1.
Set theory
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Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, the language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor, Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions among many researchers, Set theory, however, was founded by a single paper in 1874 by Georg Cantor, On a Property of the Collection of All Real Algebraic Numbers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors work initially polarized the mathematicians of his day, while Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined
2.
Hypotenuse
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In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. For example, if one of the sides has a length of 3. The length of the hypotenuse is the root of 25. The word ὑποτείνουσα was used for the hypotenuse of a triangle by Plato in the Timaeus 54d, a folk etymology says that tenuse means side, so hypotenuse means a support like a prop or buttress, but this is inaccurate. The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Using the common notation that the length of the two legs of the triangle are a and b and that of the hypotenuse is c, many computer languages support the ISO C standard function hypot, which returns the value above. The function is designed not to fail where the straightforward calculation might overflow or underflow, some scientific calculators provide a function to convert from rectangular coordinates to polar coordinates. This gives both the length of the hypotenuse and the angle the hypotenuse makes with the line at the same time when given x and y. The angle returned will normally be given by atan2. Orthographic projections, The length of the hypotenuse equals the sum of the lengths of the projections of both catheti. And The square of the length of a cathetus equals the product of the lengths of its projection on the hypotenuse times the length of this. Given the length of the c and of a cathetus b. The adjacent angle of the b, will be α = 90° – β One may also obtain the value of the angle β by the equation. Cathetus Triangle Space diagonal Nonhypotenuse number Taxicab geometry Trigonometry Special right triangles Pythagoras Hypotenuse at Encyclopaedia of Mathematics Weisstein, Eric W. Hypotenuse
3.
Right triangle
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A right triangle or right-angled triangle is a triangle in which one angle is a right angle. The relation between the sides and angles of a triangle is the basis for trigonometry. The side opposite the angle is called the hypotenuse. The sides adjacent to the angle are called legs. Side a may be identified as the adjacent to angle B and opposed to angle A, while side b is the side adjacent to angle A. If the lengths of all three sides of a triangle are integers, the triangle is said to be a Pythagorean triangle. As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the then the other is height. As a formula the area T is T =12 a b where a and b are the legs of the triangle and this formula only applies to right triangles. From this, The altitude to the hypotenuse is the mean of the two segments of the hypotenuse. Each leg of the triangle is the mean proportional of the hypotenuse, in equations, f 2 = d e, b 2 = c e, a 2 = c d where a, b, c, d, e, f are as shown in the diagram. Moreover, the altitude to the hypotenuse is related to the legs of the triangle by 1 a 2 +1 b 2 =1 f 2. For solutions of this equation in integer values of a, b, f, the altitude from either leg coincides with the other leg. Since these intersect at the vertex, the right triangles orthocenter—the intersection of its three altitudes—coincides with the right-angled vertex. The Pythagorean theorem states that, In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs. This can be stated in equation form as a 2 + b 2 = c 2 where c is the length of the hypotenuse, Pythagorean triples are integer values of a, b, c satisfying this equation. The radius of the incircle of a triangle with legs a and b. The radius of the circumcircle is half the length of the hypotenuse, thus the sum of the circumradius and the inradius is half the sum of the legs, R + r = a + b 2
4.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
5.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers
6.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
7.
Compass-and-straightedge construction
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The idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, more formally, the only permissible constructions are those granted by Euclids first three postulates. It turns out to be the case that every point constructible using straightedge, 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 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, 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 using the four basic arithmetic operations. Circles can only be starting from two given points, the centre and a point on the circle. The compass may or may not collapse when its 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 segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature and 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 Euclids Elements, although the proposition is correct, its proofs have a long and checkered history. Eyeballing it and getting close does not count as a solution and that is, it must have a finite number of steps, and not be the limit of ever closer approximations. One of the purposes of Greek mathematics was to find exact constructions for various lengths, for example. The Greeks could not find constructions for these three problems, among others, Squaring the circle, Drawing a square the same area as a given circle, doubling the cube, Drawing a cube with twice the volume of a given cube. Trisecting the angle, Dividing a given angle into three smaller angles all of the same size, for 2000 years people tried to find constructions within the limits set above, and failed. All three have now been proven under mathematical rules to be generally impossible, 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. They could also construct half of an angle, a square whose area is twice that of another square, a square having the same area as a given polygon
8.
