In mathematics, a negative number is a real number, less than zero. Negative numbers represent opposites. If positive represents a movement to the right, negative represents a movement to the left. If positive represents above sea level negative represents below sea level. If positive represents a deposit, negative represents a withdrawal, they are used to represent the magnitude of a loss or deficiency. A debt, owed may be thought of as a negative asset, a decrease in some quantity may be thought of as a negative increase. If a quantity may have either of two opposite senses one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature; the laws of arithmetic for negative numbers ensure that the common sense idea of an opposite is reflected in arithmetic.
For example, − = 3 because the opposite of an opposite is the original value. Negative numbers are written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, is pronounced "minus three" or "negative three". To help tell the difference between a subtraction operation and a negative number the negative sign is placed higher than the minus sign. Conversely, a number, greater than zero is called positive; the positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign; every real number other than zero is either negative. The positive whole numbers are referred to as natural numbers, while the positive and negative whole numbers are referred to as integers. In bookkeeping, amounts owed are represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers. Negative numbers appeared for the first time in history in the Nine Chapters on the Mathematical Art, which in its present form dates from the period of the Chinese Han Dynasty, but may well contain much older material.
Liu Hui established rules for subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of negative numbers around the middle of the 19th century. Prior to the concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd; some mathematicians like Leibniz agreed that negative numbers were false, but still used them in calculations. Negative numbers can be thought of as resulting from the subtraction of a larger number from a smaller. For example, negative three is the result of subtracting three from zero: 0 − 3 = −3. In general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers.
For example, 5 − 8 = −3since 8 − 5 = 3. The relationship between negative numbers, positive numbers, zero is expressed in the form of a number line: Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are less, thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left. Note that a negative number with greater magnitude is considered less. For example though 8 is greater than 5, written 8 > 5negative 8 is considered to be less than negative 5: −8 < −5. It follows that any negative number is less than any positive number, so −8 < 5 and −5 < 8. In the context of negative numbers, a number, greater than zero is referred to as positive, thus every real number other than zero is either positive or negative, while zero itself is not considered to have a sign. Positive numbers are sometimes written with a plus sign in front, e.g. +3 denotes a positive three. Because zero is neither positive nor negative, the term nonnegative is sometimes used to refer to a number, either positive or zero, while nonpositive is used to refer to a number, either negative or zero.
Zero is a neutral number. Goal difference in association football and hockey. Plus-minus differential in ice hockey: the difference in total goals scored for the team and against the team when a particular player is on the ice is the player’s +/− rating. Players can have a negative rating. Run differential in baseball: the run differential is negative if the team allows more runs than they scored. British football clubs are deducted points if they enter administration, thus have a negative points total until they have earned at least that many points that season. Lap times in Formula 1 may be given as the difference compared to a previous lap, will be positive if slower and negative if faster. In some athletics events, such as sprint races, the hurdles, the triple jump and the long jump, the wind assistance is measured and recorde
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular convex dodecahedron and icosahedron, differ from these in having regular pentagrammic faces or vertex figures, they can all be seen as three-dimensional analogues of the pentagram in another. These figures have pentagrams as faces or vertex figures; the small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex polygonal faces, but pentagrammic vertex figures. In all cases, two faces can intersect along a line, not an edge of either face, so that part of each face passes through the interior of the figure; such lines of intersection are not part of the polyhedral structure and are sometimes called false edges. Where three such lines intersect at a point, not a corner of any face, these points are false vertices; the images below show spheres at the true vertices, blue rods along the true edges.
For example, the small stellated dodecahedron has 12 pentagram faces with the central pentagonal part hidden inside the solid. The visible parts of each face comprise five isosceles triangles which touch at five points around the pentagon. We could treat these triangles as 60 separate faces to obtain a new, irregular polyhedron which looks outwardly identical; each edge would now be divided into three shorter edges, the 20 false vertices would become true ones, so that we have a total of 32 vertices. The hidden inner pentagons are no longer part of the polyhedral surface, can disappear. Now Euler's formula holds: 60 − 90 + 32 = 2; however this polyhedron is no longer the one described by the Schläfli symbol, so can not be a Kepler–Poinsot solid though it still looks like one from outside. A Kepler–Poinsot polyhedron covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the figures which have pentagrammic faces, the vertices in the others; because of this, they are not topologically equivalent to the sphere as Platonic solids are, in particular the Euler relation χ = V − E + F = 2 does not always hold.
