Computer science is the study of processes that interact with data and that can be represented as data in the form of programs. It enables the use of algorithms to manipulate and communicate digital information. A computer scientist studies the theory of computation and the practice of designing software systems, its fields can be divided into practical disciplines. Computational complexity theory is abstract, while computer graphics emphasizes real-world applications. Programming language theory considers approaches to the description of computational processes, while computer programming itself involves the use of programming languages and complex systems. Human–computer interaction considers the challenges in making computers useful and accessible; the earliest foundations of what would become computer science predate the invention of the modern digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, aiding in computations such as multiplication and division.
Algorithms for performing computations have existed since antiquity before the development of sophisticated computing equipment. Wilhelm Schickard designed and constructed the first working mechanical calculator in 1623. In 1673, Gottfried Leibniz demonstrated a digital mechanical calculator, called the Stepped Reckoner, he may be considered the first computer scientist and information theorist, among other reasons, documenting the binary number system. In 1820, Thomas de Colmar launched the mechanical calculator industry when he released his simplified arithmometer, the first calculating machine strong enough and reliable enough to be used daily in an office environment. Charles Babbage started the design of the first automatic mechanical calculator, his Difference Engine, in 1822, which gave him the idea of the first programmable mechanical calculator, his Analytical Engine, he started developing this machine in 1834, "in less than two years, he had sketched out many of the salient features of the modern computer".
"A crucial step was the adoption of a punched card system derived from the Jacquard loom" making it infinitely programmable. In 1843, during the translation of a French article on the Analytical Engine, Ada Lovelace wrote, in one of the many notes she included, an algorithm to compute the Bernoulli numbers, considered to be the first computer program. Around 1885, Herman Hollerith invented the tabulator, which used punched cards to process statistical information. In 1937, one hundred years after Babbage's impossible dream, Howard Aiken convinced IBM, making all kinds of punched card equipment and was in the calculator business to develop his giant programmable calculator, the ASCC/Harvard Mark I, based on Babbage's Analytical Engine, which itself used cards and a central computing unit; when the machine was finished, some hailed it as "Babbage's dream come true". During the 1940s, as new and more powerful computing machines were developed, the term computer came to refer to the machines rather than their human predecessors.
As it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. In 1945, IBM founded the Watson Scientific Computing Laboratory at Columbia University in New York City; the renovated fraternity house on Manhattan's West Side was IBM's first laboratory devoted to pure science. The lab is the forerunner of IBM's Research Division, which today operates research facilities around the world; the close relationship between IBM and the university was instrumental in the emergence of a new scientific discipline, with Columbia offering one of the first academic-credit courses in computer science in 1946. Computer science began to be established as a distinct academic discipline in the 1950s and early 1960s; the world's first computer science degree program, the Cambridge Diploma in Computer Science, began at the University of Cambridge Computer Laboratory in 1953. The first computer science degree program in the United States was formed at Purdue University in 1962.
Since practical computers became available, many applications of computing have become distinct areas of study in their own rights. Although many believed it was impossible that computers themselves could be a scientific field of study, in the late fifties it became accepted among the greater academic population, it is the now well-known IBM brand that formed part of the computer science revolution during this time. IBM released the IBM 704 and the IBM 709 computers, which were used during the exploration period of such devices. "Still, working with the IBM was frustrating if you had misplaced as much as one letter in one instruction, the program would crash, you would have to start the whole process over again". During the late 1950s, the computer science discipline was much in its developmental stages, such issues were commonplace. Time has seen significant improvements in the effectiveness of computing technology. Modern society has seen a significant shift in the users of computer technology, from usage only by experts and professionals, to a near-ubiquitous user base.
