Portal:Mathematics

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The Mathematics Portal

Mathematics is the study of numbers, quantity, space, 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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The graph of a real-valued quadratic function of a real variable x, is a parabola.
Image credit: Enoch Lau

A quadratic equation is a polynomial equation of degree two. The general form is

ax^{2}+bx+c=0,\,\!

where a ≠ 0 (if a = 0, then the equation becomes a linear equation). The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x², the linear coefficient b is the coefficient of x, and c is the constant coefficient, also called the free term.

Quadratic equations are called quadratic because quadratus is Latin for "square"; in the leading term the variable is squared.

A quadratic equation has two (not necessarily distinct) solutions, which may be real or complex, given by the quadratic formula:

x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},

These solutions are roots of the corresponding quadratic function

f(x)=ax^{2}+bx+c.\,

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animation of the classic "butterfly-shaped" Lorenz attractor seen from three different perspectives

Credit: Wikimol

The Lorenz attractor is an iconic example of a strange attractor in chaos theory. This three-dimensional fractal structure, resembling a butterfly or figure eight, reflects the long-term behavior of a set of solutions to the Lorenz system, three differential equations used by mathematician and meteorologist Edward N. Lorenz as a simple description of fluid circulation in a shallow layer heated uniformly from below and cooled uniformly from above. Analysis of the system revealed that although the solutions are completely deterministic, they develop in complex, non-repeating patterns that are highly dependent on the exact values of the parameters and initial conditions. As stated by Lorenz in his 1963 paper Deterministic Nonperiodic Flow, "Two states differing by imperceptible amounts may eventually evolve into two considerably different states". He later coined the term "butterfly effect" to describe the phenomenon. The particular solution plotted in this animation is based on the parameter values used by Lorenz ( $σ = 10$ , $ρ = 28$ , and $β = 8/3$ ). Initially developed to describe atmospheric convection, the Lorenz equations also arise in simplified models for lasers, electrical generators and motors, and chemical reactions.

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Did you know...

...that it is impossible to trisect a general angle using only a ruler and a compass?
...that in a group of 23 people, there is a more than 50% chance that two people share a birthday?
...that statistical properties dictated by Benford's Law are used in auditing of financial accounts as one means of detecting fraud?
...the hyperbolic trigonometric functions of the natural logarithm can be represented by rational algebraic fractions?
... that economists blame market failures on non-convexity?
... that, according to the pizza theorem, a circular pizza that is sliced off-center into eight equal-angled wedges can still be divided equally between two people?
... that the clique problem of programming a computer to find complete subgraphs in an undirected graph was first studied as a way to find groups of people who all know each other in social networks?