Schläfli held that all polyhedra must have χ = 2, he rejected the small stellated dodecahedron and great dodecahedron as proper polyhedra. This view was never held. A modified form of Euler's formula, using density of the vertex figures and faces was given by Arthur Cayley, holds both for convex polyhedra, the Kepler–Poinsot polyhedra: d v V − E + d f F = 2 D; the Kepler–Poinsot polyhedra exist in dual pairs. Duals have the same Petrie polygon, or more Petrie polygons with the same two dimensional projection; the following images show the two dual compounds with the same edge radius. They show that the Petrie polygons are skew. Two relationships described in the article below are easily seen in the images: That the violet edges are the same, that the green faces lie in the same planes. John Conway defines the Kepler–Poinsot polyhedra as greatenings and stellations of the convex solids. In his naming convention the small stellated dodecahedron is just the stellated dodecahedron. Stellation changes pentagonal faces into pentagrams.
Greatening maintains the type of faces and resizing them into parallel planes. The great icosahedron is one of the stellations of the icosahedron; the three others are all the stellations of the dodecahedron. The great stellated dodecahedron is a faceting of the dodecahedron; the three others are facetings of the icosahedron. If the intersections are treated as new edges and vertices, the figures obtained will not be regular, but they can still be considered stellations; the great stellated dodecahedron shares its vertices with the dodecahedron. The other three Kepler–Poinsot polyhedra share theirs with the icosahedron; the skeletons of the solids sharing vertices are topologically equivalent. The small and great stellated dodecahedron can be seen as a regular and a great dodecahedron with their edges and faces extended until they intersect; the pentagon faces of these cores are the invisible parts of the star polyhedra's pentagram faces. For the small stellated dodecahedron the hull is φ times bigger than the core, for the great it is φ + 1 = φ 2 times bigger.
Traditionally the two star polyhedra have been defined as augmentations, i.e. as dodecahedron and icosahedron with pyramids added to their faces. Kepler calls the small stellation an augmented dodecahedron. In his view the great stellation is related to the icosahedron as the small one is to the dodecahedron; these naïve definitions are still used. E.g. MathWorld states that the two star polyhedra can be constructed by adding pyramids to the faces of the Platonic solids; this is just a help to visualize the shape of these solids, not a claim that the edg
In elementary geometry, a polygon is a plane figure, described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon; the segments of a polygonal circuit are called its edges or sides, the points where two edges meet are the polygon's vertices or corners. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides. A simple polygon is one. Mathematicians are concerned only with the bounding polygonal chains of simple polygons and they define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes; the word polygon derives from the Greek adjective πολύς "much", "many" and γωνία "corner" or "angle".
It has been suggested. Polygons are classified by the number of sides. See the table below. Polygons may be characterized by their convexity or type of non-convexity: Convex: any line drawn through the polygon meets its boundary twice; as a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple: the boundary of the polygon does not cross itself. All convex polygons are simple. Concave. Non-convex and simple. There is at least one interior angle greater than 180°. Star-shaped: the whole interior is visible from at least one point, without crossing any edge; the polygon must be simple, may be convex or concave. All convex polygons are star-shaped. Self-intersecting: the boundary of the polygon crosses itself.
The term complex is sometimes used in contrast to simple, but this usage risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions. Star polygon: a polygon which self-intersects in a regular way. A polygon can not be both star-shaped. Equiangular: all corner angles are equal. Cyclic: all corners lie on a single circle, called the circumcircle. Isogonal or vertex-transitive: all corners lie within the same symmetry orbit; the polygon is cyclic and equiangular. Equilateral: all edges are of the same length; the polygon need not be convex. Tangential: all sides are tangent to an inscribed circle. Isotoxal or edge-transitive: all sides lie within the same symmetry orbit; the polygon is equilateral and tangential. Regular: the polygon is both isogonal and isotoxal. Equivalently, it is both equilateral, or both equilateral and equiangular. A non-convex regular polygon is called a regular star polygon. Rectilinear: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.
Monotone with respect to a given line L: every line orthogonal to L intersects the polygon not more than twice. Euclidean geometry is assumed throughout. Any polygon has as many corners; each corner has several angles. The two most important ones are: Interior angle – The sum of the interior angles of a simple n-gon is π radians or × 180 degrees; this is because any simple n-gon can be considered to be made up of triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular n-gon is 180 − 360 n degrees; the interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular p q -gon, each interior angle is π p radians or 180 p degrees. Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex n-gon, the angle "turned" at a corner is external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°.
