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Proof Without Words
In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or mathematically rigorous proofs due to their self-evident nature. When the diagram demonstrates a particular case of a general statement, to be a proof, it must be generalisable. A proof without words is not the same as a mathematical proof, because it omits the details of the logical argument it illustrates. However, it can provide valuable intuitions to the viewer that can help them formulate or better understand a true proof. Examples Sum of odd numbers The statement that the sum of all positive odd numbers up to 2''n'' − 1 is a perfect square—more specifically, the perfect square ''n''2—can be demonstrated by a proof without words. In one corner of a grid, a single block repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicomachus Theorem 3D
Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works '' Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in Gerasa, in the Roman province of Syria (now Jerash, Jordan). He was a Neopythagorean, who wrote about the mystical properties of numbers.Eric Temple Bell (1940), ''The development of mathematics'', page 83.Frank J. Swetz (2013), ''The European Mathematical Awakening'', page 17, Courier Life Little is known about the life of Nicomachus except that he was a Pythagorean who came from Gerasa.} Historians consider him a Neopythagorean based on his tendency to view numbers as having mystical properties. The age in which he lived (c. 100 AD) is only known because he mentions Thrasyllus in his ''Manual of Harmonics'', and because his ''Introduction to Arithmetic'' was apparently translated into Latin in the mid 2nd century by Apuleius.Henrie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jensen Graph
Jensen may refer to: People *Jensen (surname) *Jensen (given name) *Jensen (gamer), Danish professional ''League of Legends'' player Places Australia * Jensen Oval, Sydney, Australia, a soccer park * Jensen, Queensland, a suburb of Townsville Greenland *East Jensen Island in Greenland * West Jensen Island in Greenland United States *Jensen Beach, Florida, United States, a census-designated place *Jensen, Utah, United States, a census-designated place Other uses *Jensen Electronics, electronics brand owned by Audiovox Corporation *Jensen Loudspeakers *Jensen Motors, a British manufacturer of sports cars and commercial vehicles *Jensen Prize, for the best papers in the ''Journal of Financial Economics'' *''Jensen!'', Dutch television show *Jensen's alpha, financial performance index * Jensen's Device, a computer programming technique *Jensen's inequality, in mathematics, an inequality concerning the average of convex functions *''Jensen'', development codename for the DE ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Without Words
In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or mathematically rigorous proofs due to their self-evident nature. When the diagram demonstrates a particular case of a general statement, to be a proof, it must be generalisable. A proof without words is not the same as a mathematical proof, because it omits the details of the logical argument it illustrates. However, it can provide valuable intuitions to the viewer that can help them formulate or better understand a true proof. Examples Sum of odd numbers The statement that the sum of all positive odd numbers up to 2''n'' − 1 is a perfect square—more specifically, the perfect square ''n''2—can be demonstrated by a proof without words. In one corner of a grid, a single block repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is a premise or starting point for reasoning. As used in mathematics, the term ''axiom'' is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms (e.g., ) are actua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Java Applet
Java applets were small applications written in the Java programming language, or another programming language that compiles to Java bytecode, and delivered to users in the form of Java bytecode. The user launched the Java applet from a web page, and the applet was then executed within a Java virtual machine (JVM) in a process separate from the web browser itself. A Java applet could appear in a frame of the web page, a new application window, Sun's AppletViewer, or a stand-alone tool for testing applets. Java applets were introduced in the first version of the Java language, which was released in 1995. Beginning in 2013, major web browsers began to phase out support for the underlying technology applets used to run, with applets becoming completely unable to be run by 2015–2017. Java applets were deprecated by Java 9 in 2017. Java applets were usually written in Java, but other languages such as Jython, JRuby, Pascal, Scala, NetRexx, or Eiffel (via SmartEi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
USAMTS
The United States of America Mathematical Talent Search (USAMTS) is a mathematics competition open to all United States students in or below high school. History Professor George Berzsenyi initiated the contest in 1989 under the KöMaL model and under joint sponsorship of the Rose-Hulman Institute of Technology and the Consortium for Mathematics and its Applications. As of 2021, the USAMTS is sponsored by the National Security Agency and administered by the Art of Problem Solving foundation. There were 718 participants in the 2004-2005 school year, with an average score of 49.25 out of 100. Format The competition is proof and research based. Students submit proofs within the round's timeframe (usually a month), and return solutions by mail or upload their solutions in a PDF file through the USAMTS website. During this time, students are free to use any mathematical resources that are available, so long as it is not the help of another person. Carefully written justifications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
College Mathematics Journal
The ''College Mathematics Journal'' is an expository magazine aimed at teachers of college mathematics, particular those teaching the first two years. It is published by Taylor & Francis on behalf of the Mathematical Association of America and is a continuation of ''Two-Year College Mathematics Journal''. It covers all aspects of mathematics. It publishes articles intended to enhance undergraduate instruction and classroom learning, including expository articles, short notes, problems, and "mathematical ephemera" such as fallacious proofs, quotations, cartoons, poetry, and humor. Paid circulation in 2008 was 9,000 and total circulation was 9,500. The MAA gives the George Pólya Awards annually "for articles of expository excellence" published in the ''College Mathematics Journal''. References External links *''The College Mathematics Journal''at JSTOR''The College Mathematics Journal''at Taylor & Francis Taylor & Francis Group is an international company originating i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Magazine
''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a journal of mathematics rather than pedagogy. Rather than articles in the terse "theorem-proof" style of research journals, it seeks articles which provide a context for the mathematics they deliver, with examples, applications, illustrations, and historical background. Paid circulation in 2008 was 9,500 and total circulation was 10,000. ''Mathematics Magazine'' is a continuation of ''Mathematics News Letter'' (1926-1934) and ''National Mathematics Magazine'' (1934-1945.) Doris Schattschneider became the first female editor of ''Mathematics Magazine'' in 1981. .. The MAA gives the Carl B. Allendoerfer Awards annually "for articles of expository excellence" published in ''Mathematics Magazine''. See also *''American Mathematical Monthl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jensen's Inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations. Jensen's inequality generalizes the statement that the secant line of a convex function lies ''above'' the graph of the function, which is Jensen's inequality for two points: the secant line consists of weighted means of the convex function (for ''t'' ∈ ,1, :t f(x_1) + (1-t) f(x_2), whil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that 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 on the other two sides. This theorem can be written as an equation relating the lengths of the sides ''a'', ''b'' and the hypotenuse ''c'', often called the Pythagorean equation: :a^2 + b^2 = c^2 , The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proven numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
