Interpunct
An interpunct known as an interpoint, middle dot and centered dot or centred dot, is a punctuation mark consisting of a vertically centered dot used for interword separation in ancient Latin script. It appears in a variety of uses in some modern languages and is present in Unicode as code point U+00B7 · MIDDLE DOT; the multiplication dot or dot operator U+22C5 ⋅ DOT OPERATOR indicates multiplication and is optionally used instead of the styled × for multiplication of real numbers: a ⋅ b is equivalent to a × b or "a times b". The same sign is used in vector multiplication to discriminate between the scalar product and the vector cross product or exterior product; as a multiplication operator, it is encountered in symbols for compound units such as the newton-meter. The multiplication dot is a separate Unicode character, but is silently replaced by the interpunct or bullet, another similar glyph, intended for lists. Various dictionaries use the interpunct to indicate syllabification within a word with multiple syllables.
There is a separate Unicode character, U+2027 ‧ HYPHENATION POINT. In British typography, the space dot is an interpunct used as the formal decimal point, its use is advocated by laws and by academic circles such as the Cambridge University History Faculty Style Guide and is mandated by some UK-based academic journals such as The Lancet. When the British currency was decimalised in 1971, the official advice issued was to write decimal amounts with a raised point and to use a decimal point "on the line" only when typesetting constraints made it unavoidable; this usage, has been declining since the mid-1970s, as the importation of electronic typewriters and computers from the United States and Japan familiarised Britons with using full stops and made the space dot harder to typeset. The space dot may still be used in handwriting, however. In the early modern era, periods were sometimes written as interpuncts. In the Shavian alphabet, interpuncts replace capitalization as the marker of proper nouns.
The dot is placed at the beginning of a word. The punt volat is used in Catalan between two Ls in cases where each belongs to a separate syllable, for example cel·la, "cell"; this distinguishes such "geminate Ls", which are pronounced, from "double L", which are written without the flying point and are pronounced. In situations where the flying point is unavailable, periods or hyphens are used as substitutes, but this is tolerated rather than encouraged. Medieval Catalan used the symbol ⟨·⟩ as a marker for certain elisions, much like the modern apostrophe, hyphenations. There is no separate keyboard layout for Catalan: the flying point can be typed using ⇧ Shift+3 in the Spanish layout, it appears in Unicode as the letters ⟨Ŀ⟩ and ⟨ŀ⟩, but they are compatibility characters and are not used or recommended. The larger bullet may be seen but is discouraged on aesthetic grounds; the preferred Unicode representation is ⟨l·⟩. The interpunct is used in Chinese to mark divisions in transliterated foreign words names.
This is properly a full-width partition sign, although sometimes narrower forms are substituted for aesthetic reasons. In particular, the regular interpunct is more used as a computer input, although Chinese-language fonts render this as full width; when the Chinese text is romanized, the partition sign is replaced by a standard space or other appropriate punctuation. Thus, William Shakespeare is signified as 威廉·莎士比亞 or 威廉·莎士比亞, George W. Bush as 喬治·W·布殊 or 喬治·W·布什, the full name of the prophet Muhammad as 阿布·卡西木·穆罕默德·本·阿布杜拉·本·阿布杜勒-穆塔利卜·本·哈希姆. Titles and other translated words are not marked: Genghis Khan and Elizabeth II are 成吉思汗 and 伊利沙伯二世 or 伊麗莎白二世 without a partition sign; the partition sign is used to separate book and chapter titles when they are mentioned consecutively: book first and chapter. In Pe̍h-ōe-jī for Taiwanese Hokkien, middle dot is used as a workaround for dot above right diacritic because most early encoding systems did not support this diacritic; this is now encoded as U+0358 ͘ COMBINING DOT ABOVE RIGHT.
Unicode did not support this diacritic until June 2004. Newer fonts support it natively, it was derived in the late 19th century from an older barred-o with curly tail as an adaptation to the typewriter. In Tibetan the interpunct ་, called ཙེག་, is used as a morpheme delimiter; the Ge'ez language uses an interpunct of two vertically aligned dots, like a colon, but with larger dots. An example is ገድለ፡ወለተ፡ጴጥሮስ. In Franco-Provençal, the interpunct is used in order to distinguish the following graphemes: ch·, versus ch, pronounced j·, versus j, pronounced g· before e, i, versus g before e, i, pronounced Ancient Greek did not have spacing or interpuncts but instead ran all the letters together. By Late Antiquity, various
Angle
In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane. Angles are formed by the intersection of two planes in Euclidean and other spaces; these are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is used to designate the measure of an angle or of a rotation; this measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation; the word angle comes from the Latin word angulus, meaning "corner".
Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow". Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, do not lie straight with respect to each other. According to Proclus an angle must be a relationship; the first concept was used by Eudemus. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle. Lower case Roman letters are used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples. In geometric figures, angles may be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC is denoted ∠BAC or B A C ^. Sometimes, where there is no risk of confusion, the angle may be referred to by its vertex. An angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign.