This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an n-gon in general, the sum of the exterior angles can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight" or antiparallelogram, where d is the density or starriness of the polygon. See orbit. In this section, the vertices of the polygon under consideration are taken to be, ( x 1
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function. Many important function spaces are defined to consist of real-valued functions. Let X be an arbitrary set. Let F denote the set of all functions from X to real numbers R; because R is a field, F may be turned into a vector space and a commutative algebra over reals by adding the appropriate structure: f + g: x ↦ f + g – vector addition 0: x ↦ 0 – additive identity c f: x ↦ c f, c ∈ R – scalar multiplication f g: x ↦ f g – pointwise multiplicationAlso, since R is an ordered set, there is a partial order on F: f ≤ g ⟺ ∀ x: f ≤ g. F is a ordered ring; the σ-algebra of Borel sets is an important structure on real numbers. If X has its σ-algebra and a function f is such that the preimage f −1 of any Borel set B belongs to that σ-algebra f is said to be measurable. Measurable functions form a vector space and an algebra as explained above. Moreover, a set of real-valued functions on X can define a σ-algebra on X generated by all preimages of all Borel sets.
This is the way how σ-algebras arise in probability theory, where real-valued functions on the sample space Ω are real-valued random variables. Real numbers form a complete metric space. Continuous real-valued functions are important in theories of topological spaces and of metric spaces; the extreme value theorem states that for any real continuous function on a compact space its global maximum and minimum exist. The concept of metric space itself is defined with a real-valued function of two variables, the metric, continuous; the space of continuous functions on a compact Hausdorff space has a particular importance. Convergent sequences can be considered as real-valued continuous functions on a special topological space. Continuous functions form a vector space and an algebra as explained above, are a subclass of measurable functions because any topological space has the σ-algebra generated by open sets. Real numbers are used as the codomain to define smooth functions. A domain of a real smooth function can be the real coordinate space, a topological vector space, an open subset of them, or a smooth manifold.
Spaces of smooth functions are vector spaces and algebras as explained above, are a subclass of continuous functions. A measure on a set is a non-negative real-valued functional on a σ-algebra of subsets. Lp spaces on sets with a measure are defined from aforementioned real-valued measurable functions, although they are quotient spaces. More whereas a function satisfying an appropriate summability condition defines an element of Lp space, in the opposite direction for any f ∈ Lp and x ∈ X, not an atom, the value f is undefined. Though, real-valued Lp spaces still have some of the structure explicated above; each of Lp spaces is a vector space and have a partial order, there exists a pointwise multiplication of "functions" which changes p, namely ⋅: L 1 / α × L 1 / β → L 1 /, 0 ≤ α, β ≤ 1, α + β ≤ 1. For example, pointwise product of two L2 functions belongs to L1. Other contexts where real-valued functions and their special properties are used include monotonic functions, convex functions and subharmonic functions, analytic functions, algebraic functions, polynomials.
Real analysis Partial differential equations, a major user of real-valued functions Norm Scalar Weisstein, Eric W. "Real Function". MathWorld
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia
A crescent shape is a symbol or emblem used to represent the lunar phase in the first quarter, or by extension a symbol representing the Moon itself. It is used as the astrological symbol for the Moon, hence as the alchemical symbol for silver, it was the emblem of Diana/Artemis, hence represented virginity. In Roman Catholic Marian veneration, it is associated with the Virgin Mary. From its use as roof finial in Ottoman era mosques, it has become associated with Islam, the crescent was introduced as chaplain badge for Muslim chaplains in the US military in 1993; the crescent symbol is used to represent the Moon, not in a particular lunar phase. When used to represent a waxing or waning lunar phase, "crescent" or "increscent" refers to the waxing first quarter, while the symbol representing the waning final quarter is called "decrescent"; the crescent symbol was long used as a symbol of the Moon in astrology, by extension of Silver in alchemy. The astrological use of the symbol is attested in early Greek papyri containing horoscopes.
