In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.
A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.
Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.
Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
A turn is a unit of plane angle measurement equal to 2π radians, 360 degrees or 400 gradians. A turn is referred to as a cycle, complete rotation or full circle. Subdivisions of a turn include half turns, quarter turns, milliturns, etc. A turn can be divided in 100 centiturns or 1000 milliturns, with each milliturn corresponding to an angle of 0.36°, which can be written as 21′ 36″. A protractor divided in centiturns is called a percentage protractor. Binary fractions of a turn are used. Sailors have traditionally divided a turn into 32 compass points; the binary degree known as the binary radian, is 1⁄256 turn. The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte. Other measures of angle used in computing may be based on dividing one whole turn into 2n equal parts for other values of n; the notion of turn is used for planar rotations. The word turn originates via Latin and French from the Greek word τόρνος. In 1697, David Gregory used π/ρ to denote the perimeter of a circle divided by its radius.
However, earlier in 1647, William Oughtred had used δ/π for the ratio of the diameter to perimeter. The first use of the symbol π on its own with its present meaning was in 1706 by the Welsh mathematician William Jones. Euler adopted the symbol with that meaning in 1737. Percentage protractors have existed since 1922, but the terms centiturns and milliturns were introduced much by Fred Hoyle; the German standard DIN 1315 proposed the unit symbol pla for turns. Since 2011, the HP 39gII and HP Prime support the unit symbol tr for turns. In 2016, support for turns was added to newRPL for the HP 50g. In June 2017, for release 3.6, the Python programming language adopted the name tau to represent the number of radians in a turn. The standard ISO 80000-3:2006 mentions that the unit name revolution with symbol r is used with rotating machines, as well as using the term turn to mean a full rotation; the standard IEEE 260.1:2004 uses the unit name rotation and symbol r. One turn is equal to 2π radians. In 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of π, which amounts to the number of radians in half a turn, in order to make mathematics simpler and more intuitive.
His proposal used a "pi with three legs" symbol to denote the constant. In 2010, Michael Hartl proposed to use tau to represent Palais' circle constant: τ = 2π, he offered two reasons. First, τ is the number of radians in one turn, which allows fractions of a turn to be expressed more directly: for instance, a 3/4 turn would be represented as 3/4τ rad instead of 3/2π rad. Second, τ visually resembles π. Hartl's Tau Manifesto gives many examples of formulas that are asserted to be clearer where tau is used instead of pi; the τ-functionality is made available in the Google calculator and in several programming languages like Python, Perl and Nim. It has been used in at least one mathematical research article, authored by the τ-promoter P. Harremoës. However, none of these proposals have received widespread acceptance by the mathematical and scientific communities; as an angular unit, the turn or revolution is useful for large angles, such as in connection with electromagnetic coils and rotating objects.
See winding number. The angular speed of rotating machinery, such as automobile engines, is measured in revolutions per minute or RPM. Turn is used in complex dynamics for measure of internal angles; the sum of external angles of a polygon equals one turn. Angle doubling map is used. Pie charts illustrate proportions of a whole as fractions of a turn; each one percent is shown as an angle of one centiturn. In kinematics, a turn is a rotation less than a full revolution. A turn may be represented in a mathematical model that uses expressions of complex numbers or quaternions. In the complex plane every non-zero number has a polar coordinate expression z = r cis = r cos + ri sin where r > 0 and a is in [0, 2π). A turn of the complex plane arises from multiplying z = x + iy by an element u = ebi that lies on the unit circle: z ↦ uz. Frank Morley referred to elements of the unit circle as turns in the book Inversive Geometry, which he coauthored with his son Frank Vigor Morley; the Latin term for turn is versor, a quaternion that can be visualized as an arc of a great circle.
The product of two versors can be compared to a spherical triangle. For the kinematics of rotation in three dimensions, see quaternions and spatial rotation. Angle of rotation Revolutions per minute Repeating circle Spat — the 3D counterpart of the turn, equivalent to 4π steradians. Unit interval Turn Spread Modulo operation Palais, Robert. "Pi is Wrong". The Mathematical Intelligencer. New York, USA: Springer-Verlag. 23: 7–8. Doi:10.1007/bf03026846
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is one of the five Platonic solids, it has 6 faces, 12 edges, 8 vertices. The cube is a square parallelepiped, an equilateral cuboid and a right rhombohedron, it is a regular square prism in three orientations, a trigonal trapezohedron in four orientations. The cube is dual to the octahedron, it has octahedral symmetry. The cube is the only convex polyhedron; the cube has four special orthogonal projections, centered, on a vertex, edges and normal to its vertex figure. The first and third correspond to the B2 Coxeter planes; the cube can be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the Cartesian coordinates of the vertices are while the interior consists of all points with −1 < xi < 1 for all i.
In analytic geometry, a cube's surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of edge length a: As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares and second powers. A cube has the largest volume among cuboids with a given surface area. A cube has the largest volume among cuboids with the same total linear size. For a cube whose circumscribing sphere has radius R, for a given point in its 3-dimensional space with distances di from the cube's eight vertices, we have: ∑ i = 1 8 d i 4 8 + 16 R 4 9 = 2. Doubling the cube, or the Delian problem, was the problem posed by ancient Greek mathematicians of using only a compass and straightedge to start with the length of the edge of a given cube and to construct the length of the edge of a cube with twice the volume of the original cube, they were unable to solve this problem, in 1837 Pierre Wantzel proved it to be impossible because the cube root of 2 is not a constructible number.
