The Mathematics Portal
is the study of Mathematics 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.
Leonhard Euler (pronounced oiler; IPA /ˈɔɪlər/) (April 15, 1707 Basel, Switzerland - September 18, 1783 St Petersburg, Russia) was a Swiss mathematician and physicist. He is considered to be the dominant mathematician of the 18th century and one of the greatest mathematicians of all time; he is certainly among the most prolific, with collected works filling over 70 volumes.
Euler developed many important concepts and
proved numerous lasting theorems in diverse areas of mathematics, from calculus to number theory to topology. In the course of this work, he introduced many of modern mathematical terminologies, defining the concept of a , and its notation, such as function sin, cos, and tan for the trigonometric functions.
This is a graph of a portion of the complex-valued
along the Riemann zeta function critical line (the set of complex numbers having real part equal to ). More specifically, it is a graph of 1 / 2 Im ζ( versus + 1 / 2 it) Re ζ( (the imaginary part vs. the real part) for values of the real variable + 1 / 2 it) t running from 0 to 34 (the curve starts at its leftmost point, with real part approximately −1.46 and imaginary part 0). The first five zeros along the critical line are visible in this graph as the five times the curve passes through the origin (which occur at t ≈ 14.13, 21.02, 25.01, 30.42, and 32.93 — for a different perspective, see a graph of the real and imaginary parts of this function plotted separately over a wider range of values). In 1914, G. H. Hardy proved that ζ( has infinitely many zeros. According to the + 1 / 2 it) Riemann hypothesis, zeros of this form constitute the only non-trivial zeros of the full zeta function, ζ(, where s) s varies over all complex numbers. Riemann's zeta function grew out of Leonhard Euler's study of real-valued infinite series in the early 18th century. In a famous 1859 paper called " On the Number of Primes Less Than a Given Magnitude", Bernhard Riemann extended Euler's results to the complex plane and established a relation between the zeros of his zeta function and the distribution of prime numbers. The paper also contained the previously mentioned Riemann hypothesis, which is considered by many mathematicians to be the most important unsolved problem in pure mathematics. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.
Did you know…
... that as the
dimension of a hypersphere tends to infinity, its " volume" (content) tends to 0? ...that the line separating the
numerator and denominator of a fraction is called a if written as a diagonal line or a solidus if written as a horizontal line? vinculum ...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
of ménage problem 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 , a still-open generalization of the Hadwiger conjecture 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
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