In the 2nd-century Bianchini's planisphere, the personification of the Moon is shown with a crescent attached to her headdress. Its ancient association with Ishtar/Astarte and Diana is preserved in the Moon representing the female principle, virginity and female chastity. In Roman Catholic tradition, the crescent entered Marian iconography, by the association of Mary with the Woman of the Apocalypse The most well known representation of Mary as the Woman of the Apocalypse is the Virgin of Guadalupe; the crescent shape consists of a circular disk with a segment of another circle removed from its edge, so that what remains is a shape enclosed by two circular arcs of different diameters which intersect at two points. As such, it belongs to the class of figures known as lune in planar geometry; the tapering towards the points of intersection of the two arcs are known as the "horns" of the crescent. The classical crescent shape has its horns pointing upward (and is worn as horns when worn as a crown or diadem, e.g. in depictions of the lunar goddess, or in the headdress of Persian kings, etc.
The word crescent is derived etymologically from the present participle of the Latin verb crescere "to grow", technically denoting the waxing moon. As seen from the northern hemisphere, the waxing Moon tends to appear with its horns pointing towards the left, conversely the waning Moon with its horns pointing towards the right; the shape of the lit side of a spherical body that appears to be less than half illuminated by the Sun as seen by the viewer appears in a different shape from what is termed a crescent in planar geometry: Assuming the terminator lies on a great circle, the crescent Moon will appear as the figure bounded by a half-ellipse and a half-circle, with the major axis of the ellipse coinciding with a diameter of the semicircle. Unicode encodes a crescent at U+263D and a decrescent at U+263E; the Miscellaneous Symbols and Pictographs block provides variants with faces: U+1F31B FIRST QUARTER MOON WITH FACE and U+1F31C LAST QUARTER MOON WITH FACE. The crescent shape is used to represent the Moon, the Moon deity Nanna/Sin from an early time, visible in Akkadian cylinder seals as early as 2300 BC.
The crescent was well used in the iconography of the ancient Near East and was used transplanted by the Phoenicians in the 8th century BC as far as Carthage in modern Tunisia. The crescent and star appears on pre-Islamic coins of South Arabia; the combination of star and crescent arises in the ancient Near East, representing the Moon and Ishtar combined into a triad with the solar disk. It was inherited both in Hellenistic iconography. In the iconography of the Hellenistic period, the crescent became the symbol of Artemis-Diana, the virgin hunter goddess associated with the Moon. Numerous depictions show Artemis-Diana wearing the crescent Moon as part of her headdress; the related symbol of the star and crescent was the emblem of the Mithradates dynasty in the Kingdom of Pontus and was used as the emblem of Byzantium. The crescent remained in use as an emblem in Sassanid Persia, used as a Zoroastrian regal or astrological symbol. In the Crusades it came to be associated with the Orient and was used in Crusader seals and coins.
It was used as a heraldic charge by the 13th century. Anna Notaras, daughter of the last megas doux of the Byzantine Empire Loukas Notaras, after the fall of Constantinople and her emigration to Italy, made a seal with her coat of arms which included "two lions holding above the crescent a cross or a sword". From its use in Sassanid Persia, the crescent found its way into Islamic iconography after the Muslim conquest of Persia. Umar is said to have hung two crescent-shaped ornaments captured from the Sassanid capital Ctesiphon in the Kaaba; the crescent appears t
In geometry, a line segment is a part of a line, bounded by two distinct end points, contains every point on the line between its endpoints. A closed line segment includes both endpoints. Examples of line segments include the sides of a square. More when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices, or otherwise a diagonal; when the end points both lie on a curve such as a circle, a line segment is called a chord. If V is a vector space over R or C, L is a subset of V L is a line segment if L can be parameterized as L = for some vectors u, v ∈ V, in which case the vectors u and u + v are called the end points of L. Sometimes one needs to distinguish between "open" and "closed" line segments. One defines a closed line segment as above, an open line segment as a subset L that can be parametrized as L = for some vectors u, v ∈ V. Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points.
In geometry, it is sometimes defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R 2 the line segment with endpoints A = and C = is the following collection of points:. A line segment is a non-empty set. If V is a topological vector space a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More than above, the concept of a line segment can be defined in an ordered geometry. A pair of line segments can be any one of the following: intersecting, skew, or none of these; the last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane they must cross each other, but that need not be true of segments. In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line.
Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set; this is important because it transforms some of the analysis of convex sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and substitute other segments into another statement to make segments congruent. A line segment can be viewed as a degenerate case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints, the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant. A complete orbit of this ellipse traverses the line segment twice; as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, line segments appear in numerous other locations relative to other geometric shapes.
Some frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, the internal angle bisectors. In each case there are various equalities relating these segment lengths to others as well as various inequalities. Other segment