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126-498: The real Gudermannian function is typically defined for − ∞ < ψ < ∞ {\textstyle -\infty <\psi <\infty } to be the integral of the hyperbolic secant The real inverse Gudermannian function can be defined for − 1 2 π < ϕ < 1 2 π {\textstyle -{\tfrac {1}{2}}\pi <\phi <{\tfrac {1}{2}}\pi } as

252-426: A u {\displaystyle u} is called an upper bound of S . {\displaystyle S.} So, Dedekind completeness means that, if S is bounded above, it has an upper bound that is less than any other upper bound. Dedekind completeness implies other sorts of completeness (see below), but also has some important consequences. The last two properties are summarized by saying that

378-440: A , {\displaystyle D_{n-1}+a_{n}/10^{n}\leq a,} and one sets D n = D n − 1 + a n / 10 n . {\displaystyle D_{n}=D_{n-1}+a_{n}/10^{n}.} One can use the defining properties of the real numbers to show that x is the least upper bound of the D n . {\displaystyle D_{n}.} So,

504-435: A compass rose or protractor, and the corresponding directions are easily transferred from point to point, on the map, e.g. with the help of a parallel ruler . Because the linear scale of a Mercator map in normal aspect increases with latitude, it distorts the size of geographical objects far from the equator and conveys a distorted perception of the overall geometry of the planet. At latitudes greater than 70° north or south,

630-480: A decimal point , representing the infinite series For example, for the circle constant π = 3.14159 ⋯ , {\displaystyle \pi =3.14159\cdots ,} k is zero and b 0 = 3 , {\displaystyle b_{0}=3,} a 1 = 1 , {\displaystyle a_{1}=1,} a 2 = 4 , {\displaystyle a_{2}=4,} etc. More formally,

756-406: A decimal representation for a nonnegative real number x consists of a nonnegative integer k and integers between zero and nine in the infinite sequence (If k > 0 , {\displaystyle k>0,} then by convention b k ≠ 0. {\displaystyle b_{k}\neq 0.} ) Such a decimal representation specifies the real number as

882-763: A function of a complex variable , z ↦ w = gd ⁡ z {\textstyle z\mapsto w=\operatorname {gd} z} conformally maps the infinite strip | Im ⁡ z | ≤ 1 2 π {\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi } to the infinite strip | Re ⁡ w | ≤ 1 2 π , {\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi ,} while w ↦ z = gd − 1 ⁡ w {\textstyle w\mapsto z=\operatorname {gd} ^{-1}w} conformally maps

1008-443: A line called the number line or real line , where the points corresponding to integers ( ..., −2, −1, 0, 1, 2, ... ) are equally spaced. Conversely, analytic geometry is the association of points on lines (especially axis lines ) to real numbers such that geometric displacements are proportional to differences between corresponding numbers. The informal descriptions above of the real numbers are not sufficient for ensuring

1134-593: A power of ten , extending to finitely many positive powers of ten to the left and infinitely many negative powers of ten to the right. For a number x whose decimal representation extends k places to the left, the standard notation is the juxtaposition of the digits b k b k − 1 ⋯ b 0 . a 1 a 2 ⋯ , {\displaystyle b_{k}b_{k-1}\cdots b_{0}.a_{1}a_{2}\cdots ,} in descending order by power of ten, with non-negative and negative powers of ten separated by

1260-457: A real number is a number that can be used to measure a continuous one- dimensional quantity such as a distance , duration or temperature . Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion . The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in

1386-486: A total order that have the following properties. Many other properties can be deduced from the above ones. In particular: Several other operations are commonly used, which can be deduced from the above ones. The total order that is considered above is denoted a < b {\displaystyle a<b} and read as " a is less than b ". Three other order relations are also commonly used: The real numbers 0 and 1 are commonly identified with

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1512-442: A Mercator map printed in a book might have an equatorial width of 13.4 cm corresponding to a globe radius of 2.13 cm and an RF of approximately ⁠ 1 / 300M ⁠ (M is used as an abbreviation for 1,000,000 in writing an RF) whereas Mercator's original 1569 map has a width of 198 cm corresponding to a globe radius of 31.5 cm and an RF of about ⁠ 1 / 20M ⁠ . A cylindrical map projection

1638-413: A book which expounded sinh {\textstyle \sinh } and cosh {\textstyle \cosh } to a wide audience (although represented by the symbols S i n {\textstyle {\mathfrak {Sin}}} and C o s {\textstyle {\mathfrak {Cos}}} ). The notation gd {\textstyle \operatorname {gd} }

1764-452: A characterization of the real numbers.) It is not true that R {\displaystyle \mathbb {R} } is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field , and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Every uniformly complete Archimedean field must also be Dedekind-complete (and vice versa), justifying using "the" in

1890-491: A different relationship that does not diverge at  φ  = ±90°. A transverse Mercator projection tilts the cylinder axis so that it is perpendicular to Earth's axis. The tangent standard line then coincides with a meridian and its opposite meridian, giving a constant scale factor along those meridians and making the projection useful for mapping regions that are predominately north–south in extent. In its more complex ellipsoidal form, most national grid systems around

2016-523: A half-turn rotation and translation in the codomain by one of ± π , {\textstyle \pm \pi ,} and vice versa for gd − 1 : {\textstyle \operatorname {gd} ^{-1}\colon } A reflection in the domain of gd {\textstyle \operatorname {gd} } across either of the lines x ± 1 2 π i {\textstyle x\pm {\tfrac {1}{2}}\pi i} results in