However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant, no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise angle from B to C, ∠CAB to the anticlockwise angle from C to B. An angle equal to 0° or not turned is called a zero angle. Angles smaller than a right angle are called acute angles. An angle equal to 1/4 turn is called a right angle. Two lines that form a right angle are said to be orthogonal, or perpendicular. Angles larger than a right angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle. Angles larger than a straight angle but less than 1 turn are called reflex angles. An angle equal to 1 turn is called complete angle, round angle or a perigon. Angles that are not right angles or a multiple of a right angle are called oblique angles; the names and measured units are shown in a table below: Angles that have the same measure are said to be equal or congruent.
An angle is not dependent upon the lengths of the sides of the angle. Two angles which share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles. A reference angle is the acute version of any angle determined by subtracting or adding straight angle, to the results as necessary, until the magnitude of result is an acute angle, a value between 0 and 1/4 turn, 90°, or π/2 radians. For example, an angle of 30 degrees has a reference angle of 30 degrees, an angle of 150 degrees has a reference angle of 30 degrees. An angle of 750 degrees has a reference angle of 30 degrees; when two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other. A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles, they are abbreviated as vert. opp. ∠s. The equality of vertically opposite angles is called the vertical angle theorem.
Eudemus of Rhodes attributed the proof to Thales of Miletus. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical Note, w
Inner product space
In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors, they provide the means of defining orthogonality between vectors. Inner product spaces generalize Euclidean spaces to vector spaces of any dimension, are studied in functional analysis; the first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product induces an associated norm, thus an inner product space is a normed vector space. A complete space with an inner product is called a Hilbert space. An space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space.
Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C. Formally, an inner product space is a vector space V over the field F together with an inner product, i.e. with a map ⟨ ⋅, ⋅ ⟩: V × V → F that satisfies the following three axioms for all vectors x, y, z ∈ V and all scalars a ∈ F: Conjugate symmetry: ⟨ x, y ⟩ = ⟨ y, x ⟩ ¯ Linearity in the first argument: ⟨ a x, y ⟩ = a ⟨ x, y ⟩ ⟨ x + y, z ⟩ = ⟨ x, z ⟩ + ⟨ y, z ⟩ Positive-definite: ⟨ x, x ⟩ > 0, x ∈ V ∖. Positive-definiteness and linearity ensure that: ⟨ x, x ⟩ = 0 ⇒ x = 0 ⟨ 0, 0 ⟩ = ⟨ 0 x, 0 x ⟩ = 0 ⟨ x, 0 x ⟩ = 0 Notice that conjugate symmetry implies that ⟨x, x⟩ is real for all x, since we have: ⟨ x, x ⟩ = ⟨ x, x ⟩ ¯. Conjugate symmetry and linearity in the first variable imply ⟨ x, a y ⟩ = ⟨ a y, x ⟩ ¯ = a ¯ ⟨ y, x ⟩ ¯ = a ¯ ⟨ x, y ⟩ ⟨ x, y + z ⟩ = ⟨ y + z, x ⟩ ¯ = ⟨ y, x ⟩ ¯ + ⟨ z, x ⟩ ¯ = ⟨ x, y ⟩ + ⟨ x, z ⟩.
Trigonometric functions
In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications; the most familiar trigonometric functions are the sine and tangent. In the context of the standard unit circle, where a triangle is formed by a ray starting at the origin and making some angle with the x-axis, the sine of the angle gives the y-component of the triangle, the cosine gives the x-component, the tangent function gives the slope. For angles less than a right angle, trigonometric functions are defined as ratios of two sides of a right triangle containing the angle, their values can be found in the lengths of various line segments around a unit circle. Modern definitions express trigonometric functions as infinite series or as solutions of certain differential equations, allowing the extension of the arguments to the whole number line and to the complex numbers.
Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles. In this use, trigonometric functions are used, for instance, in navigation and physics. A common use in elementary physics is resolving a vector into Cartesian coordinates; the sine and cosine functions are commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, average temperature variations through the year. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another. With the last four, these relations are taken as the definitions of those functions, but one can define them well geometrically, or by other means, derive these relations; the notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides.
That is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides, it is these ratios. To define the trigonometric functions for the angle A, start with any right triangle that contains the angle A; the three sides of the triangle are named as follows: The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle; the opposite side is the side opposite in this case side a. The adjacent side is the side having both the angles in this case side b. In ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a right-angled triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation; the following definitions apply to angles in this range. They can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions.
For example, the figure shows sin for angles θ, π − θ, π + θ, 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, if the range of θ is extended to additional rotations, this behavior repeats periodically with a period 2π; the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram; the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The word comes from the Latin sinus for gulf or bay, given a unit circle, it is the side of the triangle on which the angle opens. In that case: sin A = opposite hypotenuse The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, so called because it is the sine of the complementary or co-angle, the other non-right angle; because the angle sum of a triangle is π radians, the co-angle B is equal to π/2 − A.
In that case: cos A = adjacent hypotenuse The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side, so called because it can be represented as a line segment tangent to the circle, i.e. the line that touches the circle, from Latin linea tangens or touching line. In our case: tan A = opposite adjacent Tangent may be represented in terms of sine and cosine; that is: tan A = sin A cos A = opposite
Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, deducing many other propositions from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system; the Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute metaphysical, sense.
