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Compass-and-straightedge construction
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Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.
The idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it with only one edge. The compass is assumed to collapse when lifted from the page, so may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with collapsing compass; see compass equivalence theorem.) More formally, the only permissible constructions are those granted by Euclid's first three postulates.
It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone.
The ancient Greek mathematicians first conceived compass-and-straightedge constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields.
In spite of existing proofs of impossibility, some persist in trying to solve these problems.[1] Many of these problems are easily solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometric constructions, but not possible using straightedge and compass alone.
In terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots.
Contents
[hide]
1 Compass and straightedge tools
2 History
3 The basic constructions
4 Much used compass-and-straightedge constructions
5 Constructible points and lengths
5.1 Formal proof
6 Constructible angles
7 Compass and straightedge constructions as complex arithmetic
8 Impossible constructions
8.1 Squaring the circle
8.2 Doubling the cube
8.3 Angle trisection
9 Constructing regular polygons
10 Constructing a triangle from three given characteristic points or lengths
11 Distance to an ellipse
12 Constructing with only ruler or only compass
13 Extended constructions
13.1 Solid constructions
13.2 Angle trisection
13.3 Origami
13.4 Markable rulers
14 Computation of binary digits
15 See also
16 References
17 External links
Compass and straightedge tools[edit]
Compass and straightedge
A compass
The "compass" and "straightedge" of compass and straightedge constructions are idealizations of rulers and compasses in the real world:
The compass can be opened arbitrarily wide, but (unlike some real compasses) it has no markings on it. Circles can only be drawn starting from two given points: the centre and a point on the circle. The compass may or may not collapse when it's not drawing a circle.
The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to draw a line segment between two points or to extend an existing segment.
The modern compass generally does not collapse and several modern constructions use this feature. It would appear that the modern compass is a "more powerful" instrument than the ancient collapsing compass. However, by Proposition 2 of Book 1 of Euclid's Elements, no power is lost by using a collapsing compass. Although the proposition is correct, its proofs have a long and checkered history.[2]
Each construction must be exact. "Eyeballing" it (essentially looking at the construction and guessing at its accuracy, or using some form of measurement, such as the units of measure on a ruler) and getting close does not count as a solution.
Each construction must terminate. That is, it must have a finite number of steps, and not be the limit of ever closer approximations.
Stated this way, compass and straightedge constructions appear to be a parlour game, rather than a serious practical problem; but the purpose of the restriction is to ensure that constructions can be proven to be exactly correct.
History[edit]
The ancient Greek mathematicians first attempted compass-and-straightedge constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths.[3]:p. 1 They could also construct half of a given angle, a square whose area is twice that of another square, a square having the same area as a given polygon, and a regular polygon with 3, 4, or 5 sides[3]:p. xi (or one with twice the number of sides of a given polygon[3]:pp. 49–50). But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle, or a regular polygon with other numbers of sides.[3]:p. xi Nor could they construct the side of a cube whose volume would be twice the volume of a cube with a given side.[3]:p. 29
Hippocrates and Menaechmus showed that the area of the cube could be doubled by finding the intersections of hyperbolas and parabolas, but these cannot be constructed by compass and straightedge.[3]:p. 30 In the fifth century BCE, Hippias used a curve that he called a quadratrix to both trisect the general angle and square the circle, and Nicomedes in the second century BCE showed how to use a conchoid to trisect an arbitrary angle;[3]:p. 37 but these methods also cannot be followed with just compass and straightedge.
No progress on the unsolved problems was made for two millennia, until in 1796 Gauss showed that a regular polygon with 17 sides could be constructed; five years later he showed the sufficient criterion for a regular polygon of n sides to be constructible.[3]:pp. 51 ff.
In 1837 Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle or of doubling the volume of a cube, based on the impossibility of constructing cube roots of lengths.[4] He also showed that Gauss's sufficient constructibility condition for regular polygons is also necessary.
Then in 1882 Lindemann showed that π {\displaystyle \pi } is a transcendental number, and thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle.[3]:p. 47
The basic constructions[edit]
The basic constructions
All compass and straightedge constructions consist of repeated application of five basic constructions using the points, lines and circles that have already been constructed. These are:
Creating the line through two existing points
Creating the circle through one point with centre another point
Creating the point which is the intersection of two existing, non-parallel lines
Creating the one or two points in the intersection of a line and a circle (if they intersect)
Creating the one or two points in the intersection of two circles (if they intersect).
For example, starting with just two distinct points, we can create a line or either of two circles (in turn, using each point as centre and passing through the other point). If we draw both circles, two new points are created at their intersections. Drawing lines between the two original points and one of these new points completes the construction of an equilateral triangle.
Therefore, in any geometric problem we have an initial set of symbols (points and lines), an algorithm, and some results. From this perspective, geometry is equivalent to an axiomatic algebra, replacing its elements by symbols. Probably Gauss first realized this, and used it to prove the impossibility of some constructions; only much later did Hilbert find a complete set of axioms for geometry.
Much used compass-and-straightedge constructions[edit]
The most-used compass-and-straightedge constructions include:
Constructing the perpendicular bisector from a segment
Finding the midpoint of a segment.
Drawing a perpendicular line from a point to a line.
Bisecting an angle
Mirroring a point in a line
Constructing a line through a point tangent to a circle
Constructing a circle through 3 noncollinear points
Constructible points and lengths[edit]
Formal proof[edit]
There are many different ways to prove something is impossible. A more rigorous proof would be to demarcate the limit of the possible, and show that to solve these problems one must transgress that limit. Much of what can be constructed is covered in intercept theory.
We could associate an algebra to our geometry using a Cartesian coordinate system made of two lines, and represent points of our plane by vectors. Finally we can write these vectors as complex numbers.
Using the equations for lines and circles, one can show that the points at which they intersect lie in a quadratic extension of the smallest field F containing two points on the line, the center of the circle, and the radius of the circle. That is, they are of the form x +y√k, where x, y, and k are in F.
