The decimal numeral system is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system; the way of denoting numbers in the decimal system is referred to as decimal notation. A decimal numeral, or just decimal, or casually decimal number, refers to the notation of a number in the decimal numeral system. Decimals may sometimes be identified for containing a decimal separator. "Decimal" may refer to the digits after the decimal separator, such as in "3.14 is the approximation of π to two decimals". The numbers that may be represented in the decimal system are the decimal fractions, the fractions of the form a/10n, where a is an integer, n is a non-negative integer; the decimal system has been extended to infinite decimals, for representing any real number, by using an infinite sequence of digits after the decimal separator. In this context, the decimal numerals with a finite number of non–zero places after the decimal separator are sometimes called terminating decimals.
A repeating decimal is an infinite decimal that after some place repeats indefinitely the same sequence of digits. An infinite decimal represents a rational number if and only if it is a repeating decimal or has a finite number of nonzero digits. Many numeral systems of ancient civilisations use ten and its powers for representing numbers because there are ten fingers on two hands and people started counting by using their fingers. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Chinese numerals. Large numbers were difficult to represent in these old numeral systems, only the best mathematicians were able to multiply or divide large numbers; these difficulties were solved with the introduction of the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some non-integer numbers, called decimal fractions or decimal numbers for forming the decimal numeral system. For writing numbers, the decimal system uses ten decimal digits, a decimal mark, for negative numbers, a minus sign "−".
The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. For representing a non-negative number, a decimal consists of either a sequence of digits such as 2017, or in full generality, a m a m − 1 … a 0 or two sequence of digits separated by a decimal mark such as 3.14159, 15.00, or in full generality a m a m − 1 … a 0. B 1 b 2 … b n It is assumed that, if m > 0, the first digit am is not zero, but, in some circumstances, it may be useful to have one or more 0's on the left. This does not change the value represented by the decimal. For example, 3.14 = 03.14 = 003.14. If bn =0, it may be removed, conversely, trailing zeros may be added without changing the represented number: for example, 15 = 15.0 = 15.00 and 5.2 = 5.20 = 5.200. Sometimes the extra zeros are used for indicating the accuracy of a measurement. For example, 15.00 m may indicate that the measurement error is less than one centimeter, while 15 m may mean that the length is fifteen meters, that the error may exceed 10 cm. For representing a negative number, a minus sign is placed before am.
The numeral a m a m − 1 … a 0. B 1 b 2 … b n represents the number a m 10 m + a m − 1 10 m − 1 + ⋯ + a 0 10 0 + b 1 10 1 + b 2 10 2 + ⋯ + b n 10 n Therefore, the contribution of each digit to the value of a number depends on its position in the numeral; that is, the decimal system is a positional numeral system The numbers that are represented by decimal numerals are the decimal fractions, that is, the rational numbers that may be expressed as a fraction, the denominator of, a power of ten. For example, the numerals 0.8, 14.89, 0.00024 represent the fractions 8/10, 1489/100, 24/100000. More a decimal with n digits after the separator represents the fraction with denominator 10n, whose numerator is the integer obtained by removing the separator. Expressed as a reduced fraction, the decimal numbers are those whose denominator is a product of a powe
A circle is a simple closed shape. It is the set of all points in a plane; the distance between any of the points and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. A circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior. A circle may be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations. A circle is a plane figure bounded by one line, such that all right lines drawn from a certain point within it to the bounding line, are equal; the bounding line is called the point, its centre. Annulus: a ring-shaped object, the region bounded by two concentric circles.
Arc: any connected part of a circle. Specifying two end points of an arc and a center allows for two arcs that together make up a full circle. Centre: the point equidistant from all points on the circle. Chord: a line segment whose endpoints lie on the circle, thus dividing a circle in two sements. Circumference: the length of one circuit along the circle, or the distance around the circle. Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; this is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, its length is twice the length of a radius. Disc: the region of the plane bounded by a circle. Lens: the region common to two overlapping discs. Passant: a coplanar straight line that has no point in common with the circle. Radius: a line segment joining the centre of a circle with any single point on the circle itself. Sector: a region bounded by two radii of equal length with a common center and either of the two possible arcs, determined by this center and the endpoints of the radii.
Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the center of the circle to which their arc belongs to. Secant: an extended chord, a coplanar straight line, intersecting a circle in two points. Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as center. In non-technical common usage it may mean the interior of the two dimensional region bounded by a diameter and one of its arcs, technically called a half-disc. A half-disc is a special case of a segment, namely the largest one. Tangent: a coplanar straight line that has one single point in common with a circle. All of the specified regions may be considered as open, that is, not containing their boundaries, or as closed, including their respective boundaries; the word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, meaning "hoop" or "ring".
The origins of the words circus and circuit are related. The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand; the circle is the basis for the wheel, 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 and calculus. Early science geometry and astrology and astronomy, was connected to the divine for most medieval scholars, many believed that there was something intrinsically "divine" or "perfect" that could be found in circles; some highlights in the history of the circle are: 1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as an approximate value of π. 300 BCE – Book 3 of Euclid's Elements deals with the properties of circles. In Plato's Seventh Letter there is a detailed explanation of the circle.
Plato explains the perfect circle, how it is different from any drawing, definition or explanation. 1880 CE – Lindemann proves that π is transcendental settling the millennia-old problem of squaring the circle. The ratio of a circle's circumference to its diameter is π, an irrational constant equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by: C = 2 π r = π d; as proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to π multiplied by the radius squared: A r e a = π r 2. Equivalently, denoting diameter by d, A r e
In mathematics, the gamma function is one of the extensions of the factorial function with its argument shifted down by 1, to real and complex numbers. Derived by Daniel Bernoulli, if n is a positive integer, Γ =! Although other extensions do exist, this particular definition is the most useful; the gamma function is defined for all complex numbers except the non-positive integers. For complex numbers with a positive real part, it is defined via a convergent improper integral: Γ = ∫ 0 ∞ x z − 1 e − x d x This integral function is extended by analytic continuation to all complex numbers except the non-positive integers, yielding the meromorphic function we call the gamma function, it has no zeroes, so the reciprocal gamma function 1/Γ is a holomorphic function. In fact the gamma function corresponds to the Mellin transform of the negative exponential function: Γ = The gamma function is a component in various probability-distribution functions, as such it is applicable in the fields of probability and statistics, as well as combinatorics.
The gamma function can be seen as a solution to the following interpolation problem: "Find a smooth curve that connects the points given by y =! at the positive integer values for x."A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that describes the curve, in which the number of operations does not depend on the size of x. The simple formula for the factorial, x! = 1 × 2 × … × x, cannot be used directly for fractional values of x since it is only valid when x is a natural number. There are speaking, no such simple solutions for factorials. A good solution to this is the gamma function. There are infinitely many continuous extensions of the factorial to non-integers: infinitely many curves can be drawn through any set of isolated points; the gamma function is the most useful solution in practice, being analytic, it can be characterized in several ways. However, it is not the only analytic function which extends the factorial, as adding to it any analytic function, zero on the positive integers, such as k sin mπx, will give another function with that property.
A more restrictive property than satisfying the above interpolation is to satisfy the recurrence relation defining a translated version of the factorial function, f = 1, f = x f, for x equal to any positive real number. But this would allow for multiplication by any periodic analytic function which evaluates to one on the positive integers, such as ek sin mπx. There's a final way to solve all this ambiguity: Bohr–Mollerup theorem states that when the condition that f be logarithmically convex is added, it uniquely determines f for positive, real inputs. From there, the gamma function can be extended to all real and complex values by using the unique analytic continuation of f. See Euler's infinite product definition below where the properties f = 1 and f = x f together with the asymptotic requirement that limn→+∞! nx / f = 1 uniquely define the same function. The notation Γ is due to Legendre. If the real part of the complex number z is positive the integral Γ = ∫ 0 ∞ x z − 1 e − x d x converges and is known as the Euler integral of the second kind.
