Portal:Mathematics
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Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). 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. (Full article...)
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- ... that after Archimedes first defined convex curves, mathematicians lost interest in their analysis until the 19th century, more than two millennia later?
- ... that ten-sided gaming dice have kite-shaped faces?
- ... that the prologue to The Polymath was written by Martin Kemp, a leading expert on Leonardo da Vinci?
- ... that despite published scholarship to the contrary, Andrew Planta neither received a doctorate nor taught mathematics at Erlangen?
- ... that people in Madagascar perform algebra on tree seeds in order to tell the future?
- ... that multiple mathematics competitions have made use of Sophie Germain's identity?
- ... that the British National Hospital Service Reserve trained volunteers to carry out first aid in the aftermath of a nuclear or chemical attack?
- ... that Fairleigh Dickinson's upset victory over Purdue was the biggest upset in terms of point spread in NCAA tournament history, with Purdue being a 23+1⁄2-point favorite?
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- ... 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?
- ...that some functions can be written as an infinite sum of trigonometric polynomials and that this sum is called the Fourier series of that function?
- ...that the identity elements for arithmetic operations make use of the only two whole numbers that are neither composites nor prime numbers, 0 and 1?
- ...that as of April 2010 only 35 even numbers have been found that are not the sum of two primes which are each in a Twin Primes pair? ref
- ...the Piphilology record (memorizing digits of Pi) is 70000 as of Mar 2015?
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The real part (red) and imaginary part (blue) of the critical line Re(s) = 1/2 of the Riemann zeta-function. Image credit: User:Army1987 |
The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. It has been an open question for well over a century, despite attracting concentrated efforts from many outstanding mathematicians.
The Riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta-function ζ(s). The Riemann zeta-function is defined for all complex numbers s ≠ 1. It has zeros at the negative even integers (i.e. at s=-2, s=-4, s=-6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that:
- The real part of any non-trivial zero of the Riemann zeta function is ½
Thus the non-trivial zeros should lie on the so-called critical line ½ + it with t a real number and i the imaginary unit. The Riemann zeta-function along the critical line is sometimes studied in terms of the Z-function, whose real zeros correspond to the zeros of the zeta-function on the critical line.
The Riemann hypothesis is one of the most important open problems in contemporary mathematics; a $1,000,000 prize has been offered by the Clay Mathematics Institute for a proof. Most mathematicians believe the Riemann hypothesis to be true. (J. E. Littlewood and Atle Selberg have been reported as skeptical. Selberg's skepticism, if any, waned, from his young days. In a 1989 paper, he suggested that an analogue should hold for a much wider class of functions, the Selberg class.) (Full article...)
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- ^ Kazarinoff (2003), pp. 10, 15 ; Martin (1998), p. 41, Corollary 2.16 .