Today, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, Euclidean space is a good approximation for it only over short distances. Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects; this is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas. The Elements is a systematization of earlier knowledge of geometry, its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones, they are now nearly all lost. There are 13 books in the Elements: Books I–IV and VI discuss plane geometry.
Many results about plane figures are proved, for example "In any triangle two angles taken together in any manner are less than two right angles." and the Pythagorean theorem "In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." Books V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of regions. Notions such as prime numbers and rational and irrational numbers are introduced, it is proved. Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base; the platonic solids are constructed. Euclidean geometry is an axiomatic system, in which all theorems are derived from a small number of simple axioms; until the advent of non-Euclidean geometry, these axioms were considered to be true in the physical world, so that all the theorems would be true. However, Euclid's reasoning from assumptions to conclusions remains valid independent of their physical reality.
Near the beginning of the first book of the Elements, Euclid gives five postulates for plane geometry, stated in terms of constructions: Let the following be postulated:To draw a straight line from any point to any point. To produce a finite straight line continuously in a straight line. To describe a circle with any centre and distance; that all right angles are equal to one another.: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. Although Euclid only explicitly asserts the existence of the constructed objects, in his reasoning they are implicitly assumed to be unique; the Elements include the following five "common notions": Things that are equal to the same thing are equal to one another. If equals are added to equals the wholes are equal. If equals are subtracted from equals the differences are equal.
Things that coincide with one another are equal to one another. The whole is greater than the part. Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more complete sets of axioms. To the ancients, the parallel postulate seemed less obvious than the others, they aspired to create a system of certain propositions, to them it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible, since one can construct consistent systems of geometry in which the parallel postulate is true, others in which it is false. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: his first 28 propositions are those that can be proved without it. Many alternative axioms can be formulated. For example, Playfair's axiom states: In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the giv
Three-dimensional space
Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space; when n = 3, the set of all such locations is called three-dimensional Euclidean space. It is represented by the symbol ℝ3; this serves as a three-parameter model of the physical universe. However, this space is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the three values refer to measurements in different directions, any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space. Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms width, height and length. In mathematics, analytic geometry describes every point in three-dimensional space by means of three coordinates.
Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are labeled x, y, z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, equal to the distance of that point from the plane determined by the other two axes. Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods. See Euclidean space. Below are images of the above-mentioned systems. Two distinct points always determine a line. Three distinct points determine a unique plane. Four distinct points can either coplanar or determine the entire space. Two distinct lines can either be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.
Two distinct planes are parallel. Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane. A hyperplane is a subspace of one dimension less than the dimension of the full space; the hyperplanes of a three-dimensional space are the two-dimensional subspaces. In terms of cartesian coordinates, the points of a hyperplane satisfy a single linear equation, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations, each representing a plane having this line as a common intersection. Varignon's theorem states that the midpoints of any quadrilateral in ℝ3 form a parallelogram, so, are coplanar. A sphere in 3-space consists of the set of all points in 3-space at a fixed distance r from a central point P.
The solid enclosed by the sphere is called a ball. The volume of the ball is given by V = 4 3 π r 3. Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space ℝ4. If a point has coordinates, P x2 + y2 + z2 + w2 = 1 characterizes those points on the unit 3-sphere centered at the origin. In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra. A surface generated by revolving a plane curve about a fixed line in its plane as an axis is called a surface of revolution; the plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane, perpendicular to the axis, is a circle. Simple examples occur. If the generatrix line intersects the axis line, the surface of revolution is a right circular cone with vertex the point of intersection. However, if the generatrix and axis are parallel, the surface of revolution is a circular cylinder.
In analogy with the conic sections, the set of points whose cartesian coordinates satisfy the general equation of the second degree, namely, A x 2 + B y 2 + C z 2 + F x y + G y z + H x z + J x + K y + L z + M = 0, where A, B, C, F, G, H, J, K, L and M are real numbers and not all of A, B, C, F, G and H are zero is called a quadric surface. There are six types of non-degenerate quadric surfaces: Ellipsoid Hyperboloid of one sheet Hyperboloid of two sheets Elliptic cone Elliptic paraboloid Hyperbolic paraboloidThe degenerate quadric surfaces are the empty set, a single point, a single li
Geometry
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, 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 and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. 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, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study 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 means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
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, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp
![{\displaystyle {\begin{aligned}\ [{\color {red}1,3,-5}]\cdot [{\color {blue}4,-2,-1}]&=({\color {red}1}\times {\color {blue}4})+({\color {red}3}\times {\color {blue}-2})+({\color {red}-5}\times {\color {blue}-1})\\&=4-6+5\\&=3\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be560d2c22a074c7711ae946954725d31ec77928)
![{\displaystyle {\begin{aligned}\mathbf {a} &=[a_{1},\dots ,a_{n}]=\sum _{i}a_{i}\mathbf {e} _{i}\\\mathbf {b} &=[b_{1},\dots ,b_{n}]=\sum _{i}b_{i}\mathbf {e} _{i}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b154ac2bb09512c81d917db83c273055c093571f)
