Portal:Mathematics

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The Mathematics Portal

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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Leonhard Euler
Image credit: Emanuel Handmann

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 function, and its notation, such as sin, cos, and tan for the trigonometric functions.

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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 $ω ω$ . 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 $ℵ 0$ ). 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 $ω \cdot 2$ . The ordinal numbers continue with $ω \cdot 2 + 1$ through $ω \cdot 2 + ω = ω \cdot 3$ (three-quarters of the way through the second turn, or at the "9 o'clock" position), then through $ω \cdot 4$ , and so forth, up to $ω \cdot ω = ω 2$ 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 $ω 2 + 1$ through $ω 2 + ω$ , then through $ω 2 + ω 2 = ω 2 \cdot 2$ , up to $ω 2 \cdot ω = ω 3$ 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 $ω ω$ 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 $\omega ^{\omega ^{\omega }}$ and $\omega ^{\omega ^{\omega ^{\omega }}}$ , and so on, approaching the first epsilon number, $ε 0$ . 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 $\aleph _{1}$ . 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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...that the orthocenter, circumcenter, centroid and the centre of the nine-point circle all lie on one line, the Euler line?
...that it has not been proven whether or not every even integer greater than two can be expressed as the sum of two primes?
...that the sum of the first n odd numbers divided by the sum of the next n odd numbers is always equal to one third?
...that i to the power of i, where i is the square root of -1, is a real number?
...an infinite, nonrepeating decimal can be represented using only the number 1 using continued fractions?
...that 253931039382791 and the following 18 prime numbers all end in the digit 1?