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Decagonal Number
A decagonal number is a figurate number that extends the concept of triangular and square numbers to the decagon (a ten-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the construction of decagonal numbers are not rotationally symmetrical. Specifically, the ''n''th decagonal numbers counts the number of dots in a pattern of ''n'' nested decagons, all sharing a common corner, where the ''i''th decagon in the pattern has sides made of ''i'' dots spaced one unit apart from each other. The ''n''-th decagonal number is given by the following formula : D_n = 4n^2 - 3n. The first few decagonal numbers are: : 0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326 The ''n''th decagonal number can also be calculated by adding the square of '' ... [...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 sequence of triangular numbers, starting with the 0th triangular number, is (This sequence is included in the On-Line Encyclopedia of Integer Sequences .) Formula The triangular numbers are given by the following explicit formulas: T_n= \sum_^n k = 1+2+3+ \dotsb +n = \frac = , where \textstyle is 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 first equation can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-square" arrangement of objects corresponding to the triangular numb ... [...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 numbers (other examples being Cube (algebra), cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decagon
In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. A self-intersecting ''regular decagon'' is known as a decagram. Regular decagon A '' regular decagon'' has all sides of equal length and each internal angle will always be equal to 144°. Its Schläfli symbol is and can also be constructed as a truncated pentagon, t, a quasiregular decagon alternating two types of edges. Side length The picture shows a regular decagon with side length a and radius R of the circumscribed circle. * The triangle E_E_1M has to equally long legs with length R and a base with length a * The circle around E_1 with radius a intersects ]M\,E_ _in_a_point_P_(not_designated_in_the_picture)._ *_Now_the_triangle_\;_is_a_isosceles_triangle.html" ;"title="/math> in a point P (not designated in the picture). * Now the triangle \; is a isosceles triang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
0 (number)
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usually by 10. As a number, 0 fulfills a central role in mathematics as the additive identity of the integers, real numbers, and other algebraic structures. Common names for the number 0 in English are ''zero'', ''nought'', ''naught'' (), ''nil''. In contexts where at least one adjacent digit distinguishes it from the letter O, the number is sometimes pronounced as ''oh'' or ''o'' (). Informal or slang terms for 0 include ''zilch'' and ''zip''. Historically, ''ought'', ''aught'' (), and ''cipher'', have also been used. Etymology The word ''zero'' came into the English language via French from the Italian , a contraction of the Venetian form of Italian via ''ṣafira'' or ''ṣifr''. In pre-Islamic time the word (Arabic ) had the meanin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 (number)
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
10 (number)
10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, by far the most common system of denoting numbers in both spoken and written language. It is the first double-digit number. The reason for the choice of ten is assumed to be that humans have ten fingers ( digits). Anthropology Usage and terms * A collection of ten items (most often ten years) is called a decade. * The ordinal adjective is ''decimal''; the distributive adjective is ''denary''. * Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten. * To reduce something by one tenth is to ''decimate''. (In ancient Rome, the killing of one in ten soldiers in a cohort was the punishment for cowardice or mutiny; or, one-tenth of the able-bodied men in a village as a form of retribution, thus causing a labor shortage and threat of starvation in agrarian societies.) Other * The number of kingdoms in Five Dyn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
27 (number)
27 (twenty-seven; Roman numeral XXVII) is the natural number following 26 and preceding 28. In mathematics * Twenty-seven is a cube of 3: 3^3=3\times 3\times 3. 27 is also 23 (see tetration). There are exactly 27 straight lines on a smooth cubic surface, which give a basis of the fundamental representation of the E6 Lie algebra. 27 is also a decagonal number. * In decimal, it is the first composite number not divisible by any of its digits. * It is the radix (base) of the septemvigesimal positional numeral system. * 27 is the only positive integer that is 3 times the sum of its digits. * In a prime reciprocal magic square of the multiples of , the magic constant is 27. * In the Collatz conjecture (aka the "3n+1 conjecture"), a starting value of 27 requires 111 steps to reach 1, more than any number smaller than it. * The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional. * In dec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