Since the field of constructible points is closed under square roots, it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients. By the above paragraph, one can show that any constructible point can be obtained by such a sequence of extensions. As a corollary of this, one finds that the degree of the minimal polynomial for a constructible point (and therefore of any constructible length) is a power of 2. In particular, any constructible point (or length) is an algebraic number, though not every algebraic number is constructible; for example, 3√2 is algebraic but not constructible.
Constructible angles[edit]
There is a bijection between the angles that are constructible and the points that are constructible on any constructible circle. The angles that are constructible form an abelian group under addition modulo 2π (which corresponds to multiplication of the points on the unit circle viewed as complex numbers). The angles that are constructible are exactly those whose tangent (or equivalently, sine or cosine) is constructible as a number. For example, the regular heptadecagon (the seventeen-sided regular polygon) is constructible because
cos  ( 2 π 17 ) = − 1 16 + 1 16 17 + 1 16 34 − 2 17 + 1 8 17 + 3 17 − 34 − 2 17 − 2 34 + 2 17 {\displaystyle \cos {\left({\frac {2\pi }{17}}\right)}=-{\frac {1}{16}}\;+\;{\frac {1}{16}}{\sqrt {17}}\;+\;{\frac {1}{16}}{\sqrt {34-2{\sqrt {17}}}}\;+\;{\frac {1}{8}}{\sqrt {17+3{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {34+2{\sqrt {17}}}}}}}
as discovered by Gauss.[5]
The group of constructible angles is closed under the operation that halves angles (which corresponds to taking square roots in the complex numbers). The only angles of finite order that may be constructed starting with two points are those whose order is either a power of two, or a product of a power of two and a set of distinct Fermat primes. In addition there is a dense set of constructible angles of infinite order.
Compass and straightedge constructions as complex arithmetic[edit]
Given a set of points in the Euclidean plane, selecting any one of them to be called 0 and another to be called 1, together with an arbitrary choice of orientation allows us to consider the points as a set of complex numbers.
Given any such interpretation of a set of points as complex numbers, the points constructible using valid compass and straightedge constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations (to avoid ambiguity, we can specify the square root with complex argument less than π). The elements of this field are precisely those that may be expressed as a formula in the original points using only the operations of addition, subtraction, multiplication, division, complex conjugate, and square root, which is easily seen to be a countable dense subset of the plane. Each of these six operations corresponding to a simple compass and straightedge construction. From such a formula it is straightforward to produce a construction of the corresponding point by combining the constructions for each of the arithmetic operations. More efficient constructions of a particular set of points correspond to shortcuts in such calculations.
Equivalently (and with no need to arbitrarily choose two points) we can say that, given an arbitrary choice of orientation, a set of points determines a set of complex ratios given by the ratios of the differences between any two pairs of points. The set of ratios constructible using compass and straightedge from such a set of ratios is precisely the smallest field containing the original ratios and closed under taking complex conjugates and square roots.
For example, the real part, imaginary part and modulus of a point or ratio z (taking one of the two viewpoints above) are constructible as these may be expressed as
R e ( z ) = z + z ¯ 2 {\displaystyle \mathrm {Re} (z)={\frac {z+{\bar {z}}}{2}}\;}
I m ( z ) = z − z ¯ 2 i {\displaystyle \mathrm {Im} (z)={\frac {z-{\bar {z}}}{2i}}\;}
| z | = z z ¯ . {\displaystyle \left|z\right|={\sqrt {z{\bar {z}}}}.\;}
Doubling the cube and trisection of an angle (except for special angles such as any φ such that φ/2π is a rational number with denominator not divisible by 3) require ratios which are the solution to cubic equations, while squaring the circle requires a transcendental ratio. None of these are in the fields described, hence no compass and straightedge construction for these exists.
Impossible constructions[edit]
The ancient Greeks thought that the construction problems they could not solve were simply obstinate, not unsolvable.[6] With modern methods, however, these compass-and-straightedge constructions have been shown to be logically impossible to perform. (The problems themselves, however, are solvable, and the Greeks knew how to solve them, without the constraint of working only with straightedge and compass.)
Squaring the circle[edit]
Main article: Squaring the circle
The most famous of these problems, squaring the circle, otherwise known as the quadrature of the circle, involves constructing a square with the same area as a given circle using only straightedge and compass.
Squaring the circle has been proven impossible, as it involves generating a transcendental number, that is, √π. Only certain algebraic numbers can be constructed with ruler and compass alone, namely those constructed from the integers with a finite sequence of operations of addition, subtraction, multiplication, division, and taking square roots. The phrase "squaring the circle" is often used to mean "doing the impossible" for this reason.
Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and was solved many times in antiquity.[7]
A method which comes very close to approximating the "quadrature of the circle" can be achieved using a Kepler triangle.
Doubling the cube[edit]
Main article: Doubling the cube
Doubling the cube is the construction, using only a straight-edge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots. This follows because its minimal polynomial over the rationals has degree 3. This construction is possible using a straightedge with two marks on it and a compass.
Angle trisection[edit]
Main article: Angle trisection
Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, though the angle of π/3 radians (60°) cannot be trisected, the angle 2π/5 radians (72° = 360°/5) can be trisected.[8] The general trisection problem is also easily solved when a straightedge with two marks on it is allowed (a neusis construction).
Constructing regular polygons[edit]
Main article: Constructible polygon
Construction of a regular pentagon
Some regular polygons (e.g. a pentagon) are easy to construct with straightedge and compass; others are not. This led to the question: Is it possible to construct all regular polygons with straightedge and compass?
