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Non-integer Representation
A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix ''β'' > 1, the value of :x = d_n \dots d_2d_1d_0.d_d_\dots d_ is :\begin x &= \beta^nd_n + \cdots + \beta^2d_2 + \beta d_1 + d_0 \\ &\qquad + \beta^d_ + \beta^d_ + \cdots + \beta^d_. \end The numbers ''d''''i'' are non-negative integers less than ''β''. This is also known as a ''β''-expansion, a notion introduced by and first studied in detail by . Every real number has at least one (possibly infinite) ''β''-expansion. The set of all ''β''-expansions that have a finite representation is a subset of the ring Z 'β'', ''β''−1 There are applications of ''β''-expansions in coding theory and models of quasicrystals. Construction ''β''-expansions are a generalization of decimal expansions. While infinite decimal expansions are not unique (for example, 1.000... = 0.999...), all finite decimal expansions are unique. However, even fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of 2
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Technically, it should be called the ''principal'' square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a Unit square, square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational number, irrational. The fraction (≈ 1.4142857) is sometimes used as a good Diophantine approximation, rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 60 decimal places: : History The Babylonian clay tablet YBC 7289 (–1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a Disk (mathematics), disc. The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Terminology * Annulus (mathematics), Annulus: a ring-shaped object, the region bounded by two concentric circles. * Circular arc, Arc: any Connected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions are also valid for the diameter of a sphere. In more modern usage, the length d of a diameter is also called the diameter. In this sense one speaks of diameter rather than diameter (which refers to the line segment itself), because all diameters of a circle or sphere have the same length, this being twice the radius r. :d = 2r \qquad\text\qquad r = \frac. The word "diameter" is derived from (), "diameter of a circle", from (), "across, through" and (), "measure". It is often abbreviated \text, \text, d, or \varnothing. Constructions With straightedge and compass, a diameter of a given circle can be constructed as the perpendicular bisector of an arbitrary chord. Drawing two diameters in this way can be used to locate the center of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radix Economy
In mathematics and computer science, optimal radix choice is the problem of choosing the base, or radix, that is best suited for representing numbers. Various proposals have been made to quantify the relative costs of using different radices in representing numbers, especially in computer systems. One formula is the number of digits needed to express it in that base, multiplied by the base (the number of possible values each digit could have). This expression also arises in questions regarding organizational structure, networking, and other fields. Definition The cost of representing a number ''N'' in a given base ''b'' can be defined as : E(b,N) = b \lfloor \log_b (N) +1 \rfloor \, where we use the floor function \lfloor \rfloor and the base-b logarithm \log_. If both ''b'' and ''N'' are positive integers, then the quantity E(b,N) is equal to the number of digits needed to express the number ''N'' in base ''b'', multiplied by base ''b''. This quantity thus measures the cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Common Logarithm
In mathematics, the common logarithm (aka "standard logarithm") is the logarithm with base 10. It is also known as the decadic logarithm, the decimal logarithm and the Briggsian logarithm. The name "Briggsian logarithm" is in honor of the British mathematician Henry Briggs who conceived of and developed the values for the "common logarithm". Historically', the "common logarithm" was known by its Latin name ''logarithmus decimalis'' or ''logarithmus decadis''. The mathematical notation for using the common logarithm is , , or sometimes with a capital ; on calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base ≈ 2.71828) rather than common logarithm when writing "log". Before the early 1970s, handheld electronic calculators were not available, and mechanical calculators capable of multiplication were bulky, expensive and not widely available. Instead, tables of base-10 logarithms were used in science, engineering and navi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply . Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the exponentiation, power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the Integral, area under the curve from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E (mathematical Constant)
The number is a mathematical constant approximately equal to 2.71828 that is the base of a logarithm, base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted \gamma. Alternatively, can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest. The number is of great importance in mathematics, alongside 0, 1, Pi, , and . All five appear in one formulation of Euler's identity e^+1=0 and play important and recurring roles across mathematics. Like the constant , is Irrational number, irrational, meaning that it cannot be represented as a ratio of integers, and moreover it is Transcendental number, transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supergolden Ratio
In mathematics, the supergolden ratio is a geometrical aspect ratio, proportion, given by the unique real polynomial root, solution of the equation Its decimal expansion begins with . The name ''supergolden ratio'' is by analogy with the golden ratio, the positive solution of the equation Definition Three quantities are in the supergolden ratio if \frac =\frac =\frac The ratio is commonly denoted Substituting b=\psi c \, and a=\psi b =\psi^2 c \, in the middle fraction, \psi =\frac. It follows that the supergolden ratio is the unique real solution of the cubic equation \psi^3 -\psi^2 -1 =0. The Minimal polynomial (field theory), minimal polynomial for the reciprocal root is the depressed cubic x^ +x -1, thus the simplest solution with Cubic equation#Cardano's formula, Cardano's formula, \begin w_ &=\left( 1 \pm \frac \sqrt \right) /2 \\ 1 /\psi &=\sqrt[3] +\sqrt[3] \end or, using the Cubic equation#Trigonometric and hyperbolic solutions, hyperbolic sine, : 1 /\p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Octagon
In geometry, an octagon () is an eight-sided polygon or 8-gon. A ''regular polygon, regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular Truncation (geometry), truncated square, t, which alternates two types of edges. A truncated octagon, t is a hexadecagon, . A 3D analog of the octagon can be the rhombicuboctahedron with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square. Properties The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal quadrilateral, equidiagonal and orthodiagonal quadrilateral, orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other).Dao Thanh Oai (2015), "Equila ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Silver Ratio
In mathematics, the silver ratio is a geometrical aspect ratio, proportion with exact value the positive polynomial root, solution of the equation The name ''silver ratio'' results from analogy with the golden ratio, the positive solution of the equation Although its name is recent, the silver ratio (or silver mean) has been studied since ancient times because of its connections to the square root of 2, almost-isosceles Pythagorean triple#Special cases and related equations, Pythagorean triples, square triangular numbers, Pell numbers, the octagon, and six polyhedron, polyhedra with octahedral symmetry. Definition If the ratio of two quantities is proportionate to the sum of two and their reciprocal ratio, they are in the silver ratio: \frac =\frac The ratio \frac is here denoted Substituting a=\sigma b \, in the second fraction, \sigma =\frac. It follows that the silver ratio is the positive solution of quadratic equation \sigma^2 -2\sigma -1 =0. The quadratic for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek διαγώνιος ''diagonios'', "from corner to corner" (from διά- ''dia-'', "through", "across" and γωνία ''gonia'', "corner", related to ''gony'' "knee"); it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as ''diagonus'' ("slanting line"). Polygons As applied to a polygon, a diagonal is a line segment joining any two non-consecutive vertices. Therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. For any convex polygon, all the diagonals are inside the polygon, but for re-entrant polygons, some diagonals are outside of the polygon. Any ''n''-sided polygon (''n'' ≥ 3), convex or concave, has \tfrac ''total'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