2142-585: A hyperbolic angle ψ {\textstyle \psi } , hyperbolic functions could be found by first looking up ϕ = gd ⁡ ψ {\textstyle \phi =\operatorname {gd} \psi } in a Gudermannian table and then looking up the appropriate circular function of ϕ {\textstyle \phi } , or by directly locating ψ {\textstyle \psi } in an auxiliary gd − 1 {\displaystyle \operatorname {gd} ^{-1}} column of

2268-405: A limit, without computing it, and even without knowing it. For example, the standard series of the exponential function converges to a real number for every x , because the sums can be made arbitrarily small (independently of M ) by choosing N sufficiently large. This proves that the sequence is Cauchy, and thus converges, showing that e x {\displaystyle e^{x}}

2394-459: A map in Mercator projection that correctly showed those two coordinates. Many major online street mapping services ( Bing Maps , Google Maps , Mapbox , MapQuest , OpenStreetMap , Yahoo! Maps , and others) use a variant of the Mercator projection for their map images called Web Mercator or Google Web Mercator. Despite its obvious scale variation at the world level (small scales), the projection

2520-401: A median latitude, hk = 11.7. For Australia, taking 25° as a median latitude, hk = 1.2. For Great Britain, taking 55° as a median latitude, hk = 3.04. The variation with latitude is sometimes indicated by multiple bar scales as shown below. The classic way of showing the distortion inherent in a projection is to use Tissot's indicatrix . Nicolas Tissot noted that the scale factors at

2646-459: A nonnegative real number x , one can define a decimal representation of x by induction , as follows. Define b k ⋯ b 0 {\displaystyle b_{k}\cdots b_{0}} as decimal representation of the largest integer D 0 {\displaystyle D_{0}} such that D 0 ≤ x {\displaystyle D_{0}\leq x} (this integer exists because of

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2772-713: A number of identities between hyperbolic functions of ψ {\textstyle \psi } and circular functions of ϕ . {\textstyle \phi .} These are commonly used as expressions for gd {\displaystyle \operatorname {gd} } and gd − 1 {\displaystyle \operatorname {gd} ^{-1}} for real values of ψ {\displaystyle \psi } and ϕ {\displaystyle \phi } with | ϕ | < 1 2 π . {\displaystyle |\phi |<{\tfrac {1}{2}}\pi .} For example,

2898-510: A point on a map projection, specified by the numbers h and k , define an ellipse at that point. For cylindrical projections, the axes of the ellipse are aligned to the meridians and parallels. For the Mercator projection, h  =  k , so the ellipses degenerate into circles with radius proportional to the value of the scale factor for that latitude. These circles are rendered on the projected map with extreme variation in size, indicative of Mercator's scale variations. As discussed above,

3024-473: A rational number is an equivalence class of pairs of integers, and a real number is an equivalence class of Cauchy series), and are generally harmless. It is only in very specific situations, that one must avoid them and replace them by using explicitly the above homomorphisms. This is the case in constructive mathematics and computer programming . In the latter case, these homomorphisms are interpreted as type conversions that can often be done automatically by

3150-448: A reflection in the codomain across one of the lines ± 1 2 π + y i , {\textstyle \pm {\tfrac {1}{2}}\pi +yi,} and vice versa for gd − 1 : {\textstyle \operatorname {gd} ^{-1}\colon } This is related to the identity A few specific values (where ∞ {\textstyle \infty } indicates

3276-466: A ship's bearing in sailing between locations on the chart; the region of the Earth covered by such charts was small enough that a course of constant bearing would be approximately straight on the chart. The charts have startling accuracy not found in the maps constructed by contemporary European or Arab scholars, and their construction remains enigmatic; based on cartometric analysis which seems to contradict

3402-418: A small portion of the spherical surface without otherwise distorting it, preserving angles between intersecting curves. Afterward, this cylinder is unrolled onto a flat plane to make a map. In this interpretation, the scale of the surface is preserved exactly along the circle where the cylinder touches the sphere, but increases nonlinearly for points further from the contact circle. However, by uniformly shrinking

3528-448: A straight segment. Such a course, known as a rhumb (alternately called a rhumb line or loxodrome) is preferred in marine navigation because ships can sail in a constant compass direction. This reduces the difficult, error-prone course corrections that otherwise would be necessary when sailing a different course. For small distances (compared to the radius of the Earth), the difference between

3654-412: Is R  cos  φ , the corresponding parallel on the map must have been stretched by a factor of ⁠ 1 / cos φ ⁠ = sec φ . This scale factor on the parallel is conventionally denoted by k and the corresponding scale factor on the meridian is denoted by  h . The Mercator projection is conformal . One implication of that is the "isotropy of scale factors", which means that

3780-474: Is a specific parameterization of the cylindrical equal-area projection . In response, a 1989 resolution by seven North American geographical groups disparaged using cylindrical projections for general-purpose world maps, which would include both the Mercator and the Gall–Peters. Practically every marine chart in print is based on the Mercator projection due to its uniquely favorable properties for navigation. It

3906-463: Is also commonly used by street map services hosted on the Internet, due to its uniquely favorable properties for local-area maps computed on demand. Mercator projections were also important in the mathematical development of plate tectonics in the 1960s. The Mercator projection was designed for use in marine navigation because of its unique property of representing any course of constant bearing as