Carl Friedrich Gauss in 1796 showed that a regular 17-sided polygon can be constructed, and five years later showed that a regular n-sided polygon can be constructed with straightedge and compass if the odd prime factors of n are distinct Fermat primes. Gauss conjectured that this condition was also necessary, but he offered no proof of this fact, which was provided by Pierre Wantzel in 1837.[9]
The first few constructible regular polygons have the following numbers of sides:
3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272... (sequence A003401 in the OEIS)
There are known to be an infinitude of constructible regular polygons with an even number of sides (because if a regular n-gon is constructible, then so is a regular 2n-gon and hence a regular 4n-gon, 8n-gon, etc.). However, there are only 31 known constructible regular n-gons with an odd number of sides.
Constructing a triangle from three given characteristic points or lengths[edit]
Sixteen key points of a triangle are its vertices, the midpoints of its sides, the feet of its altitudes, the feet of its internal angle bisectors, and its circumcenter, centroid, orthocenter, and incenter. These can be taken three at a time to yield 139 distinct nontrivial problems of constructing a triangle from three points.[10] Of these problems, three involve a point that can be uniquely constructed from the other two points; 23 can be non-uniquely constructed (in fact for infinitely many solutions) but only if the locations of the points obey certain constraints; in 74 the problem is constructible in the general case; and in 39 the required triangle exists but is not constructible.
Twelve key lengths of a triangle are the three side lengths, the three altitudes, the three medians, and the three angle bisectors. Together with the three angles, these give 95 distinct combinations, 63 of which give rise to a constructible triangle, 30 of which do not, and two of which are underdefined.[11]:pp. 201–203
Distance to an ellipse[edit]
The line segment from any point in the plane to the nearest point on a circle can be constructed, but the segment from any point in the plane to the nearest point on an ellipse of positive eccentricity cannot in general be constructed.[12]
Constructing with only ruler or only compass[edit]
It is possible (according to the Mohr–Mascheroni theorem) to construct anything with just a compass if it can be constructed with a ruler and compass, provided that the given data and the data to be found consist of discrete points (not lines or circles). It should be noted that the truth of this theorem depends on the truth of Archimedes' axiom,[13] which is not first-order in nature. It is impossible to take a square root with just a ruler, so some things that cannot be constructed with a ruler can be constructed with a compass; but (by the Poncelet–Steiner theorem) given a single circle and its center, they can be constructed.
Extended constructions[edit]
The ancient Greeks classified constructions into three major categories, depending on the complexity of the tools required for their solution. If a construction used only a straightedge and compass, it was called planar; if it also required one or more conic sections (other than the circle), then it was called solid; the third category included all constructions that did not fall into either of the other two categories.[14] This categorization meshes nicely with our modern algebraic point of view. A complex number that can be expressed using only the field operations and square roots (as described above) has a planar construction. A complex number that includes also the extraction of cube roots has a solid construction.
In the language of fields, a complex number that is planar has degree a power of two, and lies in a field extension that can be broken down into a tower of fields where each extension has degree two. A complex number that has a solid construction has degree with prime factors of only two and three, and lies in a field extension that is at the top of a tower of fields where each extension has degree 2 or 3.
Solid constructions[edit]
A point has a solid construction if it can be constructed using a straightedge, compass, and a (possibly hypothetical) conic drawing tool that can draw any conic with already constructed focus, directrix, and eccentricity. The same set of points can often be constructed using a smaller set of tools. For example, using a compass, straightedge, and a piece of paper on which we have the parabola y=x2 together with the points (0,0) and (1,0), one can construct any complex number that has a solid construction. Likewise, a tool that can draw any ellipse with already constructed foci and major axis (think two pins and a piece of string) is just as powerful.[15]
The ancient Greeks knew that doubling the cube and trisecting an arbitrary angle both had solid constructions. Archimedes gave a solid construction of the regular 7-gon. The quadrature of the circle does not have a solid construction.
A regular n-gon has a solid construction if and only if n=2j3km where m is a product of distinct Pierpont primes (primes of the form 2r3s+1). The set of such n is the sequence
7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 35, 36, 37, 38, 39, 42, 45, 52, 54, 56, 57, 63, 65, 70, 72, 73, 74, 76, 78, 81, 84, 90, 91, 95, 97... (sequence A051913 in the OEIS)
The set of n for which a regular n-gon has no solid construction is the sequence
11, 22, 23, 25, 29, 31, 33, 41, 43, 44, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 66, 67, 69, 71, 75, 77, 79, 82, 83, 86, 87, 88, 89, 92, 93, 94, 98, 99, 100... (sequence A048136 in the OEIS)
Like the question with Fermat primes, it is an open question as to whether there are an infinite number of Pierpont primes.
Angle trisection[edit]
What if, together with the straightedge and compass, we had a tool that could (only) trisect an arbitrary angle? Such constructions are solid constructions, but there exist numbers with solid constructions that cannot be constructed using such a tool. For example, we cannot double the cube with such a tool.[16] On the other hand, every regular n-gon that has a solid construction can be constructed using such a tool.
Origami[edit]
Main article: Huzita–Hatori axioms
The mathematical theory of origami is more powerful than compass and straightedge construction. Folds satisfying the Huzita–Hatori axioms can construct exactly the same set of points as the extended constructions using a compass and conic drawing tool. Therefore, origami can also be used to solve cubic equations (and hence quartic equations), and thus solve two of the classical problems.[17]
Markable rulers[edit]
Main article: Neusis construction
Archimedes, Nicomedes and Apollonius gave constructions involving the use of a markable ruler. This would permit them, for example, to take a line segment, two lines (or circles), and a point; and then draw a line which passes through the given point and intersects both lines, and such that the distance between the points of intersection equals the given segment. This the Greeks called neusis ("inclination", "tendency" or "verging"), because the new line tends to the point. In this expanded scheme, we can trisect an arbitrary angle (see Archimedes' trisection) or extract an arbitrary cube root (due to Nicomedes). Hence, any distance whose ratio to an existing distance is the solution of a cubic or a quartic equation is constructible. Regular polygons with solid constructions, like the heptagon, are constructible; and John H. Conway and Richard K. Guy give constructions for several of them;.[18]
The neusis construction is more powerful than a conic drawing tool, as one can construct complex numbers that do not have solid constructions. In fact, using this tool one can solve some quintics that are not solvable using radicals.[19] It is known that one cannot solve an irreducible polynomial of prime degree greater or equal to 7 using the neusis construction, so it is not possible to construct a regular 23-gon or 29-gon using this tool. Benjamin and Snyder proved that it is possible to construct the regular 11-gon, but did not give a construction.[20] It is still open as to whether a regular 25-gon or 31-gon is constructible using this tool.