Computers were quite costly, some degree of humanitarian aid was needed for efficient use—in part from professional computer operators. As computer adoption became more widespread and affordable, less human assistance was needed for common usage. Despite its short history as a formal academic discipline, computer science has made a number of fundamental contributions to science and society—in fact, along with electronics, it is
A cardioid is a plane curve traced by a point on the perimeter of a circle, rolling around a fixed circle of the same radius. It can be defined as an epicycloid having a single cusp, it is a type of sinusoidal spiral, an inverse curve of the parabola with the focus as the center of inversion. The name had been the subject of study decades beforehand. Named for its heart-like form, it is shaped more like the outline of the cross section of a round apple without the stalk. A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid, In three dimensions, the cardioid is shaped like an apple centred around the microphone, the "stalk" of the apple. Let a be the common radius of the two generating circles with midpoints, φ the rolling angle and the origin the starting point. One gets the parametric representation: x = 2 a ⋅ cos φ, y = 2 a ⋅ sin φ, 0 ≤ φ < 2 π and the representation in polar coordinates: r = 2 a. Introducing the substitutions cos φ = x / r and r = x 2 + y 2 one gets after removing the square root the implicit representation in cartesian coordinates: 2 + 4 a x − 4 a 2 y 2 = 0.proof for the parametric representation The proof can be done using complex numbers and their common description as complex plane.
The rolling movement of the black circle on the blue one can be split into two rotations. In the complex plane a rotation around point 0 by an angle φ can be performed by the multiplication of a point z by e i φ. Hence the rotation Φ + around point a is: z ↦ a + e i φ, rotation Φ − around point − a is: z ↦ − a + e i φ. A point p of the cardioid is generated by rotating the origin around point a and subsequent rotating around − a by the same angle φ: p = Φ − = Φ − = − a + e i φ = a. Herefrom one gets the parametric representation above: x = a = 2 a ⋅ cos
Archimedes of Syracuse was a Greek mathematician, engineer and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of antiquity and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including the area of a circle, the surface area and volume of a sphere, the area under a parabola. Other mathematical achievements include deriving an accurate approximation of pi, defining and investigating the spiral bearing his name, creating a system using exponentiation for expressing large numbers, he was one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics, including an explanation of the principle of the lever. He is credited with designing innovative machines, such as his screw pump, compound pulleys, defensive war machines to protect his native Syracuse from invasion.
Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, surmounted by a sphere and a cylinder, which Archimedes had requested be placed on his tomb to represent his mathematical discoveries. Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was not made until c. 530 AD by Isidore of Miletus in Byzantine Constantinople, while commentaries on the works of Archimedes written by Eutocius in the sixth century AD opened them to wider readership for the first time. The few copies of Archimedes' written work that survived through the Middle Ages were an influential source of ideas for scientists during the Renaissance, while the discovery in 1906 of unknown works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results.
Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia, located along the coast of Southern Italy. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years. In The Sand Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse. A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure, it is unknown, for instance, whether he married or had children. During his youth, Archimedes may have studied in Alexandria, where Conon of Samos and Eratosthenes of Cyrene were contemporaries, he referred to Conon of Samos as his friend, while two of his works have introductions addressed to Eratosthenes. Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two-year-long siege.
According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem; the soldier was enraged by this, killed Archimedes with his sword. Plutarch gives a lesser-known account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, was killed because the soldier thought that they were valuable items. General Marcellus was angered by the death of Archimedes, as he considered him a valuable scientific asset and had ordered that he must not be harmed. Marcellus called Archimedes "a geometrical Briareus"; the last words attributed to Archimedes are "Do not disturb my circles", a reference to the circles in the mathematical drawing that he was studying when disturbed by the Roman soldier.
This quote is given in Latin as "Noli turbare circulos meos," but there is no reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch. Valerius Maximus, writing in Memorable Doings and Sayings in the 1st century AD, gives the phrase as "...sed protecto manibus puluere'noli' inquit,'obsecro, istum disturbare'" – "... but protecting the dust with his hands, said'I beg of you, do not disturb this.'" The phrase is given in Katharevousa Greek as "μὴ μου τοὺς κύκλους τάραττε!". The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a sphere and a cylinder of the same height and diameter. Archimedes had proven that the volume and surface area of the sphere are two thirds that of the cylinder including its bases. In 75 BC, 137 years after his death, the Roman orator Cicero was serving as quaestor in Sicily, he had heard stories about the tomb of Archimedes, but none of the locals were able to give him the location.
