Multiplication is one of the four elementary mathematical operations of arithmetic. The multiplication of whole numbers may be thought as a repeated addition; the multiplier can be multiplicand second. A × b = b + ⋯ + b ⏟ a For example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together: 3 × 4 = 4 + 4 + 4 = 12 Here 3 and 4 are the factors and 12 is the product. One of the main properties of multiplication is the commutative property: adding 3 copies of 4 gives the same result as adding 4 copies of 3: 4 × 3 = 3 + 3 + 3 + 3 = 12 Thus the designation of multiplier and multiplicand does not affect the result of the multiplication; the multiplication of integers, rational numbers and real numbers is defined by a systematic generalization of this basic definition. Multiplication can be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths; the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property.
The product of two measurements is a new type of measurement, for instance multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division. For example, since 4 multiplied by 3 equals 12 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number. Multiplication is defined for other types of numbers, such as complex numbers, more abstract constructs, like matrices. For some of these more abstract constructs, the order in which the operands are multiplied together matters. A listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is written using the sign "×" between the terms. For example, 2 × 3 = 6 3 × 4 = 12 2 × 3 × 5 = 6 × 5 = 30 2 × 2 × 2 × 2 × 2 = 32 The sign is encoded in Unicode at U+00D7 × MULTIPLICATION SIGN. There are other mathematical notations for multiplication: Multiplication is denoted by dot signs a middle-position dot:5 ⋅ 2 or 5.
3 The middle dot notation, encoded in Unicode as U+22C5 ⋅ DOT OPERATOR, is standard in the United States, the United Kingdom, other countries where the period is used as a decimal point. When the dot operator character is not accessible, the interpunct is used. In other countries that use a comma as a decimal mark, either the period or a middle dot is used for multiplication. In algebra, multiplication involving variables is written as a juxtaposition called implied multiplication; the notation can be used for quantities that are surrounded by parentheses. This implicit usage of multiplication can cause ambiguity when the concatenated variables happen to match the name of another variable, when a variable name in front of a parenthesis can be confused with a function name, or in the correct determination of the order of operations. In vector multiplication, there is a distinction between the dot symbols; the cross symbol denotes the taking a cross product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar.
In computer programming, the asterisk is still the most common notation. This is due to the fact that most computers were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard; this usage originated in the FORTRAN programming language. The numbers to be multiplied are called the "factors"; the number to be multiplied is the "multiplicand", the number by which it is multiplied is the "multiplier". The multiplier is placed first and the multiplicand is placed second; as the result of a multiplication does not depend on the order of the factors, the distinction between "multiplicand" and "multiplier" is useful only at a elementary level and
In mathematics convolution is a mathematical operation on two functions to produce a third function that expresses how the shape of one is modified by the other. The term convolution refers to the process of computing it; some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, it differs from cross-correlation only in that either f or g is reflected about the y-axis. For continuous functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include probability, computer vision, natural language processing and signal processing and differential equations; the convolution can be defined for functions on Euclidean space, other groups. For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, in the design and implementation of finite impulse response filters in signal processing.
Computing the inverse of the convolution operation is known as deconvolution. The convolution of f and g is written f ∗ g, using an star, it is defined as the integral of the product of the two functions after one is shifted. As such, it is a particular kind of integral transform: An equivalent definition is: ≜ ∫ − ∞ ∞ f g d τ. While the symbol t is used above, it need not represent the time domain, but in that context, the convolution formula can be described as a weighted average of the function f at the moment t where the weighting is given by g shifted by amount t. As t changes, the weighting function emphasizes different parts of the input function. For functions f, g supported on only [0, ∞), the integration limits can be truncated, resulting in: = ∫ 0 t f g d τ for f, g: [ 0, ∞ ) → R. For the multi-dimensional formulation of convolution, see domain of definition. A common engineering convention is: f ∗ g ≜ ∫ − ∞ ∞ f g d τ ⏟, which has to be interpreted to avoid confusion. For instance, f ∗ g is equivalent to.
