1.
Non-integer representation
–
A non-integer representation uses non-integer numbers as the radix, or bases, of a positional numbering system. For a non-integer radix β >1, the value of x = d n … d 2 d 1 d 0, the numbers di are non-negative integers less than β. This is also known as a β-expansion, an introduced by Rényi. Every real number has at least one β-expansion, there are applications of β-expansions in coding theory and models of quasicrystals. β-expansions are a generalization of decimal expansions, while infinite decimal expansions are not unique, all finite decimal expansions are unique. However, even finite β-expansions are not necessarily unique, for example φ +1 = φ2 for β = φ, the golden ratio. A canonical choice for the β-expansion of a real number can be determined by the following greedy algorithm, essentially due to Rényi. Let β >1 be the base and x a non-negative real number, denote by ⌊x⌋ the floor function of x, that is, the greatest integer less than or equal to x, and let = x − ⌊x⌋ be the fractional part of x. There exists a k such that βk ≤ x < βk+1. Set d k = ⌊ x / β k ⌋ and r k =, for k −1 ≥ j > −∞, put d j = ⌊ β r j +1 ⌋, r j =. In other words, the canonical β-expansion of x is defined by choosing the largest dk such that βkdk ≤ x, then choosing the largest dk−1 such that βkdk + βk−1dk−1 ≤ x, thus it chooses the lexicographically largest string representing x. With an integer base, this defines the usual radix expansion for the number x and this construction extends the usual algorithm to possibly non-integer values of β. See Golden ratio base, 11φ = 100φ, with base e the natural logarithm behaves like the common logarithm as ln =0, ln =1, ln =2 and ln =3. This means that every integer can be expressed in base √2 without the need of a decimal point, another use of the base is to show the silver ratio as its representation in base √2 is simply 11√2. In no positional number system can every number be expressed uniquely, for example, in base ten, the number 1 has two representations,1.000. and 0.999. Another problem is to classify the real numbers whose β-expansions are periodic, let β >1, and Q be the smallest field extension of the rationals containing β. Then any real number in [0, 1) having a periodic β-expansion must lie in Q, on the other hand, the converse need not be true. The converse does hold if β is a Pisot number, although necessary and sufficient conditions are not known
2.
Power law
–
For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four. Few empirical distributions fit a power law for all their values, acoustic attenuation follows frequency power-laws within wide frequency bands for many complex media. Allometric scaling laws for relationships between biological variables are among the best known power-law functions in nature, one attribute of power laws is their scale invariance. Given a relation f = a x − k, scaling the argument x by a constant factor c causes only a proportionate scaling of the function itself and that is, f = a − k = c − k f ∝ f. That is, scaling by a constant c simply multiplies the original power-law relation by the constant c − k, thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both f and x, and the straight-line on the plot is often called the signature of a power law. With real data, such straightness is a necessary, but not sufficient, in fact, there are many ways to generate finite amounts of data that mimic this signature behavior, but, in their asymptotic limit, are not true power laws. Thus, accurately fitting and validating power-law models is an area of research in statistics. This can be seen in the thought experiment, imagine a room with your friends. Now imagine the worlds richest person entering the room, with an income of about 1 billion US$. What happens to the income in the room. Income is distributed according to a known as the Pareto distribution. On the one hand, this makes it incorrect to apply traditional statistics that are based on variance, on the other hand, this also allows for cost-efficient interventions. For example, given that car exhaust is distributed according to a power-law among cars it would be sufficient to eliminate those very few cars from the road to reduce total exhaust substantially. For instance, the behavior of water and CO2 at their boiling points fall in the universality class because they have identical critical exponents. In fact, almost all material phase transitions are described by a set of universality classes. Similar observations have made, though not as comprehensively, for various self-organized critical systems. Formally, this sharing of dynamics is referred to as universality, scientific interest in power-law relations stems partly from the ease with which certain general classes of mechanisms generate them
3.
Asymptote
–
In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, in some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος which means not falling together, + σύν together + πτωτ-ός fallen. The term was introduced by Apollonius of Perga in his work on conic sections, there are potentially three kinds of asymptotes, horizontal, vertical and oblique asymptotes. For curves given by the graph of a function y = ƒ, vertical asymptotes are vertical lines near which the function grows without bound. Asymptotes convey information about the behavior of curves in the large, the study of asymptotes of functions, construed in a broad sense, forms a part of the subject of asymptotic analysis. The idea that a curve may come close to a line without actually becoming the same may seem to counter everyday experience. The representations of a line and a curve as marks on a piece of paper or as pixels on a screen have a positive width. So if they were to be extended far enough they would seem to merge, but these are physical representations of the corresponding mathematical entities, the line and the curve are idealized concepts whose width is 0. Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience, consider the graph of the function f =1 x shown to the right. The coordinates of the points on the curve are of the form where x is an other than 0. But no matter how large x becomes, its reciprocal 1 x is never 0, so the curve extends farther and farther upward as it comes closer and closer to the y-axis. Thus, both the x and y-axes are asymptotes of the curve and these ideas are part of the basis of concept of a limit in mathematics, and this connection is explained more fully below. The asymptotes most commonly encountered in the study of calculus are of curves of the form y = ƒ and these can be computed using limits and classified into horizontal, vertical and oblique asymptotes depending on its orientation. Horizontal asymptotes are lines that the graph of the function approaches as x tends to +∞ or −∞. As the name indicate they are parallel to the x-axis, vertical asymptotes are vertical lines near which the function grows without bound. Oblique asymptotes are diagonal lines so that the difference between the curve and the line approaches 0 as x tends to +∞ or −∞, more general type of asymptotes can be defined in this case. Only open curves that have some infinite branch, can have an asymptote, no closed curve can have an asymptote
4.