Gudermannian function - Misplaced Pages Continue

4032-435: Is less than ε for all n and m that are both greater than N . This definition, originally provided by Cauchy , formalizes the fact that the x n eventually come and remain arbitrarily close to each other. A sequence ( x n ) converges to the limit x if its elements eventually come and remain arbitrarily close to x , that is, if for any ε > 0 there exists an integer N (possibly depending on ε) such that

4158-462: Is often denoted ψ . {\textstyle \psi .} In terms of latitude ϕ {\textstyle \phi } on the sphere (expressed in radians ) the isometric latitude can be written The inverse from the isometric latitude to spherical latitude is ϕ = gd ⁡ ψ . {\textstyle \phi =\operatorname {gd} \psi .} (Note: on an ellipsoid of revolution ,

4284-482: Is so that many sequences have limits . More formally, the reals are complete (in the sense of metric spaces or uniform spaces , which is a different sense than the Dedekind completeness of the order in the previous section): A sequence ( x n ) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance | x n − x m |

4410-481: Is specified by formulae linking the geographic coordinates of latitude  φ and longitude  λ to Cartesian coordinates on the map with origin on the equator and x -axis along the equator. By construction, all points on the same meridian lie on the same generator of the cylinder at a constant value of x , but the distance y along the generator (measured from the equator) is an arbitrary function of latitude, y ( φ ). In general this function does not describe

4536-422: Is the inverse of the integral of the secant function . Using Cayley's notation, He then derives "the definition of the transcendent", observing that "although exhibited in an imaginary form, [it] is a real function of u {\textstyle u} ". The Gudermannian and its inverse were used to make trigonometric tables of circular functions also function as tables of hyperbolic functions. Given

4662-535: Is the vertical coordinate of the Mercator projection . The two angle measures ϕ {\textstyle \phi } and ψ {\textstyle \psi } are related by a common stereographic projection and this identity can serve as an alternative definition for gd {\textstyle \operatorname {gd} } and gd − 1 {\textstyle \operatorname {gd} ^{-1}} valid throughout

4788-458: Is well defined for every x . The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways. First, an order can be lattice-complete . It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z , z + 1 is larger). Additionally, an order can be Dedekind-complete, see § Axiomatic approach . The uniqueness result at

4914-494: Is well-suited as an interactive world map that can be zoomed seamlessly to local (large-scale) maps, where there is relatively little distortion due to the variant projection's near- conformality . The major online street mapping services' tiling systems display most of the world at the lowest zoom level as a single square image, excluding the polar regions by truncation at latitudes of φ max  = ±85.05113°. (See below .) Latitude values outside this range are mapped using

5040-495: Is what mathematicians and physicists did during several centuries before the first formal definitions were provided in the second half of the 19th century. See Construction of the real numbers for details about these formal definitions and the proof of their equivalence. The real numbers form an ordered field . Intuitively, this means that methods and rules of elementary arithmetic apply to them. More precisely, there are two binary operations , addition and multiplication , and

5166-559: The 2-argument arctangent , u = atan2 ⁡ ( sinh ⁡ x , cos ⁡ y ) {\textstyle u=\operatorname {atan2} (\sinh x,\cos y)} .) Likewise, if x + i y = gd − 1 ⁡ ( u + i v ) , {\textstyle x+iy=\operatorname {gd} ^{-1}(u+iv),} then components x {\textstyle x} and y {\textstyle y} can be found by: Multiplying these together reveals

Gudermannian function - Misplaced Pages Continue

5292-640: The compiler . Previous properties do not distinguish real numbers from rational numbers . This distinction is provided by Dedekind completeness , which states that every set of real numbers with an upper bound admits a least upper bound . This means the following. A set of real numbers S {\displaystyle S} is bounded above if there is a real number u {\displaystyle u} such that s ≤ u {\displaystyle s\leq u} for all s ∈ S {\displaystyle s\in S} ; such

5418-529: The complex plane : We can evaluate the integral of the hyperbolic secant using the stereographic projection ( hyperbolic half-tangent ) as a change of variables : Letting ϕ = gd ⁡ ψ {\textstyle \phi =\operatorname {gd} \psi } and s = tan ⁡ 1 2 ϕ = tanh ⁡ 1 2 ψ {\textstyle s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi } we can derive

5544-566: The equator ; the closer to the poles of the Earth, the greater the distortion. Because of great land area distortions, critics like George Kellaway and Irving Fisher consider the projection unsuitable for general world maps. It has been conjectured to have influenced people's views of the world: because it shows countries near the Equator as too small when compared to those of Europe and North America, it has been supposed to cause people to consider those countries as less important. Mercator himself used

5670-1193: The exponential , then we can see that s , {\textstyle s,} exp ⁡ ϕ i , {\displaystyle \exp \phi i,} and exp ⁡ ψ {\displaystyle \exp \psi } are all Möbius transformations of each-other (specifically, rotations of the Riemann sphere ): For real values of ψ {\textstyle \psi } and ϕ {\textstyle \phi } with | ϕ | < 1 2 π {\displaystyle |\phi |<{\tfrac {1}{2}}\pi } , these Möbius transformations can be written in terms of trigonometric functions in several ways, These give further expressions for gd {\displaystyle \operatorname {gd} } and gd − 1 {\displaystyle \operatorname {gd} ^{-1}} for real arguments with | ϕ | < 1 2 π . {\displaystyle |\phi |<{\tfrac {1}{2}}\pi .} For example, As

5796-424: The globe in this section. The globe determines the scale of the map. The various cylindrical projections specify how the geographic detail is transferred from the globe to a cylinder tangential to it at the equator. The cylinder is then unrolled to give the planar map. The fraction ⁠ R / a ⁠ is called the representative fraction (RF) or the principal scale of the projection. For example,