Point (geometry)
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In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, in particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a location in Euclidean space. Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects, Euclid originally defined the point as that which has no part. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by a triplet with the additional third number representing depth. Further generalizations are represented by an ordered tuplet of n terms, many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points, As an example, a line is a set of points of the form L =. Similar constructions exist that define the plane, line segment and other related concepts, a line segment consisting of only a single point is called a degenerate line segment. In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, in spite of this, modern expansions of the system serve to remove these assumptions. There are several inequivalent definitions of dimension in mathematics, in all of the common definitions, a point is 0-dimensional. The dimension of a space is the maximum size of a linearly independent subset. In a vector space consisting of a point, there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non trivial linear combination making it zero,1 ⋅0 =0, if no such minimal n exists, the space is said to be of infinite covering dimension. A point is zero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a open set. The Hausdorff dimension of X is defined by dim H , = inf, a point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius. Although the notion of a point is considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e. g. noncommutative geometry. More precisely, such structures generalize well-known spaces of functions in a way that the operation take a value at this point may not be defined
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Two-dimensional space
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In physics and mathematics, two-dimensional space is a geometric model of the planar projection of the physical universe. The two dimensions are commonly called length and width, both directions lie in the same plane. A sequence of n numbers can be understood as a location in n-dimensional space. When n =2, the set of all locations is called two-dimensional space or bi-dimensional space. Each reference line is called an axis or just axis of the system. The coordinates can also be defined as the positions of the projections of the point onto the two axes, expressed as signed distances from the origin. The idea of system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Later, the plane was thought of as a field, where any two points could be multiplied and, except for 0, divided and this was known as the complex plane. The complex plane is called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand, although they were first described by Norwegian-Danish land surveyor, Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In mathematics, analytic geometry describes every point in space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin and they are usually labeled x and y. Another widely used system is the polar coordinate system, which specifies a point in terms of its distance from the origin. In two dimensions, there are infinitely many polytopes, the polygons, the first few regular ones are shown below, The Schläfli symbol represents a regular p-gon. The regular henagon and regular digon can be considered degenerate regular polygons and they can exist nondegenerately in non-Euclidean spaces like on a 2-sphere or a 2-torus. There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers and they are called star polygons and share the same vertex arrangements of the convex regular polygons
10.
Cartesian coordinate system
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
11.
Square root
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In mathematics, a square root of a number a is a number y such that y2 = a, in other words, a number y whose square is a. For example,4 and −4 are square roots of 16 because 42 =2 =16, every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the square root of 9 is 3, denoted √9 =3. The term whose root is being considered is known as the radicand, the radicand is the number or expression underneath the radical sign, in this example 9. Every positive number a has two roots, √a, which is positive, and −√a, which is negative. Together, these two roots are denoted ± √a, although the principal square root of a positive number is only one of its two square roots, the designation the square root is often used to refer to the principal square root. For positive a, the square root can also be written in exponent notation. Square roots of numbers can be discussed within the framework of complex numbers. In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, a method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya, has given a method for finding the root of numbers having many digits. It was known to the ancient Greeks that square roots of positive numbers that are not perfect squares are always irrational numbers, numbers not expressible as a ratio of two integers. This is the theorem Euclid X,9 almost certainly due to Theaetetus dating back to circa 380 BC, the particular case √2 is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. Mahāvīra, a 9th-century Indian mathematician, was the first to state that square roots of negative numbers do not exist, a symbol for square roots, written as an elaborate R, was invented by Regiomontanus. An R was also used for Radix to indicate square roots in Gerolamo Cardanos Ars Magna, according to historian of mathematics D. E. Smith, Aryabhatas method for finding the root was first introduced in Europe by Cataneo in 1546. According to Jeffrey A. Oaks, Arabs used the letter jīm/ĝīm, the letter jīm resembles the present square root shape. Its usage goes as far as the end of the century in the works of the Moroccan mathematician Ibn al-Yasamin. The symbol √ for the root was first used in print in 1525 in Christoph Rudolffs Coss
12.
Field (mathematics)
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In mathematics, a field is a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus an algebraic structure, which is widely used in algebra, number theory. The best known fields are the field of numbers. In addition, the field of numbers is widely used, not only in mathematics. Finite fields are used in most cryptographic protocols used for computer security, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Formally, a field is a set together with two operations the addition and the multiplication, which have the properties, called axioms of fields. An operation is a mapping that associates an element of the set to every pair of its elements, the result of the addition of a and b is called the sum of a and b and denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, associativity of addition and multiplication For all a, b and c in F, one has a + = + c and a · = · c. Commutativity of addition and multiplication For all a and b in F one has a + b = b + a and a · b = b · a. Existence of additive and multiplicative identity elements There exists an element 0 in F, called the identity, such that for all a in F. There is an element 1, different from 0 and called the identity, such that for all a in F. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element in F, denoted −a, such that a + =0. For every a ≠0 in F, there exists an element in F, denoted a−1, 1/a, or 1/a, distributivity of multiplication over addition For all a, b and c in F, one has a · = +. The elements 0 and 1 being required to be distinct, a field has, at least, for every a in F, one has − a = ⋅ a. Thus, the inverse of every element is known as soon as one knows the additive inverse of 1. A subtraction and a division are defined in every field by a − b = a +, a subfield E of a field F is a subset of F that contains 1, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. It is straightforward to verify that a subfield is indeed a field, two groups are associated to every field. The field itself is a group under addition, when considering this group structure rather the field structure, one talks of the additive group of the field
13.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0
14.