He found the tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up, was able to see the carving and read some of the verses, added as an inscription. A tomb discovered in the courtyard of the Hotel Panorama in Syracuse in the early 1960s was claimed to be that of Archimedes, but there was no compelling evidence
In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices; the vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect, or any appropriate combination of rays and lines that result in two straight "sides" meeting at one place. A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. In a polygon, a vertex is called "convex" if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians. More a vertex of a polyhedron or polytope is convex if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, concave otherwise. Polytope vertices are related to vertices of graphs, in that the 1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of the polytope, in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices.
However, in graph theory, vertices may have fewer than two incident edges, not allowed for geometric vertices. There is a connection between geometric vertices and the vertices of a curve, its points of extreme curvature: in some sense the vertices of a polygon are points of infinite curvature, if a polygon is approximated by a smooth curve there will be a point of extreme curvature near each polygon vertex. However, a smooth curve approximation to a polygon will have additional vertices, at the points where its curvature is minimal. A vertex of a plane tiling or tessellation is a point. More a tessellation can be viewed as a kind of topological cell complex, as can the faces of a polyhedron or polytope. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x. There are two types of principal vertices: mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies in P. According to the two ears theorem, every simple polygon has at least two ears.
A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedron's surface has Euler characteristic V − E + F = 2, where V is the number of vertices, E is the number of edges, F is the number of faces; this equation is known as Euler's polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, a cube has 12 edges and 6 faces, hence 8 vertices. In computer graphics, objects are represented as triangulated polyhedra in which the object vertices are associated not only with three spatial coordinates but with other graphical information necessary to render the object such as colors, reflectance properties and surface normal. Weisstein, Eric W. "Polygon Vertex". MathWorld. Weisstein, Eric W. "Polyhedron Vertex". MathWorld. Weisstein, Eric W. "Principal Vertex". MathWorld
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, measure, infinite series, analytic functions. These theories are studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary techniques of analysis. Analysis may be distinguished from geometry. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids; the explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century.
In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would be called Cavalieri's principle to find the volume of a sphere in the 5th century; the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine and arctangent. Alongside his development of the Taylor series of the trigonometric functions, he estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series, his followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. Descartes and Fermat independently developed analytic geometry, a few decades Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations and partial differential equations, Fourier analysis, generating functions.
During this period, calculus techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra used in earlier work by Euler. Instead, Cauchy formulated calculus in terms of geometric infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y, he introduced the concept of the Cauchy sequence, started the formal theory of complex analysis. Poisson, Liouville and others studied partial differential equations and harmonic analysis; the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis.
In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, introduced the "epsilon-delta" definition of limit. Mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. "monsters" began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, Baire proved the Baire category theorem.
In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, Hilbert introduced Hilbert spaces to solve integral equations; the idea of normed vector space was in the air, in the 1920s Banach created functional analysis. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. Much of analysis happens in some metric space. Examples of analysis without a metric include functional analysis. Formally, a metric space is an ordered pair where M is a set
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal, it can be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square; the term oblong is used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD; the word rectangle comes from the Latin rectangulus, a combination of rectus and angulus. A crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals, it is a special case of an antiparallelogram, its angles are not right angles. Other geometries, such as spherical and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons. A convex quadrilateral is a rectangle if and only if it is any one of the following: a parallelogram with at least one right angle a parallelogram with diagonals of equal length a parallelogram ABCD where triangles ABD and DCA are congruent an equiangular quadrilateral a quadrilateral with four right angles a quadrilateral where the two diagonals are equal in length and bisect each other a convex quadrilateral with successive sides a, b, c, d whose area is 1 4.