Computation of binary digits[edit]
In 1998 Simon Plouffe gave a ruler and compass algorithm that can be used to compute binary digits of certain numbers.[21] The algorithm involves the repeated doubling of an angle and becomes physically impractical after about 20 binary digits.
See also[edit]
Carlyle circle
Geometric cryptography
Geometrography
List of interactive geometry software, most of them show compass and straightedge constructions
Underwood Dudley, a mathematician who has made a sideline of collecting false ruler-and-compass proofs.
References[edit]
Jump up ^ Underwood Dudley (1983), "What To Do When the Trisector Comes" (PDF), The Mathematical Intelligencer, 5 (1): 20–25, doi:10.1007/bf03023502
Jump up ^ Godfried Toussaint, "A new look at Euclid’s second proposition," The Mathematical Intelligencer, Vol. 15, No. 3, (1993), pp. 12-24.
^ Jump up to: a b c d e f g h i Bold, Benjamin. Famous Problems of Geometry and How to Solve Them, Dover Publications, 1982 (orig. 1969).
Jump up ^ Wantzel, P M L (1837). "Recherches sur les moyens de reconnaître si un problème de Géométrie peut se résoudre avec la règle et le compas" (PDF). Journal de Mathématiques Pures et Appliquées. 1. 2: 366–372. Retrieved 3 March 2014.
Jump up ^ Weisstein, Eric W. "Trigonometry Angles--Pi/17". MathWorld.
Jump up ^ Stewart, Ian. Galois Theory. p. 75.
Jump up ^ *Squaring the circle at MacTutor
Jump up ^ Instructions for trisecting a 72˚ angle.
Jump up ^ Kazarinoff, Nicholas D. (2003). Ruler and the Round. Mineola, N.Y.: Dover. pp. 29–30. ISBN 0-486-42515-0.
Jump up ^ Pascal Schreck, Pascal Mathis, Vesna Marinkoviċ, and Predrag Janičiċ. "Wernick's list: A final update", Forum Geometricorum 16, 2016, pp. 69–80. http://forumgeom.fau.edu/FG2016volume16/FG201610.pdf
Jump up ^ Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
Jump up ^ Azad, H., and Laradji, A., "Some impossible constructions in elementary geometry", Mathematical Gazette 88, November 2004, 548–551.
Jump up ^ Arnon Avron, "On strict strong constructibility with a compass alone", Journal of Geometry (1990) 38: 12.
Jump up ^ T.L. Heath, "A History of Greek Mathematics, Volume I"
Jump up ^ P. Hummel, "Solid constructions using ellipses", The Pi Mu Epsilon Journal, 11(8), 429 -- 435 (2003)
Jump up ^ Gleason, Andrew: "Angle trisection, the heptagon, and the triskaidecagon", Amer. Math. Monthly 95 (1988), no. 3, 185-194.
Jump up ^ Row, T. Sundara (1966). Geometric Exercises in Paper Folding. New York: Dover.
Jump up ^ Conway, John H. and Richard Guy: The Book of Numbers
Jump up ^ A. Baragar, "Constructions using a Twice-Notched Straightedge", The American Mathematical Monthly, 109 (2), 151 -- 164 (2002).
Jump up ^ E. Benjamin, C. Snyder, "On the construction of the regular hendecagon by marked ruler and compass", Mathematical Proceedings of the Cambridge Philosophical Society, 156 (3), 409 -- 424 (2014).
Jump up ^ Simon Plouffe (1998). "The Computation of Certain Numbers Using a Ruler and Compass". Journal of Integer Sequences. 1. ISSN 1530-7638.
External links[edit]
Regular polygon constructions by Dr. Math at The Math Forum @ Drexel
Construction with the Compass Only at cut-the-knot
Angle Trisection by Hippocrates at cut-the-knot
Weisstein, Eric W. "Angle Trisection". MathWorld.
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National Financial Switch
From Wikipedia, the free encyclopedia
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National Financial Switch
Operating area
India
Members
90 Direct Members
228 Sub-Members
52 RRBs
7 WLAOs
ATMs
188600
Founded
27 August 2004
Owner
NPCI
National Financial Switch (NFS) is the largest network of shared automated teller machines (ATMs) in India.[1] It was designed, developed and deployed by the Institute for Development and Research in Banking Technology (IDRBT) in 2004, with the goal of inter-connecting the ATMs in the country and facilitating convenience banking. It is run by the National Payments Corporation of India (NPCI).
Contents
1 Background
2 History
3 Member banks
4 Services offered in NFS
5 References
6 External links
Background[edit]
The first ATM in India was set up in 1987 by HSBC in Mumbai.[2] In the following twelve years, about 1500 ATMs were set up in India. In 1997, the Indian Banks' Association (IBA) set up Swadhan, the first network of shared ATMs in India. It was managed by India Switch Company (ISC) for five years, and allowed cardholders to withdraw cash from any ATM in the network, for a fee if they did not have an account with the bank that owned the ATM. In 2002, the network connected over 1000 ATMs of the 53 member banks of the association. The network was capable of handling 250,000 transactions per day, but only 5000 transactions, worth about ₹100,000, took place each day. In contrast, ICICI Bank's network of about 640 ATMs handled transactions worth about ₹20,000,000 each day. After the contract with ISC expired, IBA failed to find a bidder to manage the operationally uneconomical network, and shut it down on 31 December 2003.
