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Freshman's Dream
The freshman's dream is a name sometimes given to the erroneous equation (x+y)^n=x^n+y^n, where n is a real number (usually a positive integer greater than 1) and x,y are nonzero real numbers. Beginning students commonly make this error in computing the power of a sum of real numbers, falsely assuming powers distribute over sums. When ''n'' = 2, it is easy to see why this is incorrect: (''x'' + ''y'')2 can be correctly computed as ''x''2 + 2''xy'' + ''y''2 using distributivity (commonly known by students as the FOIL method). For larger positive integer values of ''n'', the correct result is given by the binomial theorem. The name "freshman's dream" also sometimes refers to the theorem that says that for a prime number ''p'', if ''x'' and ''y'' are members of a commutative ring of characteristic ''p'', then (''x'' + ''y'')''p'' = ''x''''p'' + ''y''''p''. In this more exotic type of arithmetic, the "mistake" actually g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Freshman's Dream
The freshman's dream is a name sometimes given to the erroneous equation (x+y)^n=x^n+y^n, where n is a real number (usually a positive integer greater than 1) and x,y are nonzero real numbers. Beginning students commonly make this error in computing the power of a sum of real numbers, falsely assuming powers distribute over sums. When ''n'' = 2, it is easy to see why this is incorrect: (''x'' + ''y'')2 can be correctly computed as ''x''2 + 2''xy'' + ''y''2 using distributivity (commonly known by students as the FOIL method). For larger positive integer values of ''n'', the correct result is given by the binomial theorem. The name "freshman's dream" also sometimes refers to the theorem that says that for a prime number ''p'', if ''x'' and ''y'' are members of a commutative ring of characteristic ''p'', then (''x'' + ''y'')''p'' = ''x''''p'' + ''y''''p''. In this more exotic type of arithmetic, the "mistake" actually g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is that of the ring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Fallacies
In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple ''mistake'' and a ''mathematical fallacy'' in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or deception in the presentation of the proof. For example, the reason why validity fails may be attributed to a division by zero that is hidden by algebraic notation. There is a certain quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way. Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Education
In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge. Although research into mathematics education is primarily concerned with the tools, methods and approaches that facilitate practice or the study of practice, it also covers an extensive field of study encompassing a variety of different concepts, theories and methods. National and international organisations regularly hold conferences and publish literature in order to improve mathematics education. History Ancient Elementary mathematics were a core part of education in many ancient civilisations, including ancient Egypt, ancient Babylonia, ancient Greece, ancient Rome and Vedic India. In most cases, formal education was only available to male children with sufficiently high status, wealth or caste. The oldest known mathematics textbook is the Rh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sophomore's Dream
In mathematics, the sophomore's dream is the pair of identities (especially the first) :\begin \int_0^1 x^\,dx &= \sum_^\infty n^ \\ \end :\begin \int_0^1 x^x \,dx &= \sum_^\infty (-1)^n^ = - \sum_^\infty (-n)^ \end discovered in 1697 by Johann Bernoulli. The numerical values of these constants are approximately 1.291285997... and 0.7834305107..., respectively. The name "sophomore's dream"It appears in . is in contrast to the name "freshman's dream" which is given to the incorrectIncorrect in general, but correct when one is working in a commutative ring of prime characteristic ''p'' with ''n'' being a power of ''p''. The correct result in a general commutative context is given by the binomial theorem. identity . The sophomore's dream has a similar too-good-to-be-true feel, but is true. Proof The proofs of the two identities are completely analogous, so only the proof of the second is presented here. The key ingredients of the proof are: * to write ''x''''x'' = exp('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primality Test
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might more accurately be called ''compositeness tests'' instead of primality tests. Simple methods The simplest primality test is ''trial division'': given an input number, ''n'', check whether it is evenly divisible by any prime number between 2 and (i.e. that the division leaves no remainder). If so, then ''n'' is composite. Otherwise, it is prime.Riesel (1994) pp.2-3 For example, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pons Asinorum
In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the ''pons asinorum'' (, ), typically translated as "bridge of asses". This statement is Proposition 5 of Book 1 in Euclid's '' Elements'', and is also known as the isosceles triangle theorem. Its converse is also true: if two angles of a triangle are equal, then the sides opposite them are also equal. The term is also applied to the Pythagorean theorem. ''Pons asinorum'' is also used metaphorically for a problem or challenge which acts as a test of critical thinking, referring to the "ass' bridge's" ability to separate capable and incapable reasoners. Its first known usage in this context was in 1645. A persistent piece of mathematical folklore claims that an artificial intelligence program discovered an original and more elegant proof of this theorem. In fact, Marvin Minsky recounts that he had rediscovered the Pappus proof (which he was not aware of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas W
Thomas may refer to: People * List of people with given name Thomas * Thomas (name) * Thomas (surname) * Saint Thomas (other) * Thomas Aquinas (1225–1274) Italian Dominican friar, philosopher, and Doctor of the Church * Thomas the Apostle * Thomas (bishop of the East Angles) (fl. 640s–650s), medieval Bishop of the East Angles * Thomas (Archdeacon of Barnstaple) (fl. 1203), Archdeacon of Barnstaple * Thomas, Count of Perche (1195–1217), Count of Perche * Thomas (bishop of Finland) (1248), first known Bishop of Finland * Thomas, Earl of Mar (1330–1377), 14th-century Earl, Aberdeen, Scotland Geography Places in the United States * Thomas, Illinois * Thomas, Indiana * Thomas, Oklahoma * Thomas, Oregon * Thomas, South Dakota * Thomas, Virginia * Thomas, Washington * Thomas, West Virginia * Thomas County (other) * Thomas Township (other) Elsewhere * Thomas Glacier (Greenland) Arts, entertainment, and media * ''Thomas'' (Burton novel) 1969 nove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Susan Montgomery
M. Susan Montgomery (born 2 April 1943 in Lansing, MI) is a distinguished American mathematician whose current research interests concern noncommutative algebras: in particular, Hopf algebras, their structure and representations, and their actions on other algebras. Her early research was on group actions on rings. Education Montgomery received her B.A. in 1965 from the University of Michigan and her Ph.D. in Mathematics from the University of Chicago in 1969 under the supervision of I. N. Herstein. Career Upon receiving her Ph.D. from Chicago, Montgomery spent one year on the faculty at DePaul University. Montgomery joined the faculty of the University of Southern California (USC) in 1970 and was promoted to the rank of Professor in 1982. She was chair of the Department of Mathematics at USC from 1996 to 1999. Montgomery has spent sabbaticals at the Hebrew University of Jerusalem, the University of Leeds, the University of Wisconsin, the University of Munich, the Universit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stephen Cole Kleene
Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory, which subsequently helped to provide the foundations of theoretical computer science. Kleene's work grounds the study of computable functions. A number of mathematical concepts are named after him: Kleene hierarchy, Kleene algebra, the Kleene star (Kleene closure), Kleene's recursion theorem and the Kleene fixed-point theorem. He also invented regular expressions in 1951 to describe McCulloch-Pitts neural networks, and made significant contributions to the foundations of mathematical intuitionism. Biography Kleene was awarded a bachelor's degree from Amherst College in 1930. He was awarded a Ph.D. in mathematics from Princeton University in 1934, where his thesis, entitled ''A Theory of Positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