Abstract algebra
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In algebra, which is a broad division of mathematics, abstract algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, the term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra. Algebraic structures, with their homomorphisms, form mathematical categories. Category theory is a formalism that allows a way for expressing properties. Universal algebra is a subject that studies types of algebraic structures as single objects. For example, the structure of groups is an object in universal algebra. As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra, through the end of the nineteenth century, many – perhaps most – of these problems were in some way related to the theory of algebraic equations. Numerous textbooks in abstract algebra start with definitions of various algebraic structures. This creates an impression that in algebra axioms had come first and then served as a motivation. The true order of development was almost exactly the opposite. For example, the numbers of the nineteenth century had kinematic and physical motivations. An archetypical example of this progressive synthesis can be seen in the history of group theory, there were several threads in the early development of group theory, in modern language loosely corresponding to number theory, theory of equations, and geometry. Leonhard Euler considered algebraic operations on numbers modulo an integer, modular arithmetic, lagranges goal was to understand why equations of third and fourth degree admit formulae for solutions, and he identified as key objects permutations of the roots. An important novel step taken by Lagrange in this paper was the view of the roots, i. e. as symbols. However, he did not consider composition of permutations, serendipitously, the first edition of Edward Warings Meditationes Algebraicae appeared in the same year, with an expanded version published in 1782. Waring proved the theorem on symmetric functions, and specially considered the relation between the roots of a quartic equation and its resolvent cubic. Kronecker claimed in 1888 that the study of modern algebra began with this first paper of Vandermonde, cauchy states quite clearly that Vandermonde had priority over Lagrange for this remarkable idea, which eventually led to the study of group theory. Paolo Ruffini was the first person to develop the theory of permutation groups and his goal was to establish the impossibility of an algebraic solution to a general algebraic equation of degree greater than four
15.
Synthetic geometry
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Synthetic geometry is the study of geometry without the use of coordinates or formulas. It relies on the method and the tools directly related to them. Only after the introduction of methods was there a reason to introduce the term synthetic geometry to distinguish this approach to geometry from other approaches. Other approaches to geometry are embodied in analytic and algebraic geometries, geometry, as presented by Euclid in the elements, is the quintessential example of the use of the synthetic method. It was the method of Isaac Newton for the solution of geometric problems. Synthetic methods were most prominent during the 19th century when geometers rejected coordinate methods in establishing the foundations of projective geometry, for example the geometer Jakob Steiner hated analytic geometry, and always gave preference to synthetic methods. The process of logical synthesis begins with some arbitrary but definite starting point and this starting point is the introduction of primitive notions or primitives and axioms about these primitives, Primitives are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are such as points, lines and planes. Axioms are statements about these primitives, for example, any two points are incident with just one line. Axioms are assumed true, and not proven and they are the building blocks of geometric concepts, since they specify the properties that the primitives have. From a given set of axioms, synthesis proceeds as a carefully constructed logical argument, when a significant result is proved rigorously, it becomes a theorem. There is no fixed set for geometry, as more than one consistent set can be chosen. Each such set may lead to a different geometry, while there are examples of different sets giving the same geometry. With this plethora of possibilities, it is no longer appropriate to speak of geometry in the singular, historically, Euclids parallel postulate has turned out to be independent of the other axioms. Simply discarding it gives absolute geometry, while negating it yields hyperbolic geometry, other consistent axiom sets can yield other geometries, such as projective, elliptic, spherical or affine geometry. Axioms of continuity and betweeness are also optional, for example, following the Erlangen program of Klein, the nature of any given geometry can be seen as the connection between symmetry and the content of the propositions, rather than the style of development. One of the early French analysts summarized synthetic geometry this way, for example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory than is found by starting with a vector space of dimension three
16.
Plane (geometry)
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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words. Euclid set forth the first great landmark of mathematical thought, a treatment of geometry. He selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its sense is not directly given a definition anywhere in the Elements. In his work Euclid never makes use of numbers to measure length, angle, in this way the Euclidean plane is not quite the same as the Cartesian plane. This section is concerned with planes embedded in three dimensions, specifically, in R3. In a Euclidean space of any number of dimensions, a plane is determined by any of the following. A line and a point not on that line, a line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the plane must be parallel to each other. Two distinct planes perpendicular to the line must be parallel to each other. Specifically, let r0 be the vector of some point P0 =. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the plane can be described as the set of all points r such that n ⋅ =0. Expanded this becomes a + b + c =0, which is the form of the equation of a plane. This is just a linear equation a x + b y + c z + d =0 and this familiar equation for a plane is called the general form of the equation of the plane
17.
Intercept theorem
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It is equivalent to the theorem about ratios in similar triangles. Traditionally it is attributed to Greek mathematician Thales. e, if the two intersecting lines are intercepted by two arbitrary lines and | S A |, | A B | = | S C |, | C D | holds then the two intercepting lines are parallel. However the converse of the statement is not true. The intercept theorem is related to similarity. It is equivalent to the concept of similar triangles, i. e. it can be used to prove the properties of similar triangles, in a normed vector space, the axioms concerning the scalar multiplication are assuring that the intercept theorem holds. In order to them in algebraic terms using field extensions, one needs to match field operations with compass. In particular it is important to assure that for two given line segments, a new line segment can be constructed such that its length equals the product of lengths of the other two. Similarly one needs to be able to construct, for a segment of length d. The intercept theorem can be used to show that in cases such a construction is possible. According to some sources the Greek mathematician Thales applied the intercept theorem to determine the height of the Cheops pyramid. The following description illustrates the use of the theorem to compute the height of the pyramid. It does not however recount Thales original work, which was lost, Thales measured the length of the pyramids base and the height of his pole. Then at the time of the day he measured the length of the pyramids shadow. This yielded the data, height of the pole,1.63 m ⋅180 m 2 m =146.7 m The intercept theorem can be used to prove that a certain construction yields parallel line s. An elementary proof of the theorem uses triangles of area to derive the basic statements about the ratios. The other claims then follow by applying the first claim and contradiction, claim 4 can be shown by applying the intercept theorem for two lines. Similarity Construction decimal number 8.639 Proximity construction of an angle in decimal degrees Schupp, intercept Theorem at PlanetMath Alexander Bogomolny, Thales Theorems and in particular Thales Theorem at Cut-the-Knot
18.