Convolution describes the output of an important class of operations known as linear time-invariant. See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created; the existing ones are only modified. In other words, the output transform is the pointwise product of the input transform with a third transform. See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms. One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754. An expression of the type: ∫ f ⋅ g d u is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800.
Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace, Jean-Baptiste Joseph Fourier, Siméon Denis Poisson, others. The term itself did not come into wide use until the 60s. Prior to that it was sometimes known as Faltung, composition product, superposition integral, Carson's integral, yet it appears as early as 1903. The o
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, measure, infinite series, analytic functions. These theories are studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary techniques of analysis. Analysis may be distinguished from geometry. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids; the explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century.
In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would be called Cavalieri's principle to find the volume of a sphere in the 5th century; the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine and arctangent. Alongside his development of the Taylor series of the trigonometric functions, he estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series, his followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. Descartes and Fermat independently developed analytic geometry, a few decades Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations and partial differential equations, Fourier analysis, generating functions.
During this period, calculus techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra used in earlier work by Euler. Instead, Cauchy formulated calculus in terms of geometric infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y, he introduced the concept of the Cauchy sequence, started the formal theory of complex analysis. Poisson, Liouville and others studied partial differential equations and harmonic analysis; the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis.
In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, introduced the "epsilon-delta" definition of limit. Mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. "monsters" began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, Baire proved the Baire category theorem.
In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, Hilbert introduced Hilbert spaces to solve integral equations; the idea of normed vector space was in the air, in the 1920s Banach created functional analysis. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. Much of analysis happens in some metric space. Examples of analysis without a metric include functional analysis. Formally, a metric space is an ordered pair where M is a set
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is defined becomes divergent; the step-wise continuation technique may, come up against difficulties. These may have an topological nature, leading to inconsistencies, they may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities need not be isolated points, its investigation was a major reason for the development of sheaf cohomology. Suppose f is an analytic function defined on a non-empty open subset U of the complex plane C. If V is a larger open subset of C, containing U, F is an analytic function defined on V such that F = f ∀ z ∈ U F is called an analytic continuation of f. In other words, the restriction of F to U is the function f we started with.
Analytic continuations are unique in the following sense: if V is the connected domain of two analytic functions F1 and F2 such that U is contained in V and for all z in U F1 = F2 = f,then F1 = F2on all of V. This is because F1 − F2 is an analytic function which vanishes on the open, connected domain U of f and hence must vanish on its entire domain; this follows directly from the identity theorem for holomorphic functions. A common way to define functions in complex analysis proceeds by first specifying the function on a small domain only, extending it by analytic continuation. In practice, this continuation is done by first establishing some functional equation on the small domain and using this equation to extend the domain. Examples are the gamma function; the concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function. The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of Riemann surfaces.
Begin with a particular analytic function f. In this case, it's given by a power series centered at z = 1: f = ∑ k = 0 ∞ k k. By the Cauchy–Hadamard theorem, its radius of convergence is 1; that is, f is defined and analytic on the open set U = which has boundary ∂ U =. Indeed, the series diverges at z = 0 ∈ ∂ U. Pretend we don't know that f = 1 / z, focus on recentering the power series at a different point a ∈ U: f = ∑ k = 0 ∞ a k k. We'll calculate the a k's and determine whether this new power series converges in an open set V, not contained in U. If so, we will have analytically continued f to the region U ∪ V, larger than U; the distance from a to ∂ U is ρ = 1 − | a − 1 | > 0. Take 0 < r < ρ. D ∪ ∂ D ⊂ U. Using Cauchy's differentiation formula to calculate the new coefficients, a k = f k! = 1 2 π i ∫ ∂ D f d ζ k + 1 = 1 2 π i ∫
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, 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. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia
Wacław Franciszek Sierpiński was a Polish mathematician. He was known for contributions to set number theory, theory of functions and topology, he published over 50 books. Three well-known fractals are named after him, as are Sierpinski numbers and the associated Sierpiński problem. Sierpiński enrolled in the Department of Mathematics and Physics at the University of Warsaw in 1899 and graduated four years later. In 1903, while still at the University of Warsaw, the Department of Mathematics and Physics offered a prize for the best essay from a student on Voronoy's contribution to number theory. Sierpiński was awarded a gold medal for his essay, thus laying the foundation for his first major mathematical contribution. Unwilling for his work to be published in Russian, he withheld it until 1907, when it was published in Samuel Dickstein's mathematical magazine'Prace Matematyczno-Fizyczne'. After his graduation in 1904, Sierpiński worked as a school teacher of mathematics and physics in Warsaw. However, when the school closed because of a strike, Sierpiński decided to go to Kraków to pursue a doctorate.
