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Star Number
In mathematics, a star number is a centered figurate number, a centered hexagram (six-pointed star), such as the Star of David, or the board Chinese checkers is played on. The ''n''th star number is given by the formula ''Sn'' = 6''n''(''n'' − 1) + 1. The first 45 star numbers are 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361, 9841, 10333, 10837, 11353, and 11881. The digital root of a star number is always 1 or 4, and progresses in the sequence 1, 4, 1. The last two digits of a star number in base 10 are always 01, 13, 21, 33, 37, 41, 53, 61, 73, 81, or 93. Unique among the star numbers is 35113, since its prime factors (i.e., 13, 37 and 73) are also consecutive star numbers. Relationships to other kinds of numbers Geometrically, the ''n''th star number is made up of a central point and 12 copi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including Guillaume de l'Hôpital, l'Hôpital and Johann Bernoulli, Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or Magnitude (mathematics), magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Hexagonal Number
In mathematics and combinatorics, a centered hexagonal number, or centered hexagon number, is a centered polygonal number, centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. The following figures illustrate this arrangement for the first four centered hexagonal numbers: : Centered hexagonal numbers should not be confused with hexagonal number, cornered hexagonal numbers, which are figurate numbers in which the associated hexagons share a vertex. The sequence of hexagonal numbers starts out as follows : :1, 7, 19 (number), 19, 37 (number), 37, 61 (number), 61, 91 (number), 91, 127 (number), 127, 169 (number), 169, 217 (number), 217, 271 (number), 271, 331 (number), 331, 397 (number), 397, 469, 547, 631, 721, 817, 919. Formula The th centered hexagonal number is given by the formula :H(n) = n^3 - (n-1)^3 = 3n(n-1)+1 = 3n^2 - 3n +1. \, Expressing the formula as :H(n) = 1+6\left(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Series
In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation this is expressed \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms converge to 0 monotonically, but this condition is not necessary for convergence. Examples The geometric series − + − + ⋯ sums to . The alternating harmonic series has a finite sum but the harmonic series does not. The series 1-\frac+\frac-\ldots=\sum_^\infty\frac converges to \frac, but is not absolutely convergent. The Mercator series provides an analytic power series expression of the natural logarithm, given by \sum_^\infty \frac x^n = \ln (1+x),\;\;\;, x, \le1, x\ne-1. The functions si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Fraction
A unit fraction is a positive fraction with one as its numerator, 1/. It is the multiplicative inverse (reciprocal) of the denominator of the fraction, which must be a positive natural number. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. When an object is divided into equal parts, each part is a unit fraction of the whole. Multiplying two unit fractions produces another unit fraction, but other arithmetic operations do not preserve unit fractions. In modular arithmetic, unit fractions can be converted into equivalent whole numbers, allowing modular division to be transformed into multiplication. Every rational number can be represented as a sum of distinct unit fractions; these representations are called Egyptian fractions based on their use in ancient Egyptian mathematics. Many infinite sums of unit fractions are meaningful mathematically. In geometry, unit fractions can be used to characterize the curvature of triangle groups and the tangencies of Ford circles. Unit fractions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Series (mathematics)
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \cdots. The first n terms of the series sum to approximately \ln n + \gamma, where \ln is the natural logarithm and \gamma\approx0.577 is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series. Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an integral, according to the integral test for convergence. Applications of the harmonic series and its partial sums include Divergence of the sum of the reciprocals of the primes, Euler's proof that there are infinitely many prime numbers, the analysis of the coupon collector's problem on how many random trials are nee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octagonal Number
In mathematics, an octagonal number is a figurate number. The ''n''th octagonal number ''o''''n'' is the number of dots in a pattern of dots consisting of the outlines of regular octagons with sides up to ''n'' dots, when the octagons are overlaid so that they share one vertex (geometry), vertex. The octagonal number for ''n'' is given by the formula 3''n''2 − 2''n'', with ''n'' > 0. The first few octagonal numbers are : 1 (number), 1, 8 (number), 8, 21 (number), 21, 40 (number), 40, 65 (number), 65, 96 (number), 96, 133 (number), 133, 176 (number), 176, 225 (number), 225, 280 (number), 280, 341, 408, 481, 560, 645, 736, 833, 936 The octagonal number for ''n'' can also be calculated by adding the square of ''n'' to twice the (''n'' − 1)th pronic number. Octagonal numbers consistently alternate parity (mathematics), parity. Octagonal numbers are occasionally referred to as "star numbers", though that term is more commonly used to refer to centered dodecagonal numbers. Appl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of Figurate number, figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). In the Real number, real number system, square numbers are non-negative. A non-negative integer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Polygonal Number
In mathematics, the centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side in the previous layer; so starting from the second polygonal layer, each layer of a centered ''k''-gonal number contains ''k'' more dots than the previous layer. Examples Each centered ''k''-gonal number in the series is ''k'' times the previous triangular number, plus 1. This can be formalized by the expression \frac +1, where ''n'' is the series rank, starting with 0 for the initial 1. For example, each centered square number in the series is four times the previous triangular number, plus 1. This can be formalized by the expression \frac +1. These series consist of the * centered triangular numbers 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, ... (), * centered square numbers 1, 5, 13, 25, 41, 61, 85, 113, 145, 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The first 100 terms sequence of triangular numbers, starting with the 0th triangular number, are Formula The triangular numbers are given by the following explicit formulas: where \textstyle is notation for a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The fact that the nth triangular number equals n(n+1)/2 can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Number Visual Proof
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sky, night; their immense distances from Earth make them appear as fixed stars, fixed points of light. The most prominent stars have been categorised into constellations and asterism (astronomy), asterisms, and many of the brightest stars have proper names. Astronomers have assembled star catalogues that identify the known stars and provide standardized stellar designations. The observable universe contains an estimated to stars. Only about 4,000 of these stars are visible to the naked eye—all within the Milky Way galaxy. A star's life star formation, begins with the gravitational collapse of a gaseous nebula of material largely comprising hydrogen, helium, and traces of heavier elements. Its stellar mass, total mass mainly determines it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
