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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Diophantus-II-8-Fermat.jpg
Problem II.8 in the Arithmetica by Diophantus, annotated with Fermat's comment, which became Fermat's Last Theorem
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Fermat's Last Theorem is one of the most famous theorems in the history of mathematics. It states that:

has no solutions in non-zero integers , , and when is an integer greater than 2.

Despite how closely the problem is related to the Pythagorean theorem, which has infinite solutions and hundreds of proofs, Fermat's subtle variation is much more difficult to prove. Still, the problem itself is easily understood even by schoolchildren, making it all the more frustrating and generating perhaps more incorrect proofs than any other problem in the history of mathematics.

The 17th-century mathematician Pierre de Fermat wrote in 1637 in his copy of Bachet's translation of the famous Arithmetica of Diophantus: "I have a truly marvelous proof of this proposition which this margin is too narrow to contain." However, no correct proof was found for 357 years, until it was finally proven using very deep methods by Andrew Wiles in 1995 (after a failed attempt a year before).

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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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