Geometric mean theorem
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It states that the geometric mean of the two segments equals the altitude. If h denotes the altitude in a triangle and p and q the segments on the hypotenuse then the theorem can be stated as, h = p q or in term of areas. The latter version yields a method to square a rectangle with ruler and compass, for such a rectangle with sides p and q we denote its top left vertex with D. Now we extend the segment q to its left by p, then we erect a perpendicular line to the diameter in D that intersects the half circle in C. Due to Thales theorem C and the form a right triangle with the line segment DC as its altitude. The converse statement is true as well, any triangle, in which the altitude equals the geometric mean of the two line segments created by it, is a right triangle. Historically the theorem is attributed to Euclid, who stated it as a corollary to proposition 8 in book VI of his Elements, in proposition 14 of book II Euclid gives a method for squaring a rectangle, which essentially matches the method given here. Euclid however provides different slightly more complicated proof for the correctness of the rather than relying on the geometric mean theorem. e. △ A C D ∼ △ A B C ∼ △ B C D, now because of h 2 = p q we also have h p = q h. Together with ∠ A D C = ∠ C D B the triangles △ A D C and △ B D C have an angle of equal size and have corresponding pairs of legs with the same ratio. One such arrangement requires a square of area h2 to complete it, since both arrangements yield the same triangle, the areas of the square and the rectangle must be identical. Nelsen, Icons of Mathematics, An Exploration of Twenty Key Images, MAA2011, ISBN9780883853528, pp. 31–32 Ilka Agricola, Thomas Friedrich, Elementary Geometry. AMS2008, ISBN9780821843475, p.25 Hartmut Wellstein, Peter Kirsche, springer,2009, ISBN9783834808561, pp. 76-77 Euclid, Elements, book II – prop
19.
Algebraic number
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An algebraic number is any complex number that is a root of a non-zero polynomial in one variable with rational coefficients. All integers and rational numbers are algebraic, as are all roots of integers, the same is not true for all real and complex numbers because they also include transcendental numbers such as π and e. Almost all real and complex numbers are transcendental, the rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the above definition because x = a/b is the root of bx − a. The quadratic surds are algebraic numbers, if the quadratic polynomial is monic then the roots are quadratic integers. The constructible numbers are numbers that can be constructed from a given unit length using straightedge. These include all quadratic surds, all numbers, and all numbers that can be formed from these using the basic arithmetic operations. Any expression formed from algebraic numbers using any combination of the arithmetic operations. Polynomial roots that cannot be expressed in terms of the arithmetic operations. This happens with many, but not all, polynomials of degree 5 or higher, gaussian integers, those complex numbers a + bi where both a and b are integers are also quadratic integers. Trigonometric functions of rational multiples of π, that is, the trigonometric numbers, for example, each of cos π/7, cos 3π/7, cos 5π/7 satisfies 8x3 − 4x2 − 4x +1 =0. This polynomial is irreducible over the rationals, and so these three cosines are conjugate algebraic numbers. Likewise, tan 3π/16, tan 7π/16, tan 11π/16, tan 15π/16 all satisfy the irreducible polynomial x4 − 4x3 − 6x2 + 4x +1, and so are conjugate algebraic integers. Some irrational numbers are algebraic and some are not, The numbers √2 and 3√3/2 are algebraic since they are roots of polynomials x2 −2 and 8x3 −3, the golden ratio φ is algebraic since it is a root of the polynomial x2 − x −1. The numbers π and e are not algebraic numbers, hence they are transcendental, the set of algebraic numbers is countable. Hence, the set of numbers has Lebesgue measure zero. Given an algebraic number, there is a monic polynomial of least degree that has the number as a root. This polynomial is called its minimal polynomial, if its minimal polynomial has degree n, then the algebraic number is said to be of degree n. An algebraic number of degree 1 is a rational number, a real algebraic number of degree 2 is a quadratic irrational
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Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
21.