After the collapse of Swadhan, Bank of India, Union Bank of India, Indian Bank, United Bank of India and Syndicate Bank formed an ATM-sharing network called CashTree. Citibank, the Industrial Development Bank of India, Standard Chartered Bank and Axis Bank formed a similar network called Cashnet. Punjab National Bank and Canara Bank also created such networks.
In August 2003, the IDRBT announced that it would be creating the National Financial Switch (NFS) to link together the country's ATMs in a single network.
The IDRBT collaborated with Euronet Worldwide and Opus Software to build a platform to allow banks to connect their own switches to the NFS. The NFS consisted of an inter-ATM switch and a e-commerce payment gateway.
History[edit]
The National Financial Switch was launched by the IDRBT on 27 August 2004, connecting the ATMs of three banks, Corporation Bank, Bank of Baroda and ICICI Bank.[3][4][5] The IDRBT then worked towards bringing all major banks in India on board and by December 2009, the network had grown to connect 49,880 ATMs of 37 banks, thereby emerging as the largest network of shared ATMs in the country.
IDRBT decided to hive off its operational role on ATM switching to refocus on research and development, and was sought to shift the business to a national-level payment system organization. The National Payments Corporation of India (NPCI) started discussions with IDRBT on the feasibility of taking over. The Board for Regulation and Supervision of Payment and Settlement Systems (BPSS) at its meeting held on 24 September 2009 approved in-principle to issue authorisation to NPCI for operating various retail payment systems in the country. The Reserve Bank of India granted authorisation to NPCI to take over the operations of National Financial Switch (NFS) from the Institute of Development and Research in Banking Technology (IDRBT) on a ‘as is where is basis’ on 15 October 2009. NPCI deputed its officials to IDRBT Hyderabad, and the Institute handed over the National Financial Switch to the NPCI on 14 December 2009.
Any bank that provides core banking services with 24x7 transaction banking capabilities with or without ATMs may join the National Financial Switch through a sponsor bank. This allows non-scheduled Urban Co-operative Banks (UCBs) and Regional Rural Banks (RRBs) to gain access to the national network of over 103,000 ATMs in the country. Before 14 August 2011, access was limited to scheduled banks with RTGS membership.[6] The sponsorship scheme was started to increase the connectivity of ATMs across the country, and to enable customers to use ATMs across India.
The primary headquarters is located at Mumbai.
Member banks[edit]
As of April 2017, the NFS Network connects total of 2,36,190 ATMs in India. Among them 2,16,952 ATMs of 99 Direct Member banks, 4,058 ATMs of 692 Sub Member banks, 1,034 ATMs of 56 RRB Member banks and 14,146 ATMs of 8 White Label ATM providers, which is the largest number of ATMs under a single network in India.[7]
Services offered in NFS[edit]
NFS which is the largest domestic ATM network in the country member banks has been in the fore front in providing inter bank ATM services to maximum customers. Initially, the following basic transactions were available in the NFS network
Cash Withdrawal
Balance Enquiry
PIN Change
Mini Statement
To enable the member banks of NFS to offer greater utility to their customers, NPCI has introduced the below mentioned functionalities as value added services to enable customers to use these services at any participating bank ATMs.
Card to Card Fund transfer (ATM/Debit Card to ATM/Debit Card) : Using this service, a Card holder of a participating NFS Member Bank will be able to remit funds to another Card holder of a participating NFS Member Bank. The funds will be transferred basis the Beneficiary’s ‘Card Number’ which the remitter will be required to input at the time of the transaction. The main features of the service are mentioned below:
Inter Operable
Instant Fund Transfer
24/7 Availability
Paper Less
Secure
Better Fund Management
Transaction Limits - The transaction limits are in accordance with the RBI’s circular on ‘Domestic Money Transfer’. Accordingly, for cases where the Issuing and Beneficiary Cards are ‘Debit/ATM Cards’, the per transaction limit will be Rs.5000 and the monthly limit will be Rs.25,000 per remitter.
Cheque Book Request/Statement Request: At present, although statement & cheque book request options are available at ATMs, these are available only for customers of that particular bank. NFS aims to make this option inter operable wherein customers will be able to avail the aforementioned services at NFS member bank ATMs who avail this particular service from NPCI.
References[edit]
Jump up ^ "Barclays ATM network linked to National Financial Switch". The Economic Times. Mumbai: The Times Group. Press Trust of India. 2010-04-19. Archived from the original on 2014-04-29. Retrieved 2014-05-01.
Jump up ^ http://www.timescrest.com/opinion/taking-stock-3681
Jump up ^ "Common gateway to allow PSU bank customers to shop online". Economic Times. Mumbai: The Times Group. Press Trust of India. 2004-06-23. Archived from the original on 2014-04-29. Retrieved 2014-05-01.
Jump up ^ "Euronet Worldwide selected as turnkey solution provider for India's national financial switch". Leawood: Euronet Worldwide. 2004-05-20. Archived from the original on 2014-05-01. Retrieved 2014-05-01.
Jump up ^ "Banks switch to interconnectivity". Economic Times. Mumbai: The Times Group. Press Trust of India. 2004-06-23. Archived from the original on 2014-04-29. Retrieved 2004-08-31.
Jump up ^ "Any bank can now join national financial switch: NPCI". Economic Times. Mumbai: The Times Group. Press Trust of India. 2011-08-14. Archived from the original on 2014-04-29. Retrieved 2014-05-01.
Jump up ^ "NFS Member Banks ATMs". National Payments Corporation of India. Archived from the original on 2014-02-08. Retrieved 2014-05-01.
External links[edit]
Reserve Bank of india
National Financial Switch official web page
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This page was last edited on 14 February 2018, at 01:28 (UTC).