5922-547: The integral of the (circular) secant The hyperbolic angle measure ψ = gd − 1 ⁡ ϕ {\displaystyle \psi =\operatorname {gd} ^{-1}\phi } is called the anti-gudermannian of ϕ {\displaystyle \phi } or sometimes the lambertian of ϕ {\displaystyle \phi } , denoted ψ = lam ⁡ ϕ . {\displaystyle \psi =\operatorname {lam} \phi .} In

6048-407: The integral of the secant function , The function y ( φ ) is plotted alongside φ for the case R  = 1: it tends to infinity at the poles. The linear y -axis values are not usually shown on printed maps; instead some maps show the non-linear scale of latitude values on the right. More often than not the maps show only a graticule of selected meridians and parallels. The expression on

6174-422: The natural numbers 0 and 1 . This allows identifying any natural number n with the sum of n real numbers equal to 1 . This identification can be pursued by identifying a negative integer − n {\displaystyle -n} (where n {\displaystyle n} is a natural number) with the additive inverse − n {\displaystyle -n} of

6300-454: The per-component computation for the complex Gudermannian and inverse Gudermannian. In the specific case z = w , {\textstyle z=w,} double-argument identities are The Taylor series near zero, valid for complex values z {\textstyle z} with | z | < 1 2 π , {\textstyle |z|<{\tfrac {1}{2}}\pi ,} are where

6426-433: The principal branch ) or consider their domains and codomains as Riemann surfaces . If u + i v = gd ⁡ ( x + i y ) , {\textstyle u+iv=\operatorname {gd} (x+iy),} then the real and imaginary components u {\textstyle u} and v {\textstyle v} can be found by: (In practical implementation, make sure to use

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6552-570: The square roots of −1 . The real numbers include the rational numbers , such as the integer −5 and the fraction 4 / 3 . The rest of the real numbers are called irrational numbers . Some irrational numbers (as well as all the rationals) are the root of a polynomial with integer coefficients, such as the square root √2 = 1.414... ; these are called algebraic numbers . There are also real numbers which are not, such as π = 3.1415... ; these are called transcendental numbers . Real numbers can be thought of as all points on

6678-505: The Archimedean property). Then, supposing by induction that the decimal fraction D i {\displaystyle D_{i}} has been defined for i < n , {\displaystyle i<n,} one defines a n {\displaystyle a_{n}} as the largest digit such that D n − 1 + a n / 10 n ≤

6804-638: The Chinese Song dynasty may have been drafted on the Mercator projection; however, this claim was presented without evidence, and astronomical historian Kazuhiko Miyajima concluded using cartometric analysis that these charts used an equirectangular projection instead. In the 13th century, the earliest extant portolan charts of the Mediterranean sea, which are generally not believed to be based on any deliberate map projection, included windrose networks of criss-crossing lines which could be used to help set

6930-473: The Mercator projection can be found in many world maps in the centuries following Mercator's first publication. However, it did not begin to dominate world maps until the 19th century, when the problem of position determination had been largely solved. Once the Mercator became the usual projection for commercial and educational maps, it came under persistent criticism from cartographers for its unbalanced representation of landmasses and its inability to usefully show

7056-442: The Mercator projection inflates the size of lands the further they are from the equator . Therefore, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator. Nowadays the Mercator projection is widely used because, aside from marine navigation, it is well suited for internet web maps . Joseph Needham , a historian of China, speculated that some star charts of

7182-420: The Mercator projection is practically unusable, because the linear scale becomes infinitely large at the poles. A Mercator map can therefore never fully show the polar areas (but see Uses below for applications of the oblique and transverse Mercator projections). The Mercator projection is often compared to and confused with the central cylindrical projection , which is the result of projecting points from

7308-481: The Mercator projection is the unique projection which balances this East–West stretching by a precisely corresponding North–South stretching, so that at every location the scale is locally uniform and angles are preserved. The Mercator projection in normal aspect maps trajectories of constant bearing (called rhumb lines or loxodromes ) on a sphere to straight lines on the map, and is thus uniquely suited to marine navigation : courses and bearings are measured using

7434-535: The North and South poles, and the contact circle is the Earth's equator . As for all cylindrical projections in normal aspect, circles of latitude and meridians of longitude are straight and perpendicular to each other on the map, forming a grid of rectangles. While circles of latitude on the Earth are smaller the closer they are to the poles, they are stretched in an East–West direction to have uniform length on any cylindrical map projection. Among cylindrical projections,

7560-458: The Web Mercator. The Mercator projection can be visualized as the result of wrapping a cylinder tightly around a sphere, with the two surfaces tangent to (touching) each other along a circle halfway between the poles of their common axis, and then conformally unfolding the surface of the sphere outward onto the cylinder, meaning that at each point the projection uniformly scales the image of

7686-425: The additional identity The two functions can be thought of as rotations or reflections of each-other, with a similar relationship as sinh ⁡ i z = i sin ⁡ z {\textstyle \sinh iz=i\sin z} between sine and hyperbolic sine : The functions are both odd and they commute with complex conjugation . That is, a reflection across the real or imaginary axis in

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7812-529: The axiomatic definition and are thus equivalent. Real numbers are completely characterized by their fundamental properties that can be summarized by saying that they form an ordered field that is Dedekind complete . Here, "completely characterized" means that there is a unique isomorphism between any two Dedekind complete ordered fields, and thus that their elements have exactly the same properties. This implies that one can manipulate real numbers and compute with them, without knowing how they can be defined; this