Trigonometric constants expressed in real radicals
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Exact algebraic expressions for trigonometric values are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. When they are, they are more specifically in terms of square roots. For an angle of a number of degrees, which is not a multiple 3°, the values of sine, cosine. Note that 1° = π/180 radians, according to Nivens theorem, the only rational values of the sine function for which the argument is a rational number of degrees are 0, 1/2,1, −1/2, and −1. According to Bakers theorem, if the value of a sine and that is, if the angle is an algebraic, but non-rational, number of degrees, the trigonometric functions all have transcendental values. The list in this article is incomplete in several senses, first, the trigonometric functions of all angles that are integer multiples of those given can also be expressed in radicals, but some are omitted here. Second, it is possible to apply the half-angle formula to find an expression in radicals for a trigonometric function of one-half of any angle on the list, then half of that angle. This article only gives the cases based on the Fermat primes 3 and 5, thus for example cos, given in the article 17-gon, is not given here. Fourth, this article deals with trigonometric function values when the expression in radicals is in real radicals—roots of real numbers. Many other trigonometric function values are expressible in, for example, in practice, all values of sines, cosines, and tangents not found in this article are approximated using the techniques described at Generating trigonometric tables. Several different units of measure are widely used, including degrees, radians. The following table shows the conversions and values for some common angles, Values outside the range are trivially derived from these values. This is because the sum of the angles of any n-gon is 180° ×, using cos 36 ∘ =5 +14, tan 36 ∘ =5 −25, this can be simplified to, V = a 34. The derivation of sine, cosine, and tangent constants into radial forms is based upon the constructibility of right triangles, here right triangles made from symmetry sections of regular polygons are used to calculate fundamental trigonometric ratios. Each right triangle represents three points in a polygon, a vertex, an edge center containing that vertex. 2 sin θ =2 −2 cos 2 θ =2 −2 +2 cos 4 θ =2 −2 +2 +2 cos 8 θ and so on. If M =2 and N =2 then cos π17 = M −4 +28, crd is the chord function, crd θ =2 sin θ2. Thus sin 18 ∘ =11 +5 =5 −14, similarly crd 108 ∘ = crd = b a =1 +52, so sin 54 ∘ = cos 36 ∘ =1 +54
22.
Ancient Greece
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Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages of the 12th-9th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and this was followed by the period of Classical Greece, an era that began with the Greco-Persian Wars, lasting from the 5th to 4th centuries BC. Due to the conquests by Alexander the Great of Macedonia, Hellenistic civilization flourished from Central Asia to the end of the Mediterranean Sea. Classical Greek culture, especially philosophy, had a influence on ancient Rome. For this reason Classical Greece is generally considered to be the culture which provided the foundation of modern Western culture and is considered the cradle of Western civilization. Classical Antiquity in the Mediterranean region is considered to have begun in the 8th century BC. Classical Antiquity in Greece is preceded by the Greek Dark Ages and this period is succeeded, around the 8th century BC, by the Orientalizing Period during which a strong influence of Syro-Hittite, Jewish, Assyrian, Phoenician and Egyptian cultures becomes apparent. The end of the Dark Ages is also dated to 776 BC. The Archaic period gives way to the Classical period around 500 BC, Ancient Periods Astronomical year numbering Dates are approximate, consult particular article for details The history of Greece during Classical Antiquity may be subdivided into five major periods. The earliest of these is the Archaic period, in which artists made larger free-standing sculptures in stiff, the Archaic period is often taken to end with the overthrow of the last tyrant of Athens and the start of Athenian Democracy in 508 BC. It was followed by the Classical period, characterized by a style which was considered by observers to be exemplary, i. e. classical, as shown in the Parthenon. This period saw the Greco-Persian Wars and the Rise of Macedon, following the Classical period was the Hellenistic period, during which Greek culture and power expanded into the Near and Middle East. This period begins with the death of Alexander and ends with the Roman conquest, Herodotus is widely known as the father of history, his Histories are eponymous of the entire field. Herodotus was succeeded by authors such as Thucydides, Xenophon, Demosthenes, Plato, most of these authors were either Athenian or pro-Athenian, which is why far more is known about the history and politics of Athens than those of many other cities. Their scope is limited by a focus on political, military and diplomatic history, ignoring economic. In the 8th century BC, Greece began to emerge from the Dark Ages which followed the fall of the Mycenaean civilization, literacy had been lost and Mycenaean script forgotten, but the Greeks adopted the Phoenician alphabet, modifying it to create the Greek alphabet. The Lelantine War is the earliest documented war of the ancient Greek period and it was fought between the important poleis of Chalcis and Eretria over the fertile Lelantine plain of Euboea. Both cities seem to have suffered a decline as result of the long war, a mercantile class arose in the first half of the 7th century BC, shown by the introduction of coinage in about 680 BC
23.
Doubling the cube
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Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a cube whose volume is double that of the first, using only the tools of a compass. As with the problems of squaring the circle and trisecting the angle. The Egyptians, Indians, and particularly the Greeks were aware of the problem and made futile attempts at solving what they saw as an obstinate. However, the nonexistence of a solution was finally proven by Pierre Wantzel in 1837, in algebraic terms, doubling a unit cube requires the construction of a line segment of length x, where x3 =2, in other words, x = 3√2. This is because a cube of side length 1 has a volume of 13 =1, the impossibility of doubling the cube is therefore equivalent to the statement that 3√2 is not a constructible number. This implies that the degree of the extension generated by a constructible point must be a power of 2. The field extension generated by 3√2, however, is of degree 3 and we begin with the unit line segment defined by points and in the plane. We are required to construct a line segment defined by two separated by a distance of 3√2. Any newly defined point either arises as the result of the intersection of two circles, as the intersection of a circle and a line, or as the intersection of two lines. Restated in more abstract terminology, the new x- and y-coordinates have minimal polynomials of degree at most 2 over the subfield of ℝ generated by the previous coordinates, therefore, the degree of the field extension corresponding to each new coordinate is 2 or 1. By Gausss Lemma, p is irreducible over ℚ, and is thus a minimal polynomial over ℚ for 3√2. The field extension ℚ, ℚ is therefore of degree 3. But this is not a power of 2, so by the above, 3√2 is not the coordinate of a point, and thus a line segment of 3√2 cannot be constructed. The problem owes its name to a story concerning the citizens of Delos, the oracle responded that they must double the size of the altar to Apollo, which was a regular cube. This may be why the problem is referred to in the 350s BC by the author of the pseudo-Platonic Sisyphus as still unsolved, however another version of the story says that all three found solutions but they were too abstract to be of practical value. In modern notation, this means that given segments of lengths a and 2a, the duplication of the cube is equivalent to finding segments of lengths r and s so that a r = r s = s 2 a. In turn, this means that r = a ⋅23 But Pierre Wantzel proved in 1837 that the root of 2 is not constructible
24.