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
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Hyperinflation in the Weimar Republic
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Piles of new banknotes awaiting distribution at the Reichsbank, during the hyperinflation.
During a period between 1918 and January 1924, the German mark suffered hyperinflation. It caused considerable internal political instability in the country, the occupation of the Ruhr by foreign troops as well as misery for the general populace.
Contents
[hide]
1 Background
2 Hyperinflation
3 Stabilization
4 Revaluation
5 Analysis
6 Aftermath
7 See also
8 References
9 Sources
Background[edit]
To pay for the large costs of the ongoing First World War, Germany suspended the gold standard (the convertibility of its currency to gold) when the war broke out. Unlike the French Third Republic, which imposed its first income tax to pay for the war, German Emperor Wilhelm II and the German parliament decided unanimously to fund the war entirely by borrowing,[1] a decision criticized by financial experts such as Hjalmar Schacht as a dangerous risk for currency devaluation.[2]
The government believed that it would be able to pay off the debt by winning the war, and it would be able to annex resource-rich industrial territory in the west and east. Also, it would be able to impose massive reparations on the defeated Allies.[3] The exchange rate of the mark against the US dollar thus steadily devalued from 4.2 to 7.9 marks per dollar, a preliminary to the extreme postwar inflation.[4]
The strategy failed when Germany lost the war. The new Weimar Republic was saddled with a massive war debt that it could not afford. That was worsened by the fact that it was printing money without economic resources to back it.[3] The Treaty of Versailles further accelerated the decline in the value of the mark, so that 48 paper marks were required to buy a US dollar by late 1919.[5]
German currency was relatively stable at about 90 marks per dollar during the first half of 1921.[6] Because the Western Front was mostly in France and Belgium, Germany came out of the war with most of its industrial infrastructure intact. It was in a better position to become the dominant economic force on the European continent.[7]
The London Ultimatum in May 1921, demanded World War I reparations in gold or foreign currency to be paid in annual installments of 2 billion gold marks, plus 26% of the value of Germany's exports.[8]
The first payment was made when it came due in June 1921.[9] It marked the beginning of an increasingly rapid devaluation of the mark, which fell in value to approximately 330 marks per dollar.[5] The total reparations demanded were 132 billion gold marks, but Germany had to pay only 50 billion marks.[10]
Since reparations were required to be repaid in hard currency, not the rapidly depreciating paper mark, one strategy that Germany used was the mass printing of bank notes to buy foreign currency, which was then used to pay reparations. That greatly exacerbated the inflation of the paper mark.[11][12]
Hyperinflation[edit]
Weimar Republic hyperinflation from one to a trillion paper marks per gold mark; values on logarithmic scale.
From August 1921, Germany began to buy foreign currency with marks at any price, but that only increased the speed of breakdown in the value of the mark.[13] As the mark sank in international markets, more and more marks were required to buy the foreign currency that was demanded by the Reparations Commission.[12]
In the first half of 1922, the mark stabilized at about 320 marks per dollar.[5] International reparations conferences were being held. One, in June 1922, was organized by US investment banker J. P. Morgan, Jr.[14] The meetings produced no workable solution, and inflation erupted into hyperinflation, the mark falling to 7,400 marks per US dollar by December 1922.[5] The cost-of-living index was 41 in June 1922 and 685 in December, a 15-fold increase.
By fall 1922, Germany found itself unable to make reparations payments.[15] The mark was by now practically worthless, making it impossible for Germany to buy foreign exchange or gold using paper marks. Instead, reparations were to be paid in goods such as coal. In January 1923, French and Belgian troops occupied the industrial region of Germany in the Ruhr valley to ensure reparations payments. Inflation was exacerbated when workers in the Ruhr went on a general strike and the German government printed more money to continue paying for their passive resistance.[16] By November 1923, the US dollar was worth 4,210,500,000,000 German marks.[17]
50,000 marks, Aachen, 1923
500,000 marks, Leipzig, 1923
5,000,000 marks, Danzig, 1923
50,000,000 marks, Trier, 1923
500,000,000 marks, Dresden, 1923
5,000,000,000 marks, Berlin, 1923
50,000,000,000 marks, Plauen, 1923
500,000,000,000 marks, Berlin, 1923
5,000,000,000,000 marks, Stuttgart, 1923
50,000,000,000,000 marks Eschweiler, 1923
Stabilization[edit]
The hyperinflation crisis led prominent economists and politicians to seek a means to stabilize German currency. In August 1923, an economist, Karl Helfferich, proposed a plan to issue a new currency, the "Roggenmark" ("rye mark"), to be backed by mortgage bonds indexed to the market price of rye grain. The plan was rejected because of the greatly fluctuating price of rye in paper marks.
Agriculture Minister Hans Luther proposed a plan that substituted gold for rye and led to the issuance of the Rentenmark ("mortgage mark"), backed by bonds indexed to the market price of gold.[18] The gold bonds were defined at the rate of 2790 gold marks per kilogram of gold, the same as the prewar gold marks. Rentenmarks were not redeemable in gold but only indexed to the gold bonds. The plan was adopted in monetary reform decrees, on October 13–15, 1923. A new bank, the Rentenbank, was set up and controlled by new German Finance Minister Hans Luther.
Two Rentenmark notes, issued in line with the Decree of 15 October 1923
After November 12, 1923, when Hjalmar Schacht became currency commissioner, Germany's central bank (the Reichsbank) was not allowed to discount any further government Treasury bills, which meant the corresponding issue of paper marks also ceased.[19] The discounting of commercial trade bills was allowed and the amount of Rentenmarks expanded, but the issue was strictly controlled to conform to current commercial and government transactions. The Rentenbank refused credit to the government and to speculators who were not able to borrow Rentenmarks, because Rentenmarks were not legal tender.[20]
On November 16, 1923, the new Rentenmark was introduced to replace the worthless paper marks issued by the Reichsbank. Twelve zeros were cut from prices, and the prices quoted in the new currency remained stable.