A convex quadrilateral with successive sides a, b, c, d whose area is 1 2. A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a special case of a trapezium in which both pairs of opposite sides are parallel and equal in length. A trapezium is a convex quadrilateral. A convex quadrilateral is Simple: The boundary does not cross itself. Star-shaped: The whole interior is visible from a single point, without crossing any edge. De Villiers defines a rectangle more as any quadrilateral with axes of symmetry through each pair of opposite sides; this definition crossed rectangles. Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, another, the perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry for either side that it bisects. Quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides.
These quadrilaterals crossed isosceles trapezia. A rectangle is cyclic: all corners lie on a single circle, it is equiangular: all its corner angles are equal. It is isogonal or vertex-transitive: all corners lie within the same symmetry orbit, it has two lines of reflectional symmetry and rotational symmetry of order 2. The dual polygon of a rectangle is a rhombus; the figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa. A rectangle is rectilinear: its sides meet at right angles. A rectangle in the plane can be defined by five independent degrees of freedom consisting, for example, of three for position, one for shape, one for overall size. Two rectangles, neither of which will fit inside the other, are said to be incomparable. If a rectangle has length ℓ and width w it has area A = ℓ w, it has perimeter P = 2 ℓ + 2 w = 2, each diagonal has length d = ℓ 2 + w 2, when ℓ = w, the rectangle is a square; the isoperimetric theorem for rectangles states that among all rectangles of a given perimeter, the square has the largest area.
The midpoints of the sides of any quadrilateral with perpendicular diagonals form a rectangle. A parallelogram with equal diagonals is a rectangle; the Japanese theorem for cyclic quadrilaterals states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle. The British flag theorem states that with vertices denoted A, B, C, D, for any point P on the same plane of a rectangle: 2 + 2 = 2 + 2
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, C is denoted △ A B C. In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. In other words, there is only one plane that contains that triangle, every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; this article is about triangles in Euclidean geometry, in particular, the Euclidean plane, except where otherwise noted. Triangles can be classified according to the lengths of their sides: An equilateral triangle has all sides the same length. An equilateral triangle is a regular polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length. An isosceles triangle has two angles of the same measure, namely the angles opposite to the two sides of the same length; some mathematicians define an isosceles triangle to have two equal sides, whereas others define an isosceles triangle as one with at least two equal sides.
The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths. Equivalently, it has all angles of different measure. Hatch marks called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of short line segments in the form of tally marks. In a triangle, the pattern is no more than 3 ticks. An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, a scalene triangle has different patterns on all sides since no sides are equal. Patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles. An equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, a scalene triangle has different patterns on all angles since no angles are equal.
Triangles can be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°; the side opposite to the right angle is the longest side of the triangle. The other two sides are called the catheti of the triangle. Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 + 42 = 52. In this situation, 3, 4, 5 are a Pythagorean triple; the other one is an isosceles triangle. Triangles that do not have an angle measuring 90° are called oblique triangles. A triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle.
If c is the length of the longest side a2 + b2 > c2, where a and b are the lengths of the other sides. A triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side a2 + b2 < c2, where a and b are the lengths of the other sides. A triangle with an interior angle of 180° is degenerate. A right degenerate triangle has collinear vertices. A triangle that has two angles with the same measure has two sides with the same length, therefore it is an isosceles triangle, it follows that in a triangle where all angles have the same measure, all three sides have the same length, such a triangle is therefore equilateral. Triangles are assumed to be two-dimensional plane figures. In rigorous treatments, a triangle is therefore called a 2-simplex. Elementary facts about triangles were presented by Euclid in books 1–4 of his Elements, around 300 BC; the sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees.
This fact is equivalent to Euclid's parallel postulate. This allows determination of the measure of the third angle of any triangle given the measure of two angles. An exterior angle of a triangle is an angle, a linear pair to an interior angle; the measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it. The sum of the measures of the three exterior angles of any triangle is 360 degrees. Two triangles are said to be similar if every angle of one triangle has the same measure as the corresponding angle in the other triangle; the corresponding sides of similar triangles have lengths that are in the same proportion, this property is sufficient to establish similarity. Some basic theorems about similar triangles are: If and only if one pair of internal angles of two triangles have the sam