Angle trisection
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Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an equal to one third of a given arbitrary angle. The problem as stated is generally impossible to solve, as proved by Pierre Wantzel in 1837, however, although there is no way to trisect an angle in general with just a compass and a straightedge, some special angles can be trisected. For example, it is straightforward to trisect a right angle. It is possible to trisect an angle by using tools other than straightedge. For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, other techniques were developed by mathematicians over the centuries. Because it is defined in terms, but complex to prove unsolvable. These solutions often involve mistaken interpretations of the rules, or are simply incorrect, three problems proved elusive, specifically, trisecting the angle, doubling the cube, and squaring the circle. The problem of angle trisection reads, Construct an angle equal to one-third of an arbitrary angle. Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle in 1837. Wantzels proof, restated in modern terminology, uses the algebra of field extensions. However Wantzel published these results earlier than Galois and did not use the connection between field extensions and groups that is the subject of Galois theory itself. The problem of constructing an angle of a given measure θ is equivalent to constructing two segments such that the ratio of their length is cos θ. From a solution to one of two problems, one may pass to a solution of the other by a compass and straightedge construction. The triple-angle formula gives an expression relating the cosines of the angle and its trisection. It follows that, given a segment that is defined to have unit length and this equivalence reduces the original geometric problem to a purely algebraic problem. Every irrational number which is constructible in a step from some given numbers is a root of a polynomial of degree 2 with coefficients in the field generated by these numbers. Therefore, any number which is constructible by a sequence of steps is a root of a polynomial whose degree is a power of two
25.
Squaring the circle
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Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the area as a given circle by using only a finite number of steps with compass. It may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail the existence of such a square. It had been known for decades before then that the construction would be impossible if π were transcendental. Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a number of steps. The expression squaring the circle is used as a metaphor for trying to do the impossible. The term quadrature of the circle is used to mean the same thing as squaring the circle. Methods to approximate the area of a circle with a square were known already to Babylonian mathematicians. Indian mathematicians also found a method, though less accurate. Archimedes showed that the value of pi lay between 3 + 1/7 and 3 + 10/71, see Numerical approximations of π for more on the history. The first known Greek to be associated with the problem was Anaxagoras, Hippocrates of Chios squared certain lunes, in the hope that it would lead to a solution — see Lune of Hippocrates. Even then there were skeptics—Eudemus argued that magnitudes cannot be divided up without limit, the problem was even mentioned in Aristophaness play The Birds. It is believed that Oenopides was the first Greek who required a plane solution, james Gregory attempted a proof of its impossibility in Vera Circuli et Hyperbolae Quadratura in 1667. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of pi and it was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility. The Victorian-age mathematician, logician and author, Charles Lutwidge Dodgson also expressed interest in debunking illogical circle-squaring theories, in one of his diary entries for 1855, Dodgson listed books he hoped to write including one called Plain Facts for Circle-Squarers. The value my friend selected for Pi was 3.2, more than a score of letters were interchanged before I became sadly convinced that I had no chance. A ridiculing of circle-squaring appears in Augustus de Morgans A Budget of Paradoxes published posthumously by his widow in 1872, originally published as a series of articles in the Athenæum, he was revising them for publication at the time of his death. Circle squaring was very popular in the century, but hardly anyone indulges in it today
26.
Plato
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Plato was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world. He is widely considered the most pivotal figure in the development of philosophy, unlike nearly all of his philosophical contemporaries, Platos entire work is believed to have survived intact for over 2,400 years. Along with his teacher, Socrates, and his most famous student, Aristotle, Plato laid the foundations of Western philosophy. Alfred North Whitehead once noted, the safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato. In addition to being a figure for Western science, philosophy. Friedrich Nietzsche, amongst other scholars, called Christianity, Platonism for the people, Plato was the innovator of the written dialogue and dialectic forms in philosophy, which originate with him. He was not the first thinker or writer to whom the word “philosopher” should be applied, few other authors in the history of Western philosophy approximate him in depth and range, perhaps only Aristotle, Aquinas and Kant would be generally agreed to be of the same rank. Due to a lack of surviving accounts, little is known about Platos early life, the philosopher came from one of the wealthiest and most politically active families in Athens. Ancient sources describe him as a bright though modest boy who excelled in his studies, the exact time and place of Platos birth are unknown, but it is certain that he belonged to an aristocratic and influential family. Based on ancient sources, most modern scholars believe that he was born in Athens or Aegina between 429 and 423 BCE. According to a tradition, reported by Diogenes Laertius, Ariston traced his descent from the king of Athens, Codrus. Platos mother was Perictione, whose family boasted of a relationship with the famous Athenian lawmaker, besides Plato himself, Ariston and Perictione had three other children, these were two sons, Adeimantus and Glaucon, and a daughter Potone, the mother of Speusippus. The brothers Adeimantus and Glaucon are mentioned in the Republic as sons of Ariston, and presumably brothers of Plato, but in a scenario in the Memorabilia, Xenophon confused the issue by presenting a Glaucon much younger than Plato. Then, at twenty-eight, Hermodorus says, went to Euclides in Megara, as Debra Nails argues, The text itself gives no reason to infer that Plato left immediately for Megara and implies the very opposite. Thus, Nails dates Platos birth to 424/423, another legend related that, when Plato was an infant, bees settled on his lips while he was sleeping, an augury of the sweetness of style in which he would discourse about philosophy. Ariston appears to have died in Platos childhood, although the dating of his death is difficult. Perictione then married Pyrilampes, her mothers brother, who had served many times as an ambassador to the Persian court and was a friend of Pericles, Pyrilampes had a son from a previous marriage, Demus, who was famous for his beauty. Perictione gave birth to Pyrilampes second son, Antiphon, the half-brother of Plato and these and other references suggest a considerable amount of family pride and enable us to reconstruct Platos family tree
27.