Using integration by parts, one sees that: Γ = ∫ 0 ∞ x z e − x d x = 0 ∞ + ∫ 0 ∞ z x z − 1 e − x d x = lim x →
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More each point of an n-dimensional manifold has a neighbourhood, homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold. One-dimensional manifolds include circles, but not figure eights. Two-dimensional manifolds are called surfaces. Examples include the plane, the sphere, the torus, which can all be embedded in three dimensional real space, but the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space. Although a manifold locally resembles Euclidean space, meaning that every point has a neighbourhood homeomorphic to an open subset of Euclidean space, globally it may not: manifolds in general are not homeomorphic to Euclidean space. For example, the surface of the sphere is not homeomorphic to the Euclidean plane, because it has the global topological property of compactness that Euclidean space lacks, but in a region it can be charted by means of map projections of the region into the Euclidean plane.
When a region appears in two neighbouring charts, the two representations do not coincide and a transformation is needed to pass from one to the other, called a transition map. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described and understood in terms of the simpler local topological properties of Euclidean space. Manifolds arise as solution sets of systems of equations and as graphs of functions. Manifolds can be equipped with additional structure. One important class of manifolds is the class of differentiable manifolds. A Riemannian metric on a manifold allows angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. A surface is a two dimensional manifold, meaning that it locally resembles the Euclidean plane near each point. For example, the surface of a globe can be described by a collection of maps, which together form an atlas of the globe.
Although no individual map is sufficient to cover the entire surface of the globe, any place in the globe will be in at least one of the charts. Many places will appear in more than one chart. For example, a map of North America will include parts of South America and the Arctic circle; these regions of the globe will be described in full in separate charts, which in turn will contain parts of North America. There is a relation between adjacent charts, called a transition map that allows them to be patched together to cover the whole of the globe. Describing the coordinate charts on surfaces explicitly requires knowledge of functions of two variables, because these patching functions must map a region in the plane to another region of the plane. However, one-dimensional examples of manifolds can be described with functions of a single variable only. Manifolds have applications in computer-graphics and augmented-reality given the need to associate pictures to coordinates. In an augmented reality setting, a picture can be seen as something associated with a coordinate and by using sensors for detecting movements and rotation one can have knowledge of how the picture is oriented and placed in space.
After a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Consider, for instance, the top part of the unit circle, x2 + y2 = 1, where the y-coordinate is positive. Any point of this arc can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous, invertible, mapping from the upper arc to the open interval: χ t o p = x; such functions along with the open regions they map are called charts. There are charts for the bottom and right parts of the circle: χ b o t t o m = x χ l e f t = y χ r i g h t = y. Together, these parts cover the four charts form an atlas for the circle; the top and right charts, χ t o
In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting and bending, but not tearing or gluing. An n-dimensional topological space is a space with certain properties of connectedness and compactness; the space discrete. It can be closed. Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space and transformation; such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems; the term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics. Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.
Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realised the importance of the edges of a polyhedron; this led to his polyhedron formula, V − E + F = 2. Some authorities regard this analysis as the first theorem. Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print; the English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". The term "topologist" in the sense of a specialist in topology was used in 1905 in the magazine Spectator.
Their work was corrected and extended by Henri Poincaré. In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space. A topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. Modern topology depends on the ideas of set theory, developed by Georg Cantor in the part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series.
For further developments, see point-set topology and algebraic topology. Topology can be formally defined as "the study of qualitative properties of certain objects that are invariant under a certain kind of transformation those properties that are invariant under a certain kind of invertible transformation." Topology is used to refer to a structure imposed upon a set X, a structure that characterizes the set X as a topological space by taking proper care of properties such as convergence and continuity, upon transformation. Topological spaces show up in every branch of mathematics; this has made topology one of the great unifying ideas of mathematics. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside.
