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Even And Odd Numbers
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity Of 5 And 6 Cuisenaire Rods
Parity may refer to: * Parity (computing) ** Parity bit in computing, sets the parity of data for the purpose of error detection ** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the result of the last operation ** Parity file in data processing, created in conjunction with data files and used to check data integrity and assist in data recovery * Parity (mathematics), indicates whether a number is even or odd ** Parity of a permutation, indicates whether a permutation has an even or odd number of inversions ** Parity function, a Boolean function whose value is 1 if the input vector has an odd number of ones ** Parity learning, a problem in machine learning ** Parity of even and odd functions * Parity (physics), a symmetry property of physical quantities or processes under spatial inversion * Parity (biology), the number of times a female has given birth; gravidity and parity represent pregnancy and viability, resp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number ( RSA-240) utilizing approximately 900 core-years of computing power. The researchers estimated that a 1024-bit R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Definition and first examples Definition A ''ring'' is a set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even And Odd Ordinals
In mathematics, even and odd ordinals extend the concept of parity from the natural numbers to the ordinal numbers. They are useful in some transfinite induction proofs. The literature contains a few equivalent definitions of the parity of an ordinal α: *Every limit ordinal (including 0) is even. The successor of an even ordinal is odd, and vice versa. *Let α = λ + ''n'', where λ is a limit ordinal and ''n'' is a natural number. The parity of α is the parity of ''n''. *Let ''n'' be the finite term of the Cantor normal form of α. The parity of α is the parity of ''n''. *Let α = ωβ + ''n'', where ''n'' is a natural number. The parity of α is the parity of ''n''. *If α = 2β, then α is even. Otherwise α = 2β + 1 and α is odd. Unlike the case of even integers, one cannot go on to characterize even ordinals as ordinal numbers of the form Ordinal multiplication In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutilated Chessboard Problem
The mutilated chessboard problem is a tiling puzzle posed by Max Black in 1946 that asks: Suppose a standard 8×8 chessboard (or checkerboard) has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2×1 so as to cover all of these squares? It is an impossible puzzle: there is no domino tiling meeting these conditions. One proof of its impossibility uses the fact that, with the corners removed, the chessboard has 32 squares of one color and 30 of the other, but each domino must cover equally many squares of each color. More generally, if any two squares are removed from the chessboard, the rest can be tiled by dominoes if and only if the removed squares are of different colors. This problem has been used as a test case for automated reasoning, creativity, and the philosophy of mathematics. History The mutilated chessboard problem is an instance of domino tiling of grids and polyominoes, also known as "dimer models" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knight (chess)
The knight (♘, ♞) is a piece in the game of chess, represented by a horse's head and neck. It moves two squares vertically and one square horizontally, or two squares horizontally and one square vertically, jumping over other pieces. Each player starts the game with two knights on the b- and g-, each located between a rook and a bishop. Movement Compared to other chess pieces, the knight's movement is unique: it moves two squares vertically and one square horizontally, or two squares horizontally and one square vertically (with both forming the shape of a capital L). When moving, the knight can jump over pieces to reach its destination. Knights capture in the same way, replacing the enemy piece on the square and removing it from the board. A knight can have up to eight available moves at once. Knights and pawns are the only pieces that can be moved in the chess starting position. Value Knights and bishops, also known as , have a value of about three pawns. Bishops util ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bishop (chess)
The bishop (♗, ♝) is a piece in the game of chess. It moves and captures along without jumping over intervening pieces. Each player begins the game with two bishops. One starts between the and the king, the other between the and the queen. The starting squares are c1 and f1 for White's bishops, and c8 and f8 for Black's bishops. Placement and movement The king's bishop is placed between the king and the king's knight, f1 for White and f8 for Black; the queen's bishop is placed between the queen and the queen's knight, c1 for White and c8 for Black. The bishop has no restrictions in distance for each move but is limited to diagonal movement. It cannot jump over other pieces. A bishop captures by occupying the square on which an enemy piece stands. As a consequence of its diagonal movement, each bishop always remains on one square color. Due to this, it is common to refer to a bishop as a light-squared or dark-squared bishop. Comparison – other pieces Versus rook A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chess
Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to distinguish it from related games, such as xiangqi (Chinese chess) and shogi (Japanese chess). The recorded history of chess goes back at least to the emergence of a similar game, chaturanga, in seventh-century India. The rules of chess as we know them today emerged in Europe at the end of the 15th century, with standardization and universal acceptance by the end of the 19th century. Today, chess is one of the world's most popular games, played by millions of people worldwide. Chess is an abstract strategy game that involves no hidden information and no use of dice or cards. It is played on a chessboard with 64 squares arranged in an eight-by-eight grid. At the start, each player controls sixteen pieces: one king, one queen, two rooks, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice (group)
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension n which spans the vector space \mathbb^n. For any basis of \mathbb^n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a regu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Crystal System
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of these crystals: *Primitive cubic (abbreviated ''cP'' and alternatively called simple cubic) *Body-centered cubic (abbreviated ''cI'' or bcc) *Face-centered cubic (abbreviated ''cF'' or fcc, and alternatively called ''cubic close-packed'' or ccp) Each is subdivided into other variants listed below. Although the ''unit cells'' in these crystals are conventionally taken to be cubes, the primitive unit cells often are not. Bravais lattices The three Bravais lattices in the cubic crystal system are: The primitive cubic lattice (cP) consists of one lattice point on each corner of the cube; this means each simple cubic unit cell has in total one lattice point. Each atom at a lattice point is then shared equally between eight adjacent cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the '' Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knight (chess)
The knight (♘, ♞) is a piece in the game of chess, represented by a horse's head and neck. It moves two squares vertically and one square horizontally, or two squares horizontally and one square vertically, jumping over other pieces. Each player starts the game with two knights on the b- and g-, each located between a rook and a bishop. Movement Compared to other chess pieces, the knight's movement is unique: it moves two squares vertically and one square horizontally, or two squares horizontally and one square vertically (with both forming the shape of a capital L). When moving, the knight can jump over pieces to reach its destination. Knights capture in the same way, replacing the enemy piece on the square and removing it from the board. A knight can have up to eight available moves at once. Knights and pawns are the only pieces that can be moved in the chess starting position. Value Knights and bishops, also known as , have a value of about three pawns. Bishops util ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