52 (number)
52 (fifty-two) is the natural number following 51 and preceding 53. In mathematics Fifty-two is * the 5th Bell number, the number of ways to partition a set of 5 objects. * a decagonal number. * an untouchable number, since it is never the sum of proper divisors of any number, and it is a noncototient since it is not equal to ''x'' − φ(''x'') for any ''x''. * a vertically symmetrical number. In science *The atomic number of tellurium Astronomy *Messier object M52, a magnitude 8.0 open cluster in the constellation Cassiopeia, also known as NGC 7654. *The New General Cataloguebr>objectNGC 52, a spiral galaxy in the constellation Pegasus. In other fields Fifty-two is: *The approximate number of weeks in a year. 52 weeks is 364 days, while the tropical year is 365.24 days long. According to ISO 8601, most years have 52 weeks while some have 53. *A significant number in the Maya calendar *On the modern piano, the number of white keys (notes in the C major scale) *The number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
85 (number)
85 (eighty-five) is the natural number following 84 and preceding 86. In mathematics 85 is: * the product of two prime numbers (5 and 17), and is therefore a semiprime; specifically, the 24th biprime not counting perfect squares. Together with 86 and 87, it forms the second cluster of three consecutive biprimes. * an octahedral number. * a centered triangular number. * a centered square number. * a decagonal number. * the smallest number that can be expressed as a sum of two squares, with all squares greater than 1, in two ways, 85 = 92 + 22 = 72 + 62. * the length of the hypotenuse of four Pythagorean triangles. * a Smith number in decimal. In astronomy * Messier object M85 is a magnitude 10.5 lenticular galaxy in the constellation Coma Berenices * NGC 85 is a galaxy in the constellation Andromeda * 85 Io is a large main belt asteroid * 85 Pegasi is a multiple star system in constellation of Pegasus * 85 Ceti is a variable star in the constellation of Cetus * 85D/Boethin i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
126 (number)
126 (one hundred ndtwenty-six) is the natural number following 125 and preceding 127. In mathematics As the binomial coefficient \tbinom, 126 is a central binomial coefficient and a pentatope number. It is also a decagonal number, a Harshad number and a pentagonal pyramidal number. As 125 + 1 it is σ3(5), the fifth value of the sum of cubed divisors function, and is a sum of two cubes. There are exactly 126 crossing points among the diagonals of a regular nonagon, 126 binary strings of length seven that are not repetitions of a shorter string, 126 different semigroups on four elements (up to isomorphism and reversal), and 126 different ways to partition a decagon into even polygons by diagonals. There are exactly 126 positive integers that are not solutions of the equation :x=abc+abd+acd+bcd, where ''a'', ''b'', ''c'', and ''d'' must themselves all be positive integers. It is the fifth Granville number, and the third such not to be a perfect number. Also, it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
175 (number)
175 (one hundred ndseventy-five) is the natural number following 174 and preceding 176. In mathematics Raising the decimal digits of 175 to the powers of successive integers produces 175 back again: 175 is a figurate number for a rhombic dodecahedron, the difference of two consecutive fourth powers: It is also a decagonal number and a decagonal pyramid number, the smallest number after 1 that has both properties. In other fields In the Book of Genesis 25:7-8, Abraham is said to have lived to be 175 years old. 175 is the fire emergency number in Lebanon Lebanon ( , ar, لُبْنَان, translit=lubnān, ), officially the Republic of Lebanon () or the Lebanese Republic, is a country in Western Asia. It is located between Syria to Lebanon–Syria border, the north and east and Israel to Blue .... See also * The year AD 175 or 175 BC * List of highways numbered 175 * References {{DEFAULTSORT:175 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4000 (number)
4000 (four thousand) is the natural number following 3000 (number)#3900 to 3999, 3999 and preceding 4001. It is a decagonal number. Selected numbers in the range 4001–4999 4001 to 4099 * 4005 – triangular number * 4007 – safe prime * 4010 – magic constant of ''n'' × ''n'' normal magic square and Eight queens puzzle, ''n''-queens problem for ''n'' = 20. * 4013 – balanced prime * 4019 – Sophie Germain prime * 4027 – super-prime * 4028 – sum of the first 45 primes * 4030 – third weird number * 4031 – sum of the cubes of the first six primes * 4032 – pronic number * 4033 – sixth super-Poulet number; strong pseudoprime in base 2 * 4060 – tetrahedral number * 4073 – Sophie Germain prime * 4079 – safe prime * 4091 – super-prime * 4092 – an occasional glitch in the game The Legend of Zelda: Ocarina of Time causes the Gossip Stones to say this number * 4095 – triangular number and odd abundant number; number of divisors in the sum of the fifth and lar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
