Wolfram Research Mathworld, Repository hosted at UIUC
(13.662 recursos)
Wolfram Research's MathWorld is a comprehensive and interactive mathematics encyclopedia intended for students, educators, math enthusiasts, and researchers. It has been assembled by internet encyclopedist Eric W.ÊWeisstein with assistance from the mathematics and internet communities. Like the discipline of mathematics, this site is continuously updated to include new material and incorporate new discoveries. This is a free service for the mathematical community provided by Wolfram Research, makers of Mathematica, with additional support from the National Science Foundation.
Mostrando recursos 1 - 20 de 1.584
1.
1 -- from MathWorld - Weisstein, Eric W.
The number one (1), also called "unity" is the first positive integer. It is an odd number. Although the number 1 used to be considered a prime number, it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own (Wells 1986, p. 31). The number 1 is sometimes also called "unity," so the nth roots of 1 are often called the nth roots of unity. Fractions having 1 as a...
2.
10 -- from MathWorld - Weisstein, Eric W.
The number 10 (ten) is the basis for the decimal system of notation. In this system, each "decimal place" consists of a digit 0-9 arranged such that each digit is multiplied by a power of 10, decreasing from left to right, and with a decimal place indicating the 10^0=1s place. For example, the number 1234.56 specifies 1\times 10^3+2\times 10^2+3\times 10^1+4\times 10^0+5\times 10^{-1}+6\times 10^{-2}. The decimal places to the left of the decimal point are 1, 10, 100, 1000, 10000,...
4.
12 -- from MathWorld - Weisstein, Eric W.
One dozen, or a twelfth of a gross. See also: Dozen, Gross
5.
13 -- from MathWorld - Weisstein, Eric W.
A number traditionally associated with bad luck. A so-called baker's dozen is equal to 13. Fear of the number 13 is called triskaidekaphobia. There are 13 Archimedean solids. Mazur and Tate (1973/74) proved that there is no elliptic curve over the rationals \mathbb{Q} having a rational point of order 13. See also: Baker's Dozen, Triskaidekaphobia
7.
144 -- from MathWorld - Weisstein, Eric W.
A dozen dozen, also called a gross. 144 is a square number and a sum-product number. See also: Dozen
8.
15 -- from MathWorld - Weisstein, Eric W.
See also: 15 Puzzle, Fifteen Theorem
9.
163 -- from MathWorld - Weisstein, Eric W.
The number 163 is very important in number theory, since d = 163 is the largest number such that the imaginary quadratic field \mathbb{Q}(\sqrt{-d}\,) has class number h(-d)=1. It also satisfies the curious identities 163 = \sum_{i=0}^4 {8\choose i} = {1\over 2}\left[{4^4+{8\choose 4}}\right] = {1\over 2}\left[{4^4+\sum_{i=0}^4{4\choose i}^2}\right], where {n\choose k} is a binomial coefficient (Stoschek). An approximation due to Stoschek is given by \pi\approx {2^9\over 163}={512\over...
10.
17 -- from MathWorld - Weisstein, Eric W.
17 is a Fermat prime, which means that the 17-sided regular polygon (the heptadecagon) is constructible using compass and straightedge (as proved by Gauss ). See also: 17-gon, Constructible Polygon, Fermat Prime, Heptadecagon, Thomson Cubic
11.
1729 -- from MathWorld - Weisstein, Eric W.
1729 is sometimes called the Hardy-Ramanujan number. It is the smallest taxicab number, i.e., the smallest number which can be expressed as the sum of two cubes in two different ways: 1729=1^3+12^3=9^3+10^3. See also: Hardy-Ramanujan Number, Taxicab Number
13.
196-Algorithm -- from MathWorld - Weisstein, Eric W.