In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg that would cross each of its seven bridges once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks; this Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory. The hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is convincing to most people though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of 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
A meridian is the half of an imaginary great circle on the Earth's surface, terminated by the North Pole and the South Pole, connecting points of equal longitude, as measured in angular degrees east or west of the Prime Meridian. The position of a point along the meridian is given by that longitude and its latitude, measured in angular degrees north or south of the Equator; each meridian is perpendicular to all circles of latitude. Each is the same length, being half of a great circle on the Earth's surface and therefore measuring 20,003.93 km. The first prime meridian was set by Eratosthenes in 200 BCE; this prime meridian was used to provide measurement of the earth, but had many problems because of the lack of latitude measurement. Many years around the 19th century there was still concerns of the prime meridian; the idea of having one prime meridian came from William Parker Snow, because he realized the confusion of having multiple prime meridian locations. Many of theses geographical locations were traced back to the ancient Greeks, others were created by several nations.
Multiple locations for the geographical meridian meant that there was inconsistency, because each country had their own guidelines for where the prime meridian was located. The term meridian comes from the spanish meridies, meaning "midday"; the Sun crosses the celestial meridian at the same time. The same Latin stem gives rise to the terms a.m. and p.m. used to disambiguate hours of the day when utilizing the 12-hour clock. Toward the ending of the 12th century there were two main locations that were acknowledged as the geographic location of the meridian and Britain; these two locations conflicted and a settlement was reached only after there was an International Meridian Conference held, in which Greenwich was recognized as the 0° location. The meridian through Greenwich, called the Prime Meridian, was set at zero degrees of longitude, while other meridians were defined by the angle at the center of the earth between where it and the prime meridian cross the equator; as there are 360 degrees in a circle, the meridian on the opposite side of the earth from Greenwich, the antimeridian, forms the other half of a circle with the one through Greenwich, is at 180° longitude near the International Date Line.
The meridians from West of Greenwich to the antimeridian define the Western Hemisphere and the meridians from East of Greenwich to the antimeridian define the Eastern Hemisphere. Most maps show the lines of longitude; the position of the prime meridian has changed a few times throughout history due to the transit observatory being built next door to the previous one. Such changes had no significant practical effect; the average error in the determination of longitude was much larger than the change in position. The adoption of WGS84 as the positioning system has moved the geodetic prime meridian 102.478 metres east of its last astronomic position. The position of the current geodetic prime meridian is not identified at all by any kind of sign or marking in Greenwich, but can be located using a GPS receiver, it was in the best interests of the nations to agree to one standard meridian to benefit their fast growing economy and production. The disorganized system they had before was not sufficient for their increasing mobility.
The coach services in England had erratic timing before the GWT. U. S. and Canada were improving their railroad system and needed a standard time as well. With a standard meridian, stage coach and trains were able to be more efficient; the argument of which meridian is more scientific was set aside in order to find the most convenient for practical reasons. They were able to agree that the universal day was going to be the mean solar day, they agreed that the days would begin at midnight and the universal day would not impact the use of local time. In the "Transactions of the Royal Society of Canada a report was submitted, dated 10 May 1894. Therefore, a compass needle will be parallel to the magnetic meridian. However, a compass needle will not be steady in the magnetic meridian, because of the longitude from east to west being complete geodesic; the angle between the magnetic and the true meridian is the magnetic declination, relevant for navigating with a compass. Navigators were able to use the azimuth of the rising and setting Sun to measure the magnetic variation.
The true meridian is the plane that passes through true north poles and true south poles at the spot of the observer. The difference between true meridian and magnetic meridian is that the true meridian is fixed while the magnetic meridian is formed through the movement of the needle. True bearing is the horizontal angle between a line. Henry D. Thoreau classified this true meridian