Portal:Mathematics
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Mathematics is the study of numbers, quantity, space, pattern, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.
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Mathematics department in Göttingen where Hilbert worked from 1895 until his retirement in 1930 Image credit: Daniel Schwen |
David Hilbert (January 23, 1862, Wehlau, Prussia–February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. He established his reputation as a great mathematician and scientist by inventing or developing a broad range of ideas, such as invariant theory, the axiomization of geometry, and the notion of Hilbert space, one of the foundations of functional analysis. Hilbert and his students supplied significant portions of the mathematic infrastructure required for quantum mechanics and general relativity. He is one of the founders of proof theory, mathematical logic, and the distinction between mathematics and metamathematics, and warmly defended Cantor's set theory and transfinite numbers. A famous example of his world leadership in mathematics is his 1900 presentation of a set of problems that set the course for much of the mathematical research of the 20th century.
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The four conic sections arise when a plane cuts through a double cone in different ways. If the plane cuts through parallel to the side of the cone (case 1), a parabola results (to be specific, the parabola is the shape of the planar graph that is formed by the set of points of intersection of the plane and the cone). If the plane is perpendicular to the cone's axis of symmetry (case 2, lower plane), a circle results. If the plane cuts through at some angle between these two cases (case 2, upper plane) — that is, if the angle between the plane and the axis of symmetry is larger than that between the side of the cone and the axis, but smaller than a right angle — an ellipse results. If the plane is parallel to the axis of symmetry (case 3), or makes a smaller positive angle with the axis than the side of the cone does (not shown), a hyperbola results. In all of these cases, if the plane passes through the point at which the two cones meet (the vertex), a degenerate conic results. First studied by the ancient Greeks in the 4th century BCE, conic sections were still considered advanced mathematics by the time Euclid (fl. c. 300 BCE) created his Elements, and so do not appear in that famous work. Euclid did write a work on conics, but it was lost after Apollonius of Perga (d. c. 190 BCE) collected the same information and added many new results in his Conics. Other important results on conics were discovered by the medieval Persian mathematician Omar Khayyám (d. 1131 CE), who used conic sections to solve algebraic equations.
In the news
- 19 March 2019 –
- The Norwegian Academy of Science and Letters awards this year's Abel Prize to Karen Uhlenbeck for "her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems." Uhlenbeck is the first woman to win this prize. (The New York Times via MSN.com)
Did you know…
- ...that the primality of a number can be determined using only a single division using Wilson's Theorem?
- ...that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type the complete works of William Shakespeare?
- ... that there are 115,200 solutions to the ménage problem of permuting six female-male couples at a twelve-person table so that men and women alternate and are seated away from their partners?
- ... that mathematician Paul Erdős called the Hadwiger conjecture, a still-open generalization of the four-color problem, "one of the deepest unsolved problems in graph theory"?
- ...that the six permutations of the vector (1,2,3) form a regular hexagon in 3d space, the 24 permutations of (1,2,3,4) form a truncated octahedron in four dimensions, and both are examples of permutohedra?
- ...that Ostomachion is a mathematical treatise attributed to Archimedes on a 14-piece tiling puzzle similar to tangram?
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