7938-441: The axioms of Zermelo–Fraenkel set theory including the axiom of choice (ZFC)—the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it. As a topological space, the real numbers are separable . This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have

8064-420: The cardinality of the power set of the set of the natural numbers. The statement that there is no subset of the reals with cardinality strictly greater than ℵ 0 {\displaystyle \aleph _{0}} and strictly smaller than c {\displaystyle {\mathfrak {c}}} is known as the continuum hypothesis (CH). It is neither provable nor refutable using

8190-411: The classical definitions of limits , continuity and derivatives . The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold R , often using blackboard bold , ⁠ R {\displaystyle \mathbb {R} } ⁠ . The adjective real , used in the 17th century by René Descartes , distinguishes real numbers from imaginary numbers such as

8316-671: The complex plane, these functions can be correctly written as: For the gd {\textstyle \operatorname {gd} } and gd − 1 {\textstyle \operatorname {gd} ^{-1}} functions to remain invertible with these extended domains, we might consider each to be a multivalued function (perhaps Gd {\textstyle \operatorname {Gd} } and Gd − 1 {\textstyle \operatorname {Gd} ^{-1}} , with gd {\textstyle \operatorname {gd} } and gd − 1 {\textstyle \operatorname {gd} ^{-1}}

8442-439: The construction of the reals from surreal numbers , since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield. The set of all real numbers is uncountable , in the sense that while both the set of all natural numbers {1, 2, 3, 4, ...} and the set of all real numbers are infinite sets , there exists no one-to-one function from

8568-462: The context of geodesy and navigation for latitude ϕ {\textstyle \phi } , k gd − 1 ⁡ ϕ {\displaystyle k\operatorname {gd} ^{-1}\phi } (scaled by arbitrary constant k {\textstyle k} ) was historically called the meridional part of ϕ {\displaystyle \phi } ( French : latitude croissante ). It

8694-652: The correctness of proofs of theorems involving real numbers. The realization that a better definition was needed, and the elaboration of such a definition was a major development of 19th-century mathematics and is the foundation of real analysis , the study of real functions and real-valued sequences . A current axiomatic definition is that real numbers form the unique ( up to an isomorphism ) Dedekind-complete ordered field . Other common definitions of real numbers include equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts , and infinite decimal representations . All these definitions satisfy

8820-417: The distance | x n − x | is less than ε for n greater than N . Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete. The set of rational numbers is not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), where each term adds a digit of the decimal expansion of

8946-420: The domain results in the same reflection in the codomain: The functions are periodic , with periods 2 π i {\textstyle 2\pi i} and 2 π {\textstyle 2\pi } : A translation in the domain of gd {\textstyle \operatorname {gd} } by ± π i {\textstyle \pm \pi i} results in

9072-487: The end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way. These two notions of completeness ignore

9198-515: The equal-area sinusoidal projection to show relative areas. However, despite such criticisms, the Mercator projection was, especially in the late 19th and early 20th centuries, perhaps the most common projection used in world maps. Atlases largely stopped using the Mercator projection for world maps or for areas distant from the equator in the 1940s, preferring other cylindrical projections , or forms of equal-area projection . The Mercator projection is, however, still commonly used for areas near

9324-458: The equator where distortion is minimal. It is also frequently found in maps of time zones. Arno Peters stirred controversy beginning in 1972 when he proposed what is now usually called the Gall–Peters projection to remedy the problems of the Mercator, claiming it to be his own original work without referencing prior work by cartographers such as Gall's work from 1855. The projection he promoted

9450-427: The field structure. However, an ordered group (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness ; the description in § Completeness is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces , since the definition of metric space relies on already having

9576-800: The first decimal representation, all a n {\displaystyle a_{n}} are zero for n > h , {\displaystyle n>h,} and, in the second representation, all a n {\displaystyle a_{n}} 9. (see 0.999... for details). In summary, there is a bijection between the real numbers and the decimal representations that do not end with infinitely many trailing 9. The preceding considerations apply directly for every numeral base B ≥ 2 , {\displaystyle B\geq 2,} simply by replacing 10 with B {\displaystyle B} and 9 with B − 1. {\displaystyle B-1.} A main reason for using real numbers

9702-561: The form of the Web Mercator projection . Today, the Mercator can be found in marine charts, occasional world maps, and Web mapping services, but commercial atlases have largely abandoned it, and wall maps of the world can be found in many alternative projections. Google Maps , which relied on it since 2005, still uses it for local-area maps but dropped the projection from desktop platforms in 2017 for maps that are zoomed out of local areas. Many other online mapping services still exclusively use

9828-418: The geometrical projection (as of light rays onto a screen) from the centre of the globe to the cylinder, which is only one of an unlimited number of ways to conceptually project a cylindrical map. Since the cylinder is tangential to the globe at the equator, the scale factor between globe and cylinder is unity on the equator but nowhere else. In particular since the radius of a parallel, or circle of latitude,

9954-431: The geometry of corresponding small elements on the globe and map. The figure below shows a point P at latitude  φ and longitude  λ on the globe and a nearby point Q at latitude φ  +  δφ and longitude λ  +  δλ . The vertical lines PK and MQ are arcs of meridians of length Rδφ . The horizontal lines PM and KQ are arcs of parallels of length R (cos  φ ) δλ . The corresponding points on