At the Jagiellonian University in Kraków he attended. He studied astronomy and philosophy, he received his doctorate and was appointed to the University of Lwów in 1908. In 1907 Sierpiński first became interested in set theory when he came across a theorem which stated that points in the plane could be specified with a single coordinate, he wrote to Tadeusz Banachiewicz. He received the one-word reply'Cantor'. Sierpiński began to study set theory and, in 1909, he gave the first lecture course devoted to the subject. Sierpiński maintained an output of research books. During the years 1908 to 1914, when he taught at the University of Lwów, he published three books in addition to many research papers; these books were The Theory of Irrational Numbers, Outline of Set Theory, The Theory of Numbers. When World War I began in 1914, Sierpiński and his family were in Russia. To avoid the persecution, common for Polish foreigners, Sierpiński spent the rest of the war years in Moscow working with Nikolai Luzin.
Together they began the study of analytic sets. In 1916, Sierpiński gave the first example of an normal number; when World War I ended in 1918, Sierpiński returned to Lwów. However shortly after taking up his appointment again in Lwów he was offered a post at the University of Warsaw, which he accepted. In 1919 he was promoted to a professor, he spent the rest of his life in Warsaw. During the Polish–Soviet War, Sierpiński helped break Soviet Russian ciphers for the Polish General Staff's cryptological agency. In 1920, Sierpiński, together with Zygmunt Janiszewski and his former student Stefan Mazurkiewicz, founded the mathematical journal Fundamenta Mathematicae. Sierpiński edited the journal. During this period, Sierpiński worked predominantly on set theory, but on point set topology and functions of a real variable. In set theory he made contributions on the continuum hypothesis, he proved that Zermelo–Fraenkel set theory together with the Generalized continuum hypothesis imply the Axiom of choice.
He worked on what is now known as the Sierpinski curve. Sierpiński continued to collaborate with Luzin on investigations of projective sets, his work on functions of a real variable includes results on functional series, differentiability of functions and Baire's classification. Sierpiński retired in 1960 as professor at the University of Warsaw, but continued until 1967 to give a seminar on the Theory of Numbers at the Polish Academy of Sciences, he continued editorial work as editor-in-chief of Acta Arithmetica, as an editorial-board member of Rendiconti del Circolo Matematico di Palermo, Composito Matematica, Zentralblatt für Mathematik. Sierpiński is interred at the Powązki Cemetery in Poland. Honorary Degrees: Lwów, St. Marks of Lima, Tarta, Prague, Wrocław, Moscow. For high involvement with the development of mathematics in Poland, Sierpiński was honored with election to the Polish Academy of Learning in 1921 and that same year was made dean of the faculty at the University of Warsaw. In 1928, he became vice-chairman of the Warsaw Scientific Society, that same year was elected chairman of the Polish Mathematical Society.
He was elected to the Geographic Society of Lima, the Royal Scientific Society of Liège, the Bulgarian Academy of Sciences, the National Academy of Lima, the Royal Society of Sciences of Naples, the Accademia dei Lincei of Rome, the Germany Academy of Sciences, the United States National Academy of Sciences, the Paris Academy, the Royal Dutch Academy, the Academy of Science of Brussels, the London Mathematical Society, the Romanian Academy and the Papal Academy of Sciences. In 1949 Sierpiński was awarded Poland's Scientific Prize, first degree. Sierpiński authored 50 books. W. Sierpiński. Elementary theory of numbers. Monografie Matematyczne. 42. ISBN 0-444-86662-0. Arity theorem List of Poles Menger sponge
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