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Warn moderators| Author : | Topic: notation | Bottom |
| saucer admin Posts : 673 A Good Tautology is Hard to Find!  |
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| very very small tic Posts : 70  |
++ .....Before introducing the integral symbol, Leibniz wrote omn. for "omnia" in front of the term to be integrated. The integral symbol was first used by Gottfried Wilhelm Leibniz (1646-1716) on October 29, 1675, in an unpublished manuscrïpt. Several weeks later, on Nov. 21, he first placed dx after the integral symbol (Burton, page 359). Later in 1675, he proposed the use of the symbol in a letter to Henry Oldenburg, secretary of the Royal Society: "Utile erit scribi pro omnia, ut l = omn. l, id est summa ipsorum l" [It will be useful to write for omn. so that l = omn. l, or the sum of all the l's.] The first appearance of the integral symbol in print was in a paper by Leibniz in the Acta Eruditorum. The integral symbol was actually a long letter S for "summa." In his Quadratura curvarum of 1704, Newton wrote a small vertical bar above x to indicate the integral of x. He wrote two side-by-side vertical bars over x to indicate the integral of (x with a single bar over it). Another notation he used was to enclose the term in a rectangle to indicate its integral. Cajori writes that Newton's symbolism for integration was defective because the x with a bar could be misinterpreted as x-prime and the placement of a rectangle about the term was difficult for the printer, and that therefore Newton's symbolism was never popular, even in England.. http://members.aol.com/jeff570/mathsym.html very very small tic |
| 360pan gold Posts : 14  |
Number theory/ fractions Number theory is one of the oldest branches of pure mathematics, and one of the largest. Of course, it concerns questions about numbers, usually meaning whole numbers or rational numbers (fractions). Elementary number theory involves divisibility among integers -- the division "algorithm", the Euclidean algorithm (and thus the existence of greatest common divisors), elementary properties of primes (the unique factorization theorem, the infinitude of primes), congruences (and the structure of the sets Z/nZ as commutative rings), including Fermat's little theorem and Euler's theorem extending it. But the term "elementary" is usually used in this setting only to mean that no advanced tools from other areas are used -- not that the results themselves are simple. Indeed, a course in "elementary" number theory usually includes classic and elegant results such as Quadratic Reciprocity; counting results using the Möbius Inversion Formula (and other multiplicative number-theoretic functions); and even the Prime Number Theorem, asserting the approximate density of primes among the integers, which has difficult but "elementary" proofs. Other topics in elementary number theory -- the solutions of sets of linear congruence equations (the Chinese Remainder Theorem), or solutions of single binary quadratic equations (Pell's equations and continued fractions), or the generation of Fibonacci numbers or Pythagorean triples -- turn out in retrospect to be harbingers of sophisticated tools and themes in other areas. 360 |
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