10080-405: The globe radius R . It is often convenient to work directly with the map width W  = 2 π R . For example, the basic transformation equations become The ordinate y of the Mercator projection becomes infinite at the poles and the map must be truncated at some latitude less than ninety degrees. This need not be done symmetrically. Mercator's original map is truncated at 80°N and 66°S with

10206-556: The identification of natural numbers with some real numbers is justified by the fact that Peano axioms are satisfied by these real numbers, with the addition with 1 taken as the successor function . Formally, one has an injective homomorphism of ordered monoids from the natural numbers N {\displaystyle \mathbb {N} } to the integers Z , {\displaystyle \mathbb {Z} ,} an injective homomorphism of ordered rings from Z {\displaystyle \mathbb {Z} } to

10332-436: The impossibility of determining the longitude at sea with adequate accuracy and the fact that magnetic directions, instead of geographical directions , were used in navigation. Only in the middle of the 18th century, after the marine chronometer was invented and the spatial distribution of magnetic declination was known, could the Mercator projection be fully adopted by navigators. Despite those position-finding limitations,

10458-461: The infinite strip | Re ⁡ w | ≤ 1 2 π {\textstyle \left|\operatorname {Re} w\right|\leq {\tfrac {1}{2}}\pi } to the infinite strip | Im ⁡ z | ≤ 1 2 π . {\textstyle \left|\operatorname {Im} z\right|\leq {\tfrac {1}{2}}\pi .} Analytically continued by reflections to

10584-451: The isotropy condition implies that h = k = sec φ . Consider a point on the globe of radius R with longitude λ and latitude φ . If φ is increased by an infinitesimal amount, dφ , the point moves R dφ along a meridian of the globe of radius R , so the corresponding change in y , dy , must be hR dφ = R  sec  φ dφ . Therefore y′ ( φ ) =  R  sec  φ . Similarly, increasing λ by dλ moves

10710-408: The least upper bound of the decimal fractions that are obtained by truncating the sequence: given a positive integer n , the truncation of the sequence at the place n is the finite partial sum The real number x defined by the sequence is the least upper bound of the D n , {\displaystyle D_{n},} which exists by Dedekind completeness. Conversely, given

10836-687: The limit at one end of the infinite strip): As the Gudermannian and inverse Gudermannian functions can be defined as the antiderivatives of the hyperbolic secant and circular secant functions, respectively, their derivatives are those secant functions: By combining hyperbolic and circular argument-addition identities, with the circular–hyperbolic identity , we have the Gudermannian argument-addition identities: Further argument-addition identities can be written in terms of other circular functions, but they require greater care in choosing branches in inverse functions. Notably, which can be used to derive

10962-416: The map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata : "A new and augmented description of Earth corrected for the use of sailors". This title, along with an elaborate explanation for using the projection that appears as a section of text on the map, shows that Mercator understood exactly what he had achieved and that he intended the projection to aid navigation. Mercator never explained

11088-410: The mathematical principle of the rhumb line or loxodrome, a path with constant bearing as measured relative to true north, which can be used in marine navigation to pick which compass bearing to follow. In 1537, he proposed constructing a nautical atlas composed of several large-scale sheets in the equirectangular projection as a way to minimize distortion of directions. If these sheets were brought to

11214-404: The maximum latitude attained must correspond to y  = ± ⁠ W / 2 ⁠ , or equivalently ⁠ y / R ⁠  =  π . Any of the inverse transformation formulae may be used to calculate the corresponding latitudes: The relations between y ( φ ) and properties of the projection, such as the transformation of angles and the variation in scale, follow from

11340-531: The method of construction or how he arrived at it. Various hypotheses have been tendered over the years, but in any case Mercator's friendship with Pedro Nunes and his access to the loxodromic tables Nunes created likely aided his efforts. English mathematician Edward Wright published the first accurate tables for constructing the projection in 1599 and, in more detail, in 1610, calling his treatise "Certaine Errors in Navigation". The first mathematical formulation

11466-605: The metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals form a contractible (hence connected and simply connected ), separable and complete metric space of Hausdorff dimension  1. The real numbers are locally compact but not compact . There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to

11592-654: The numbers E k {\textstyle E_{k}} are the Euler secant numbers , 1, 0, -1, 0, 5, 0, -61, 0, 1385 ... (sequences A122045 , A000364 , and A028296 in the OEIS ). These series were first computed by James Gregory in 1671. Because the Gudermannian and inverse Gudermannian functions are the integrals of the hyperbolic secant and secant functions, the numerators E k {\textstyle E_{k}} and | E k | {\textstyle |E_{k}|} are same as

11718-480: The numerators of the Taylor series for sech and sec , respectively, but shifted by one place. The reduced unsigned numerators are 1, 1, 1, 61, 277, ... and the reduced denominators are 1, 6, 24, 5040, 72576, ... (sequences A091912 and A136606 in the OEIS ). The function and its inverse are related to the Mercator projection . The vertical coordinate in the Mercator projection is called isometric latitude , and

11844-729: The numerically well-behaved formulas (Note, for | ϕ | > 1 2 π {\displaystyle |\phi |>{\tfrac {1}{2}}\pi } and for complex arguments, care must be taken choosing branches of the inverse functions.) We can also express ψ {\textstyle \psi } and ϕ {\textstyle \phi } in terms of s : {\textstyle s\colon } If we expand tan ⁡ 1 2 {\textstyle \tan {\tfrac {1}{2}}} and tanh ⁡ 1 2 {\textstyle \tanh {\tfrac {1}{2}}} in terms of