Plutarch
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Plutarch was a Greek biographer and essayist, known primarily for his Parallel Lives and Moralia. He is classified as a Middle Platonist, Plutarchs surviving works were written in Greek, but intended for both Greek and Roman readers. Plutarch was born to a prominent family in the town of Chaeronea, about 80 km east of Delphi. The name of Plutarchs father has not been preserved, but based on the common Greek custom of repeating a name in alternate generations, the name of Plutarchs grandfather was Lamprias, as he attested in Moralia and in his Life of Antony. His brothers, Timon and Lamprias, are mentioned in his essays and dialogues. Rualdus, in his 1624 work Life of Plutarchus, recovered the name of Plutarchs wife, Timoxena, from internal evidence afforded by his writings. A letter is still extant, addressed by Plutarch to his wife, bidding her not to grieve too much at the death of their two-year-old daughter, interestingly, he hinted at a belief in reincarnation in that letter of consolation. The exact number of his sons is not certain, although two of them, Autobulus and the second Plutarch, are often mentioned. Plutarchs treatise De animae procreatione in Timaeo is dedicated to them, another person, Soklarus, is spoken of in terms which seem to imply that he was Plutarchs son, but this is nowhere definitely stated. Plutarch studied mathematics and philosophy at the Academy of Athens under Ammonius from 66 to 67, at some point, Plutarch took Roman citizenship. He lived most of his life at Chaeronea, and was initiated into the mysteries of the Greek god Apollo. For many years Plutarch served as one of the two priests at the temple of Apollo at Delphi, the site of the famous Delphic Oracle, twenty miles from his home. By his writings and lectures Plutarch became a celebrity in the Roman Empire, yet he continued to reside where he was born, at his country estate, guests from all over the empire congregated for serious conversation, presided over by Plutarch in his marble chair. Many of these dialogues were recorded and published, and the 78 essays, Plutarch held the office of archon in his native municipality, probably only an annual one which he likely served more than once. He busied himself with all the matters of the town. The Suda, a medieval Greek encyclopedia, states that Emperor Trajan made Plutarch procurator of Illyria, however, most historians consider this unlikely, since Illyria was not a procuratorial province, and Plutarch probably did not speak Illyrian. Plutarch spent the last thirty years of his serving as a priest in Delphi. He thus connected part of his work with the sanctuary of Apollo, the processes of oracle-giving
28.
Archytas
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Archytas was an Ancient Greek philosopher, mathematician, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the founder of mathematical mechanics. Archytas was born in Tarentum, Magna Graecia and was the son of Mnesagoras or Histiaeus, for a while, he was taught by Philolaus, and was a teacher of mathematics to Eudoxus of Cnidus. Archytas and Eudoxus student was Menaechmus, as a Pythagorean, Archytas believed that only arithmetic, not geometry, could provide a basis for satisfactory proofs. Archytas is believed to be the founder of mathematical mechanics and this machine, which its inventor called The pigeon, may have been suspended on a wire or pivot for its flight. Archytas also wrote some lost works, as he was included by Vitruvius in the list of the authors of works of mechanics. Thomas Winter has suggested that the pseudo-Aristotelian Mechanical Problems is an important mechanical work by Archytas, not lost after all, Archytas named the harmonic mean, important much later in projective geometry and number theory, though he did not invent it. According to Eutocius, Archytas solved the problem of doubling the cube in his manner with a geometric construction, hippocrates of Chios before, reduced this problem to finding mean proportionals. Archytas theory of proportions is treated in book VIII of Euclids Elements, the Archytas curve, which he used in his solution of the doubling the cube problem, is named after him. Politically and militarily, Archytas appears to have been the dominant figure in Tarentum in his generation, the Tarentines elected him strategos, general, seven years in a row – a step that required them to violate their own rule against successive appointments. He was allegedly undefeated as a general, in Tarentine campaigns against their southern Italian neighbors, the Seventh Letter of Plato asserts that Archytas attempted to rescue Plato during his difficulties with Dionysius II of Syracuse. In his public career, Archytas had a reputation for virtue as well as efficacy, some scholars have argued that Archytas may have served as one model for Platos philosopher king, and that he influenced Platos political philosophy as expressed in The Republic and other works. Archytas may have drowned in a shipwreck in the shore of Mattinata, the poem, however, is difficult to interpret and it is not certain that the shipwrecked and Archytas are in fact the same person. The crater Archytas on the Moon is named in his honour and this rotation will cut out a portion of the cylinder forming the Archytas curve. A cone can go through the procedures also producing the Archytas curve. Archytas used his curve to determine the construction of a cube with a volume of half of that of a given cube, on line Huffman, Carl A. Archytas of Tarentum, Cambridge University Press,2005, ISBN 0-521-83746-4 Huffman, Carl. OConnor, John J. Robertson, Edmund F. Archytas, MacTutor History of Mathematics archive, texto en PDF, mediante registro, en Hybris, revista de filosofía
29.