Take any positive integer of two digits or more, reverse the digits, and add to the original number. This is the operation of the reverse-then-add sequence. Now repeat the procedure with the sum so obtained. This procedure quickly produces palindromic numbers for most integers. For example, starting with the number 5280 produces the sequence 5280, 6105, 11121, 23232. The end results of applying the algorithm to 1, 2, 3, ... are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99,...
14.
2 -- from MathWorld - Weisstein, Eric W.
The number two (2) is the second positive integer and the first prime number. It is even, and is the only even prime (the primes other than 2 are called the odd primes). The number 2 is also equal to its factorial since 2!=2. A quantity taken to the power 2 is said to be squared. The number of times k a given binary number b_n\cdots b_2 b_1 b_0 is divisible by 2 is given by the position of the first b_k=1, counting from the right. For example, 12 = 1100 is divisible by 2 twice, and 13 = 1101...
15.
2187 -- from MathWorld - Weisstein, Eric W.
The digits in the number 2187 form the two vampire numbers: 21\times 87=1827 and 2187=27\times 81. 2187 is also given by 37. See also: Vampire Number
16.
239 -- from MathWorld - Weisstein, Eric W.
Some interesting properties (as well as a few arcane ones not reiterated here) of the number 239 are discussed in Beeler et al. (1972, Item 63). 239 appears in Machin's formula {{1\over 4}}\pi = 4\tan^{-1}({{1\over 5}}) - \tan^{-1}({{1\over 239}}), which is related to the fact that 2 \cdot 13^4 - 1 = 239^2, which is why 239/169 is the 7th convergent of \sqrt{2}\,. Another pair of inverse tangent formulas involving 239 is \tan^{-1}({{1\over 239}}) = \tan^{-1}({{1\over 70}}) -...
17.
24 -- from MathWorld - Weisstein, Eric W.
The number 24 is equal to 4! (four factorial). A number puzzle asks to construct 24 in as many ways possible using elementary mathematical operations on three copies of the same digit. Example solutions include 24 = 2+\left\langle{2,2}\right\rangle{} = \left\langle{2,2^2}\right\rangle{} = \left\langle{2,2+2}\right\rangle{} = \left\langle{2,2\cdot 2}\right\rangle{} = 3\cdot 3!+3 = 4!+(4-4) = 4!^{4/4} = 4!{4\over 4} = 8+8+8, where the not necessarily elementary "operation" of digit...
18.
243 -- from MathWorld - Weisstein, Eric W.
Feynman (1997) noticed the curious fact that the decimal expansion {{1\over 243}}=0.004115226337448559\dots repeats pairs of the digits 0, 1, 2, 3, ... separated by the digits 4, 5, 6, 7, .... Just after this point, the pattern breaks, since the fraction is given exactly by the repeating decimal {{1\over 243}}=0.\overline{004115226337448559670781893}. This pattern is related to the fact that {{1\over 9}}=0.\overline{1} and {{1\over 81}}=0.\overline{012345679}.
19.
3 -- from MathWorld - Weisstein, Eric W.
3 is the only integer which is the sum of the preceding positive integers (1 + 2 = 3) and the only number which is the sum of the factorials of the preceding positive integers (1!+2!=3). It is also the first odd prime. A quantity taken to the power 3 is said to be cubed. The sequence 1, 31, 331, 3331, 33331, ... (Sloane's A033175) consisting of n = 0, 1, ... 3s followed by a 1 has its nth term is given by a(n)={10^{n+1}-7\over 3}. The result is prime for n = 1, 2, 3, 4, 5, 6, 7,...
20.
4 -- from MathWorld - Weisstein, Eric W.
The smallest positive composite number and the first even perfect square. Four is the smallest even number appearing in a Pythagorean triple: 3, 4, 5. In the numerology of the Pythagorean school, it was the number of justice. The sacred tetraktýs (10) was the sum of the first four numbers, depicted as a triangle with two equal sides of length 4. 4 is the highest degree for which an algebraic equation is always solvable by radicals. It is the smallest order of a field which is not a prime...
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