11970-418: The oblique Mercator in order to keep scale variation low along the surface projection of the cylinder's axis. Although the surface of Earth is best modelled by an oblate ellipsoid of revolution , for small scale maps the ellipsoid is approximated by a sphere of radius a , where a is approximately 6,371 km. This spherical approximation of Earth can be modelled by a smaller sphere of radius R , called

12096-464: The phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way. But the original use of the phrase "complete Archimedean field" was by David Hilbert , who meant still something else by it. He meant that

12222-435: The point R cos φ dλ along a parallel of the globe, so dx = kR cos φ dλ = R dλ . That is, x′ ( λ ) =  R . Integrating the equations with x ( λ 0 ) = 0 and y (0) = 0, gives x(λ) and y(φ) . The value λ 0 is the longitude of an arbitrary central meridian that is usually, but not always, that of Greenwich (i.e., zero). The angles λ and φ are expressed in radians. By

12348-464: The point scale factor is independent of direction, so that small shapes are preserved by the projection. This implies that the vertical scale factor, h , equals the horizontal scale factor, k . Since k = sec φ , so must h . The graph shows the variation of this scale factor with latitude. Some numerical values are listed below. The area scale factor is the product of the parallel and meridian scales hk = sec φ . For Greenland, taking 73° as

12474-428: The polar regions. The criticisms leveled against inappropriate use of the Mercator projection resulted in a flurry of new inventions in the late 19th and early 20th century, often directly touted as alternatives to the Mercator. Due to these pressures, publishers gradually reduced their use of the projection over the course of the 20th century. However, the advent of Web mapping gave the projection an abrupt resurgence in

12600-399: The positive square root of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positive square root of 2). The completeness property of the reals is the basis on which calculus , and more generally mathematical analysis , are built. In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has

12726-492: The rational numbers Q , {\displaystyle \mathbb {Q} ,} and an injective homomorphism of ordered fields from Q {\displaystyle \mathbb {Q} } to the real numbers R . {\displaystyle \mathbb {R} .} The identifications consist of not distinguishing the source and the image of each injective homomorphism, and thus to write These identifications are formally abuses of notation (since, formally,

12852-533: The rational numbers an ordered subfield of the real numbers R . {\displaystyle \mathbb {R} .} The Dedekind completeness described below implies that some real numbers, such as 2 , {\displaystyle {\sqrt {2}},} are not rational numbers; they are called irrational numbers . The above identifications make sense, since natural numbers, integers and real numbers are generally not defined by their individual nature, but by defining properties ( axioms ). So,

12978-464: The real number identified with n . {\displaystyle n.} Similarly a rational number p / q {\displaystyle p/q} (where p and q are integers and q ≠ 0 {\displaystyle q\neq 0} ) is identified with the division of the real numbers identified with p and q . These identifications make the set Q {\displaystyle \mathbb {Q} } of

13104-436: The real numbers form a real closed field . This implies the real version of the fundamental theorem of algebra , namely that every polynomial with real coefficients can be factored into polynomials with real coefficients of degree at most two. The most common way of describing a real number is via its decimal representation , a sequence of decimal digits each representing the product of an integer between zero and nine times

13230-417: The real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R {\displaystyle \mathbb {R} } . Thus R {\displaystyle \mathbb {R} } is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to

13356-429: The real numbers to the natural numbers. The cardinality of the set of all real numbers is denoted by c . {\displaystyle {\mathfrak {c}}.} and called the cardinality of the continuum . It is strictly greater than the cardinality of the set of all natural numbers (denoted ℵ 0 {\displaystyle \aleph _{0}} and called 'aleph-naught' ), and equals

13482-403: The reals. Mercator projection The Mercator projection ( / m ər ˈ k eɪ t ər / ) is a conformal cylindrical map projection first presented by Flemish geographer and mapmaker Gerardus Mercator in 1569. In the 18th century, it became the standard map projection for navigation due to its property of representing rhumb lines as straight lines. When applied to world maps,

13608-463: The relation between geodetic latitude and isometric latitude is slightly more complicated.) Gerardus Mercator plotted his celebrated map in 1569, but the precise method of construction was not revealed. In 1599, Edward Wright described a method for constructing a Mercator projection numerically from trigonometric tables, but did not produce a closed formula. The closed formula was published in 1668 by James Gregory . The Gudermannian function per se

13734-409: The result that European countries were moved toward the centre of the map. The aspect ratio of his map is ⁠ 198 / 120 ⁠ = 1.65. Even more extreme truncations have been used: a Finnish school atlas was truncated at approximately 76°N and 56°S, an aspect ratio of 1.97. Much Web-based mapping uses a zoomable version of the Mercator projection with an aspect ratio of one. In this case

13860-410: The resulting flat map, as a final step, any pair of circles parallel to and equidistant from the contact circle can be chosen to have their scale preserved, called the standard parallels ; then the region between chosen circles will have its scale smaller than on the sphere, reaching a minimum at the contact circle. This is sometimes visualized as a projection onto a cylinder which is secant to (cuts)

13986-496: The resulting sequence of digits is called a decimal representation of x . Another decimal representation can be obtained by replacing ≤ x {\displaystyle \leq x} with < x {\displaystyle <x} in the preceding construction. These two representations are identical, unless x is a decimal fraction of the form m 10 h . {\textstyle {\frac {m}{10^{h}}}.} In this case, in