The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123. The cube has three classes of symmetry, which can be represented by vertex-transitive coloring the faces; the highest octahedral symmetry Oh has all the faces the same color. The dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color; the lowest symmetry D2h is a prismatic symmetry, with sides alternating colors, so there are three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol. A cube has eleven nets: that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the same color, one would need at least three colors; the cube is the cell of the only regular tiling of three-dimensional Euclidean space. It is unique among the Platonic solids in having faces with an number of sides and it is the only member of that group, a zonohedron; the cube can be cut into six identical square pyramids.
If these square pyramids are attached to the faces of a second cube, a rhombic dodecahedron is obtained. The analogue of a cube in four-dimensional Euclidean space has a special name—a tesseract or hypercube. More properly, a hypercube is the analogue of the cube in n-dimensional Euclidean space and a tesseract is the order-4 hypercube. A hypercube is called a measure polytope. There are analogues of the cube in lower dimensions too: a point in dimension 0, a line segment in one dimension and a square in two dimensions; the quotient of the cube by the antipodal map yields the hemicube. If the original cube has edge length 1, its dual polyhedron has edge length 2 / 2; the cube is a special case in various classes of general polyhedra: The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron. These two together form the stella octangula; the int
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
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is quantified numerically using the SI derived unit, the cubic metre; the volume of a container is understood to be the capacity of the container. Three dimensional mathematical shapes are assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, circular shapes can be calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space; the volume of a solid can be determined by fluid displacement. Displacement of liquid can be used to determine the volume of a gas; the combined volume of two substances is greater than the volume of just one of the substances. However, sometimes one substance dissolves in the other and in such cases the combined volume is not additive.
In differential geometry, volume is expressed by means of the volume form, is an important global Riemannian invariant. In thermodynamics, volume is a fundamental parameter, is a conjugate variable to pressure. Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube. In the International System of Units, the standard unit of volume is the cubic metre; the metric system includes the litre as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus 1 litre = 3 = 1000 cubic centimetres = 0.001 cubic metres. Small amounts of liquid are measured in millilitres, where 1 millilitre = 0.001 litres = 1 cubic centimetre. In the same way, large amounts can be measured in megalitres, where 1 million litres = 1000 cubic metres = 1 megalitre. Various other traditional units of volume are in use, including the cubic inch, the cubic foot, the cubic yard, the cubic mile, the teaspoon, the tablespoon, the fluid ounce, the fluid dram, the gill, the pint, the quart, the gallon, the minim, the barrel, the cord, the peck, the bushel, the hogshead, the acre-foot and the board foot.
Capacity is defined by the Oxford English Dictionary as "the measure applied to the content of a vessel, to liquids, grain, or the like, which take the shape of that which holds them". Capacity is not identical in meaning to volume, though related. Units of capacity are the SI litre and its derived units, Imperial units such as gill, pint and others. Units of volume are the cubes of units of length. In SI the units of volume and capacity are related: one litre is 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial; the density of an object is defined as the ratio of the mass to the volume. The inverse of density is specific volume, defined as volume divided by mass. Specific volume is a concept important in thermodynamics where the volume of a working fluid is an important parameter of a system being studied; the volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time. In calculus, a branch of mathematics, the volume of a region D in R3 is given by a triple integral of the constant function f = 1 and is written as: ∭ D 1 d x d y d z.
The volume integral in cylindrical coordinates is ∭ D r d r d θ d z, the volume integral in spherical coordinates has the form ∭ D ρ 2 sin ϕ d ρ d θ d ϕ. The above formulas can be used to show that the volumes of a cone and cylinder of the same radius and height are in the ratio 1: 2: 3, as follows. Let the radius be r and the height be h the volume of cone is 1 3 π r 2 h = 1 3 π r 2 = × 1, the volume of the sphere
In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are: hyperbolic sine "sinh", hyperbolic cosine "cosh",from which are derived: hyperbolic tangent "tanh", hyperbolic cosecant "csch" or "cosech" hyperbolic secant "sech", hyperbolic cotangent "coth",corresponding to the derived trigonometric functions; the inverse hyperbolic functions are: area hyperbolic sine "arsinh" and so on. Just as the points form a circle with a unit radius, the points form the right half of the equilateral hyperbola; the hyperbolic functions take. The size of a hyperbolic angle is twice the area of its hyperbolic sector; the hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. Hyperbolic functions occur in the solutions of many linear differential equations, of some cubic equations, in calculations of angles and distances in hyperbolic geometry, of Laplace's equation in Cartesian coordinates.
Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, special relativity. In complex analysis, the hyperbolic functions arise as the imaginary parts of cosine; the hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane. By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc. and Cc. to refer to circular functions and Sh. and Ch. to refer to hyperbolic functions. Lambert altered the abbreviations to what they are today; the abbreviations sh, ch, th, cth are at disposition, their use depending more on personal preference of mathematics of influence than on the local language. There are various equivalent ways for defining the hyperbolic functions.
They may be defined in terms of the exponential function: Hyperbolic sine: the odd part of the exponential function, sinh x = e x − e − x 2 = e 2 x − 1 2 e x = 1 − e − 2 x 2 e − x. Hyperbolic cosine: the part of the exponential function, cosh x = e x + e − x 2 = e 2 x + 1 2 e x = 1 + e − 2 x 2 e − x. Hyperbolic tangent: tanh x = sinh x cosh x = e x − e − x e x + e − x = e 2 x − 1 e 2 x + 1. Hyperbolic cotangent: for x ≠ 0, coth x = cosh x sinh x = e x + e − x e x − e − x = e 2 x + 1 e 2 x − 1. Hyperbolic secant: sech x = 1 cosh x = 2 e x + e − x = 2 e x e 2 x + 1. Hyperbolic cosecant: for x ≠ 0, csch x = 1 sinh x = 2