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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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spiral figure representing both finite and transfinite ordinal numbers
spiral figure representing both finite and transfinite ordinal numbers
This spiral diagram represents all ordinal numbers less than ωω. The first (outermost) turn of the spiral represents the finite ordinal numbers, which are the regular counting numbers starting with zero. As the spiral completes its first turn (at the top of the diagram), the ordinal numbers approach infinity, or more precisely ω, the first transfinite ordinal number (identified with the set of all counting numbers, a "countably infinite" set, the cardinality of which corresponds to the first transfinite cardinal number, called 0). The ordinal numbers continue from this point in the second turn of the spiral with ω + 1, ω + 2, and so forth. (A special ordinal arithmetic is defined to give meaning to these expressions, since the + symbol here does not represent the addition of two real numbers.) Halfway through the second turn of the spiral (at the bottom) the numbers approach ω + ω, or ω · 2. The ordinal numbers continue with ω · 2 + 1 through ω · 2 + ω = ω · 3 (three-quarters of the way through the second turn, or at the "9 o'clock" position), then through ω · 4, and so forth, up to ω · ω = ω2 at the top. (As with addition, the multiplication and exponentiation operations have definitions that work with transfinite numbers.) The ordinals continue in the third turn of the spiral with ω2 + 1 through ω2 + ω, then through ω2 + ω2 = ω2 · 2, up to ω2 · ω = ω3 at the top of the third turn. Continuing in this way, the ordinals increase by one power of ω for each turn of the spiral, approaching ωω in the middle of the diagram, as the spiral makes a countably infinite number of turns. This process can actually continue (not shown in this diagram) through and , and so on, approaching the first epsilon number, ε0. Each of these ordinals is still countable, and therefore equal in cardinality to ω. After uncountably many of these transfinite ordinals, the first uncountable ordinal is reached, corresponding to only the second infinite cardinal, . The identification of this larger cardinality with the cardinality of the set of real numbers can neither be proved nor disproved within the standard version of axiomatic set theory called Zermelo–Fraenkel set theory, whether or not one also assumes the axiom of choice.

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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

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 known by that name 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:

If the discriminant , then the quadratic equation has two distinct real solutions; if , the equation has two real solutions which are equal; if , the equation has two complex solutions.

These solutions are roots of the corresponding quadratic function

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Topics in mathematics

General Foundations Number theory Discrete mathematics


Algebra Analysis Geometry and topology Applied mathematics
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