Pure geometry
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Synthetic geometry is the study of geometry without the use of coordinates or formulas. It relies on the method and the tools directly related to them. Only after the introduction of methods was there a reason to introduce the term synthetic geometry to distinguish this approach to geometry from other approaches. Other approaches to geometry are embodied in analytic and algebraic geometries, geometry, as presented by Euclid in the elements, is the quintessential example of the use of the synthetic method. It was the method of Isaac Newton for the solution of geometric problems. Synthetic methods were most prominent during the 19th century when geometers rejected coordinate methods in establishing the foundations of projective geometry, for example the geometer Jakob Steiner hated analytic geometry, and always gave preference to synthetic methods. The process of logical synthesis begins with some arbitrary but definite starting point and this starting point is the introduction of primitive notions or primitives and axioms about these primitives, Primitives are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are such as points, lines and planes. Axioms are statements about these primitives, for example, any two points are incident with just one line. Axioms are assumed true, and not proven and they are the building blocks of geometric concepts, since they specify the properties that the primitives have. From a given set of axioms, synthesis proceeds as a carefully constructed logical argument, when a significant result is proved rigorously, it becomes a theorem. There is no fixed set for geometry, as more than one consistent set can be chosen. Each such set may lead to a different geometry, while there are examples of different sets giving the same geometry. With this plethora of possibilities, it is no longer appropriate to speak of geometry in the singular, historically, Euclids parallel postulate has turned out to be independent of the other axioms. Simply discarding it gives absolute geometry, while negating it yields hyperbolic geometry, other consistent axiom sets can yield other geometries, such as projective, elliptic, spherical or affine geometry. Axioms of continuity and betweeness are also optional, for example, following the Erlangen program of Klein, the nature of any given geometry can be seen as the connection between symmetry and the content of the propositions, rather than the style of development. One of the early French analysts summarized synthetic geometry this way, for example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory than is found by starting with a vector space of dimension three
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Eratosthenes
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Eratosthenes of Cyrene was a Greek mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the librarian at the Library of Alexandria. He invented the discipline of geography, including the terminology used today and he is best known for being the first person to calculate the circumference of the Earth, which he did by applying a measuring system using stadia, a standard unit of measure during that time period. He was also the first to calculate the tilt of the Earths axis, additionally, he may have accurately calculated the distance from the Earth to the Sun and invented the leap day. He created the first map of the world, incorporating parallels, Eratosthenes was the founder of scientific chronology, he endeavored to revise the dates of the chief literary and political events from the conquest of Troy. In number theory, he introduced the sieve of Eratosthenes, an efficient method of identifying prime numbers and he was a figure of influence in many fields. According to an entry in the Suda, his critics scorned him, nonetheless, his devotees nicknamed him Pentathlos after the Olympians who were well rounded competitors, for he had proven himself to be knowledgeable in every area of learning. Eratosthenes yearned to understand the complexities of the entire world, the son of Aglaos, Eratosthenes was born in 276 BC in Cyrene. Alexander the Great conquered Cyrene in 332 BC, and following his death in 323 BC, its rule was given to one of his generals, Ptolemy I Soter, the founder of the Ptolemaic Kingdom. Under Ptolemaic rule the economy prospered, based largely on the export of horses and silphium, Cyrene became a place of cultivation, where knowledge blossomed. Eratosthenes went to Athens to further his studies, there he was taught Stoicism by its founder, Zeno of Citium, in philosophical lectures on living a virtuous life. He then studied under Ariston of Chios, who led a more cynical school of philosophy and he also studied under the head of the Platonic Academy, who was Arcesilaus of Pitane. His interest in Plato led him to write his very first work at a level, Platonikos. Eratosthenes was a man of many perspectives and investigated the art of poetry under Callimachus and he was a talented and imaginative poet. He wrote poems, one in hexameters called Hermes, illustrating the life history. He wrote Chronographies, a text that scientifically depicted dates of importance and this work was highly esteemed for its accuracy. George Syncellus was later able to preserve from Chronographies a list of 38 kings of the Egyptian Thebes, Eratosthenes also wrote Olympic Victors, a chronology of the winners of the Olympic Games. It is not known when he wrote his works, but they highlighted his abilities and these works and his great poetic abilities led the pharaoh Ptolemy III Euergetes to seek to place him as a librarian at the Library of Alexandria in the year 245 BC
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