When the president of the Reichsbank, Rudolf Havenstein, died on November 20, 1923, Schacht was appointed to replace him. By November 30, 1923, there were 500,000,000 Rentenmarks in circulation, which increased to 1,000,000,000 by January 1, 1924 and to 1,800,000,000 Rentenmarks by July 1924. Meanwhile, the old paper Marks continued in circulation. The total paper marks increased to 1.2 sextillion (1,200,000,000,000,000,000,000) in July 1924 and continued to fall in value to a third of their conversion value in Rentenmarks.[20]
On August 30, 1924, a monetary law permitted the exchange of a 1-trillion paper mark note to a new Reichsmark, worth the same as a Rentenmark. By 1924 one dollar was equivalent to 4.2 Rentenmark.
Revaluation[edit]
Conversion Table
Eventually, some debts were reinstated to compensate creditors partially for the catastrophic reduction in the value of debts that had been quoted in paper marks before the hyperinflation. A decree of 1925 reinstated some mortgages at 25% of face value in the new currency, effectively 25,000,000,000 times their value in the old paper marks, if they had been held for at least five years. Similarly, some government bonds were reinstated at 2.5% of face value, to be paid after reparations were paid.[21]
Mortgage debt was reinstated at much higher rates than government bonds were. The reinstatement of some debts and a resumption of effective taxation in a still-devastated economy triggered a wave of corporate bankruptcies.
One of the important issues of the stabilization of a hyperinflation is the revaluation. The term normally refers to the raising of the exchange rate of one national currency against other currencies. As well, it can mean revalorization, the restoration of the value of a currency depreciated by inflation. The German government had the choice of a revaluation law to finish the hyperinflation quickly or of allowing sprawling and the political and violent disturbances on the streets. The government argued in detail that the interests of creditors and debtors had to be fair and balanced. Neither the living standard price index nor the share price index was judged as relevant.
The calculation of the conversion relation was considerably judged to the dollar index as well as to the wholesale price index. In principle, the German government followed the line of market-oriented reasoning that the dollar index and the wholesale price index would roughly indicate the true price level in general over the period of high inflation and hyperinflation. In addition, the revaluation was bound on the exchange rate mark and United States dollar to obtain the value of the Goldmark.[22]
Finally, the Law on the Revaluation of Mortgages and other Claims of 16 July 1925 (Gesetz über die Aufwertung von Hypotheken und anderen Ansprüchen or Aufwertungsgesetze) included only the ratio of the paper mark to the gold mark for the period from January 1, 1918, to November 30, 1923, and the following days.[23] The galloping inflation thus caused the end of a principle, "a mark is worth a mark", which had been recognized, the nominal value principle.[24]
The law was challenged in the Supreme Court of the German Reich (Reichsgericht), but its 5th Senate ruled, on November 4, 1925, that the law was constitutional, even according to the Bill of Rights and Duties of Germans (Articles 109, 134, 152 and 153 of the Constitution).[25][26][27] The case set a precedent for judicial review in German jurisprudence.[28]
Analysis[edit]
The hyperinflation episode in the Weimar Republic in the early 1920s was not the first or even the most severe instance of inflation in history (the Hungarian pengő and Zimbabwean dollar, for example, have been even more inflated). However, it has been the subject of the most scholarly economic analysis and debate. The hyperinflation drew significant interest, as many of the dramatic and unusual economic behaviors now associated with hyperinflation were first documented systematically: exponential increases in prices and interest rates, redenomination of the currency, consumer flight from cash to hard assets and the rapid expansion of industries that produced those assets.
German monetary economics was at that time heavily influenced by Chartalism and the German Historical School, which conditioned the way the hyperinflation was analyzed.[29]
John Maynard Keynes described the situation in The Economic Consequences of the Peace: "The inflationism of the currency systems of Europe has proceeded to extraordinary lengths. The various belligerent Governments, unable, or too timid or too short-sighted to secure from loans or taxes the resources they required, have printed notes for the balance."
It was during then that French and British economic experts began to claim that Germany deliberately destroyed its economy to avoid war reparations, but both governments had conflicting views on how to handle the situation. The French declared that Germany should keep paying reparations, but Britain sought to grant a moratorium to allow financial reconstruction.[7]
Reparations accounted for about a third of the German deficit from 1920 to 1923[30] and so were cited by the German government as one of the main causes of hyperinflation. Other causes cited included bankers and speculators (particularly foreign). Hyperinflation reached its peak by November 1923[31] but ended when a new currency (the Rentenmark) was introduced. To make way for the new currency, banks "turned the marks over to junk dealers by the ton"[32] to be recycled as paper.
Aftermath[edit]
Germany, 1923: banknotes had lost so much value that they were used as wallpaper.
Since the hyperinflation, German monetary policy has retained a central concern with the maintenance of a sound currency, a concern that still affects Germany's attitude to the European sovereign debt crisis from 2009.[33]
The hyperinflated, worthless marks became widely collected abroad. The Los Angeles Times estimated in 1924 that more of the decommissioned notes were spread about the US than existed in Germany.[32]
The cause of the immense acceleration of prices seemed unclear and unpredictable to those who lived through it, but in retrospect, it was relatively simple. The Treaty of Versailles imposed a huge debt on Germany that could be paid only in gold or foreign currency. With its gold depleted, the German government attempted to buy foreign currency with German currency,[13] equivalent to selling German currency in exchange for payment in foreign currency, but the resulting increase in the supply of German marks on the market caused the German mark to fall rapidly in value, which greatly increased the number of marks needed to buy more foreign currency.