14112-455: The rhumb and the great circle course is negligible. Even for longer distances, the simplicity of the constant bearing makes it attractive. As observed by Mercator, on such a course, the ship would not arrive by the shortest route, but it will surely arrive. Sailing a rhumb meant that all that the sailors had to do was keep a constant course as long as they knew where they were when they started, where they intended to be when they finished, and had

14238-469: The right of the second equation defines the Gudermannian function ; i.e., φ  = gd( ⁠ y / R ⁠ ): the direct equation may therefore be written as y  =  R ·gd ( φ ). There are many alternative expressions for y ( φ ), all derived by elementary manipulations. Corresponding inverses are: For angles expressed in degrees: The above formulae are written in terms of

14364-425: The same cardinality as the reals. The real numbers form a metric space : the distance between x and y is defined as the absolute value | x − y | . By virtue of being a totally ordered set, they also carry an order topology ; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in

14490-416: The same scale and assembled, they would approximate the Mercator projection. In 1541, Flemish geographer and mapmaker Gerardus Mercator included a network of rhumb lines on a terrestrial globe he made for Nicolas Perrenot . In 1569, Mercator announced a new projection by publishing a large world map measuring 202 by 124 cm (80 by 49 in) and printed in eighteen separate sheets. Mercator titled

14616-463: The scholarly consensus, they have been speculated to have originated in some unknown pre-medieval cartographic tradition, possibly evidence of some ancient understanding of the Mercator projection. German polymath Erhard Etzlaub engraved miniature "compass maps" (about 10×8 cm) of Europe and parts of Africa that spanned latitudes 0°–67° to allow adjustment of his portable pocket-size sundials . The projection found on these maps, dating to 1511,

14742-431: The sphere onto a tangent cylinder along straight radial lines, as if from a light source placed at the Earth's center. Both have extreme distortion far from the equator and cannot show the poles. However, they are different projections and have different properties. As with all map projections , the shapes or sizes are distortions of the true layout of the Earth's surface. The Mercator projection exaggerates areas far from

14868-405: The sphere, though this picture is misleading insofar as the standard parallels are not spaced the same distance apart on the map as the shortest distance between them through the interior of the sphere. The original and most common aspect of the Mercator projection for maps of the Earth is the normal aspect, for which the axis of the cylinder is the Earth's axis of rotation which passes through

14994-430: The trigonometric table. The Gudermannian function can be thought of mapping points on one branch of a hyperbola to points on a semicircle. Points on one sheet of an n -dimensional hyperboloid of two sheets can be likewise mapped onto a n -dimensional hemisphere via stereographic projection. The hemisphere model of hyperbolic space uses such a map to represent hyperbolic space. Real number In mathematics ,

15120-568: The whole complex plane, w ↦ z = gd − 1 ⁡ w {\textstyle w\mapsto z=\operatorname {gd} ^{-1}w} is a periodic function of period 2 π {\textstyle 2\pi } which sends any infinite strip of "width" 2 π {\textstyle 2\pi } onto the strip − π < Im ⁡ z ≤ π . {\textstyle -\pi <\operatorname {Im} z\leq \pi .} For all points in

15246-554: The whole complex plane, z ↦ w = gd ⁡ z {\textstyle z\mapsto w=\operatorname {gd} z} is a periodic function of period 2 π i {\textstyle 2\pi i} which sends any infinite strip of "height" 2 π i {\textstyle 2\pi i} onto the strip − π < Re ⁡ w ≤ π . {\textstyle -\pi <\operatorname {Re} w\leq \pi .} Likewise, extended to

15372-522: The world use the transverse Mercator, as does the Universal Transverse Mercator coordinate system . An oblique Mercator projection tilts the cylinder axis away from the Earth's axis to an angle of one's choosing, so that its tangent or secant lines of contact are circles that are also tilted relative to the Earth's parallels of latitude. Practical uses for the oblique projection, such as national grid systems, use ellipsoidal developments of

15498-470: Was introduced by Johann Heinrich Lambert in the 1760s at the same time as the hyperbolic functions . He called it the "transcendent angle", and it went by various names until 1862 when Arthur Cayley suggested it be given its current name as a tribute to Christoph Gudermann 's work in the 1830s on the theory of special functions. Gudermann had published articles in Crelle's Journal that were later collected in

15624-664: Was introduced by Cayley who starts by calling ϕ = gd ⁡ u {\textstyle \phi =\operatorname {gd} u} the Jacobi elliptic amplitude am ⁡ u {\textstyle \operatorname {am} u} in the degenerate case where the elliptic modulus is m = 1 , {\textstyle m=1,} so that 1 + m sin 2 ϕ {\textstyle {\sqrt {1+m\sin \!^{2}\,\phi }}} reduces to cos ⁡ ϕ . {\textstyle \cos \phi .} This

15750-538: Was publicized around 1645 by a mathematician named Henry Bond ( c.  1600 –1678). However, the mathematics involved were developed but never published by mathematician Thomas Harriot starting around 1589. The development of the Mercator projection represented a major breakthrough in the nautical cartography of the 16th century. However, it was much ahead of its time, since the old navigational and surveying techniques were not compatible with its use in navigation. Two main problems prevented its immediate application:

15876-424: Was stated by John Snyder in 1987 to be the same projection as Mercator's. However, given the geometry of a sundial, these maps may well have been based on the similar central cylindrical projection , a limiting case of the gnomonic projection , which is the basis for a sundial. Snyder amended his assessment to "a similar projection" in 1993. Portuguese mathematician and cosmographer Pedro Nunes first described

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