Portal:Mathematics

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Mathematics is the study of numbers, quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. However, mathematical proofs are less formal and painstaking than proofs in mathematical logic. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions.

Through the use of abstraction and logical reasoning, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.

Galileo Galilei (1564–1642) said, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth". Carl Friedrich Gauss (1777–1855) referred to mathematics as "the queen of sciences". The mathematician Benjamin Peirce (1809–1880) called the discipline, "the science that draws necessary conclusions". David Hilbert said of it: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise." Albert Einstein (1879–1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality".

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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Image credit: User:Melchoir

The real number denoted by the recurring decimal 0.999… is exactly equal to 1. In other words, "0.999…" represents the same number as the symbol "1". Various proofs of this identity have been formulated with varying rigour, preferred development of the real numbers, background assumptions, historical context, and target audience.

The equality has long been taught in textbooks, and in the last few decades, researchers of mathematics education have studied the reception of this equation among students, who often reject the equality. The students' reasoning is typically based on one of a few common erroneous intuitions about the real numbers; for example, a belief that each unique decimal expansion must correspond to a unique number, an expectation that infinitesimal quantities should exist, that arithmetic may be broken, an inability to understand limits or simply the belief that 0.999… should have a last 9. These ideas are false with respect to the real numbers, which can be proven by explicitly constructing the reals from the rational numbers, and such constructions can also prove that 0.999… = 1 directly.

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spiral figure representing both finite and transfinite ordinal numbers
Credit: Pop-up casket & Fool

This spiral diagram represents all ordinal numbers less than \omega^\omega. 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 \omega, 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 \aleph_0). The ordinal numbers continue from this point in the second turn of the spiral with \omega+1, \omega+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 \omega+\omega, or \omega\cdot2. The ordinal numbers continue with \omega\cdot2+1 through \omega\cdot2+\omega=\omega\cdot3 (three-quarters of the way through the second turn, or at the "9 o'clock" position), then through \omega\cdot4, and so forth, up to \omega\cdot\omega=\omega^2 at the top. (As with addition, the multiplication and exponentiation operations have definitions that work with transfinite numbers.) As one would expect, the ordinals continue in the third turn of the spiral with \omega^2+1 through \omega^2+\omega, then through \omega^2+\omega^2=\omega^2\cdot2, up to \omega^2\cdot\omega=\omega^3 at the top of the third turn. Continuing in this way, the ordinals increase by one power of \omega for each turn of the spiral, approaching \omega^\omega in the middle of the diagram, as the spiral makes a countably infinite number of turns. This process can actually continue through \omega^{\omega^\omega} and \omega^{\omega^{\omega^\omega}}, and so on, approaching the first uncountable ordinal number, which (assuming the axiom of choice) corresponds to only the second transfinite cardinal number, \aleph_1, the cardinality (according to the continuum hypothesis) of the set of real numbers.

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