That caused German prices of goods to rise rapidly, increasing the cost of operating the German government, which could not be financed by raising taxes because those taxes would be payable in the ever-falling German currency. The alternative was some combination of running a budget deficit and simply creating more money, both increasing the supply of German currency on the market and reduced that currency's price. When the German people realized that their money was rapidly losing value, they tried to spend it quickly. That increased monetary velocity and caused an ever-faster increase in prices, creating a vicious cycle.[34]
The government and the banks had two unacceptable alternatives. If they stopped inflation, there would be immediate bankruptcies, unemployment, strikes, hunger, violence, collapse of civil order, insurrection and possibly even revolution.[35] If they continued the inflation, they would default on their foreign debt.
However, attempting to avoid both unemployment and insolvency ultimately failed when Germany had both.[35]
See also[edit]
Weimar Republic portal
Wikimedia Commons has media related to Hyperinflation in the Weimar Republic.
German gold mark
Rudolf Havenstein
Andreas Hermes
Karl Helfferich
Zero stroke
References[edit]
Jump up ^ Fergusson, When Money Dies; p. 10
Jump up ^ Fergusson; When Money Dies; p. 11
^ Jump up to: a b Evans, p. 103.
Jump up ^ Officer, Lawrence. "Exchange Rates Between the United States Dollar and Forty-one Currencies". MeasuringWorth. Retrieved 2015-01-28.
^ Jump up to: a b c d Board of Governors of the Federal Reserve System (1943). Banking and Monetary Statistics 1914-1941. Washington, DC. p. 671.
Jump up ^ Laursen and Pedersen, page 134
^ Jump up to: a b Marks, page 53
Jump up ^ Kolb, Eberhard (2012). The Weimar Republic. Translated by P.S. Falla (2nd ed.). Routledge. p. 92. ISBN 0-415-09077-6.
Jump up ^ Fergusson, page 38.
Jump up ^ Marks (1978), p. 237
Jump up ^ Fergusson, page 36
^ Jump up to: a b Shapiro, page 187
^ Jump up to: a b Fergusson; When Money Dies; p. 40
Jump up ^ Balderston, page 21
Jump up ^ Evans, p. 104.
Jump up ^ Civilisation in the West, Seventh Edition, Kishlansky, Geary, and O'Brien, New York, page 807.
Jump up ^ Coffin; "Western Civilizations"; p. 918
Jump up ^ The Rentenmark Miracle, Gustavo H.B. Franco, page 16
Jump up ^ Guttmann, pages 208-211
^ Jump up to: a b Fergusson, Chapter 13
Jump up ^ Fergusson, Chapter 14
Jump up ^ Fischer 2010, p. 83.
Jump up ^ Fischer 2010, p. 84.
Jump up ^ Fischer 2010, p. 87.
Jump up ^ Friedrich 1928, p. 197.
Jump up ^ RGZ III, 325
Jump up ^ Fischer 2010, p. 89.
Jump up ^ Friedrich 1928, pp. 196-197.
Jump up ^ Monetary Explanations of the Weimar Republic's Hyperinflation: Some Neglected Contributions in Contemporary German Literature, David E.W. Laidler & George W. Stadler, Journal of Money, Credit and Banking, vol. 30, pages 816, 818
Jump up ^ The Economics of Inflation, Costantino Bresciani-Turroni, page 93
Jump up ^ Fischer 2010, p. 64.
^ Jump up to: a b Americans With Marks Out of Luck, Cable and Associated Press, Los Angeles Times, 15 Nov 1924
Jump up ^ Greece bailout: What's the future of the euro?, Ben Quinn, Christian Science Monitor, 28 March 2010
Jump up ^ Parsson; Dying of Money; p. 116–117
^ Jump up to: a b Fergusson; When Money Dies; p. 254
Sources[edit]
Ahamed, Liaquat (2009). Lords of Finance: The Bankers Who Broke the World. Penguin Books. ISBN 978-1-59420-182-0.
Balderston, Theo, prepared for the Economic History Society (2002). Economics and politics in the Weimar Republic (1. publ. ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 0-521-77760-7.
Costantino Bresciani-Turroni, The Economics of Inflation (English transl.), Northampton, England: Augustus Kelly Publishers, 1937, on the German 1919-1923 inflation. [1]
Evans, Richard J. (2003). The Coming of the Third Reich. New York City: Penguin Press. ISBN 978-0141009759.
Feldman, Gerald D. (1996). The great disorder politics, economics, and society in the German inflation, 1914 - 1924 ([Nachdruck] ed.). New York, NY [u.a.]: Oxford Univ. Press. ISBN 0-19-510114-6.
Fergusson, Adam (2010). When money dies : the nightmare of deficit spending, devaluation, and hyperinflation in Weimar Germany (1st [U.S.] ed.). New York: PublicAffairs. ISBN 1-58648-994-1.
Fischer, Wolfgang Chr., ed. (2010). German Hyperinflation 1922/23: A Law and Economics Approach. Eul-Verlag Köln. ISBN 978-3-89936-931-1.
Friedrich, Carl Joachim (June 1928). "The Issue of Judicial Review in Germany". Political Science Quarterly. 43 (2). JSTOR 2143300.
When Money Buys Little - Jerry Jensen Study of the 1923 German postage stamps
Karsten Laursen and Jorgen Pedersen, The German Inflation, North-Holland Publishing Co., Amsterdam, 1964.
Marks, Sally (September 1978). "The Myths of Reparations". Central European History. Cambridge University Press. 11 (3): 231–255. doi:10.1017/s0008938900018707. JSTOR 4545835.
Marks, Sally (2003). The Illusion of Peace. New York: Palgrave Macmillan.
Parsson, Jens O. (1974). Dying of Money : Lessons of the Great German and American Inflations. Boston: Wellspring Press.
Shapiro, Max (1980). The penniless billionaires. New York: Times Books. ISBN 0-8129-0923-2.
Widdig, Bernd (2001). Culture and inflation in Weimar Germany ([Online-Ausg.] ed.). Berkeley: University of California Press. ISBN 0-520-22290-3.
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