Devlin earned a BSc (special) in mathematics at King's College London in 1968, and a mathematics PhD in logic at the University of Bristol in 1971 under the supervision of Frederick Rowbottom.Career Later he got a position as a scientific assistant in mathematics at the University of Oslo, Norway, from August till December 1972. In 1974 he became a scientific assist ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanford University
Leland Stanford Junior University, commonly referred to as Stanford University, is a Private university, private research university in Stanford, California, United States. It was founded in 1885 by railroad magnate Leland Stanford (the eighth List of governors of California, governor of and then-incumbent List of United States senators from California, United States senator representing California) and his wife, Jane Stanford, Jane, in memory of their only child, Leland Stanford Jr., Leland Jr. The university admitted its first students in 1891, opening as a Mixed-sex education, coeducational and non-denominational institution. It struggled financially after Leland died in 1893 and again after much of the campus was damaged by the 1906 San Francisco earthquake. Following World War II, university Provost (education), provost Frederick Terman inspired an entrepreneurship, entrepreneurial culture to build a self-sufficient local industry (later Silicon Valley). In 1951, Stanfor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imaginary Unit
The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of in a complex number is Imaginary numbers are an important mathematical concept; they extend the real number system \mathbb to the complex number system \mathbb, in which at least one Root of a function, root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term ''imaginary'' is used because there is no real number having a negative square (algebra), square. There are two complex square roots of and , just as there are two complex square roots of every real number other than zero (which has one multiple root, double square root). In contexts in which use of the letter is ambiguous or problematic, the le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E (mathematical Constant)
The number is a mathematical constant approximately equal to 2.71828 that is the base of a logarithm, base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted \gamma. Alternatively, can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest. The number is of great importance in mathematics, alongside 0, 1, Pi, , and . All five appear in one formulation of Euler's identity e^+1=0 and play important and recurring roles across mathematics. Like the constant , is Irrational number, irrational, meaning that it cannot be represented as a ratio of integers, and moreover it is Transcendental number, transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a Disk (mathematics), disc. The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Terminology * Annulus (mathematics), Annulus: a ring-shaped object, the region bounded by two concentric circles. * Circular arc, Arc: any Connected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Identity
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an (often denoted as 0) and an identity with respect to multiplication is called a (often denoted as 1). The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Identity
In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element in the set, yields . One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings. Elementary examples * The additive identity familiar from elementary mathematics is zero, denoted 0. For example, *:5+0 = 5 = 0+5. * In the natural numbers (if 0 is included), the integers the rational numbers the real numbers and the complex numbers the additive identity is 0. This says that for a number belonging to any of these sets, *:n+0 = n = 0+n. Formal definition Let be a group that is closed under the operation of addition, denoted +. An additive identity for , denoted , is an element in such that for any element in , :e+n = n = n+e. Further examples * In a group, the additive identi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