Logarithm
–
In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, in simple cases the logarithm counts factors in multiplication. For example, the base 10 logarithm of 1000 is 3, the logarithm of x to base b, denoted logb, is the unique real number y such that by = x. For example, log2 =6, as 64 =26, the logarithm to base 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the e as its base, its use is widespread in mathematics and physics. The binary logarithm uses base 2 and is used in computer science. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations and they were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the function in the 18th century. Logarithmic scales reduce wide-ranging quantities to tiny scopes, for example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios. In chemistry, pH is a measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and they describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant, it has uses in public-key cryptography, the idea of logarithms is to reverse the operation of exponentiation, that is, raising a number to a power. For example, the power of 2 is 8, because 8 is the product of three factors of 2,23 =2 ×2 ×2 =8. It follows that the logarithm of 8 with respect to base 2 is 3, the third power of some number b is the product of three factors equal to b. More generally, raising b to the power, where n is a natural number, is done by multiplying n factors equal to b. The n-th power of b is written bn, so that b n = b × b × ⋯ × b ⏟ n factors, exponentiation may be extended to by, where b is a positive number and the exponent y is any real number. For example, b−1 is the reciprocal of b, that is, the logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x
5.
Base (exponentiation)
–
In exponentiation, the base is the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b and it is more commonly expressed as the nth power of b, b to the nth power or b to the power n. For example, the power of 10 is 10,000 because 104 =10 ×10 ×10 ×10 =10,000. The term power strictly refers to the expression, but is sometimes used to refer to the exponent. When the nth power of b equals a number a, or a = bn, for example,10 is a fourth root of 10,000. The inverse function to exponentiation with base b is called the logarithm to base b, for example, log1010,000 =4
6.
Mathematical constant
–
A mathematical constant is a special number, usually a real number, that is significantly interesting in some way. Constants arise in areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory. The more popular constants have been studied throughout the ages and computed to many decimal places, all mathematical constants are definable numbers and usually are also computable numbers. These are constants which one is likely to encounter during pre-college education in many countries, however, its ubiquity is not limited to pure mathematics. It appears in many formulas in physics, and several physical constants are most naturally defined with π or its reciprocal factored out and it is debatable, however, if such appearances are fundamental in any sense. For example, the textbook nonrelativistic ground state wave function of the atom is ψ =11 /2 e − r / a 0. This formula contains a π, but it is unclear if that is fundamental in a physical sense, furthermore, this formula gives only an approximate description of physical reality, as it omits spin, relativity, and the quantal nature of the electromagnetic field itself. The numeric value of π is approximately 3.1415926535, memorizing increasingly precise digits of π is a world record pursuit. The constant e also has applications to probability theory, where it arises in a way not obviously related to exponential growth, suppose a slot machine with a one in n probability of winning is played n times. Then, for large n the probability that nothing will be won is approximately 1/e, another application of e, discovered in part by Jacob Bernoulli along with French mathematician Pierre Raymond de Montmort, is in the problem of derangements, also known as the hat check problem. Here n guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into labelled boxes, the butler does not know the name of the guests, and so must put them into boxes selected at random. The problem of de Montmort is, what is the probability that none of the hats gets put into the right box, the answer is p n =1 −11. + ⋯ + n 1 n. and as n tends to infinity, the numeric value of e is approximately 2.7182818284. The square root of 2, often known as root 2, radical 2, or Pythagorass constant, and written as √2, is the algebraic number that. It is more called the principal square root of 2. Geometrically the square root of 2 is the length of a diagonal across a square sides of one unit of length. It was probably the first number known to be irrational and its numerical value truncated to 65 decimal places is,1.41421356237309504880168872420969807856967187537694807317667973799. The quick approximation 99/70 for the root of two is frequently used
7.
Irrational number
–
In mathematics, the irrational numbers are all the real numbers, which are not rational numbers, the latter being the numbers constructed from ratios of integers. Irrational numbers may also be dealt with via non-terminating continued fractions, for example, the decimal representation of the number π starts with 3.14159265358979, but no finite number of digits can represent π exactly, nor does it repeat. Mathematicians do not generally take terminating or repeating to be the definition of the concept of rational number, as a consequence of Cantors proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of numbers is usually attributed to a Pythagorean. The then-current Pythagorean method would have claimed that there must be sufficiently small. However, Hippasus, in the 5th century BC, was able to deduce that there was in no common unit of measure. His reasoning is as follows, Start with a right triangle with side lengths of integers a, b. The ratio of the hypotenuse to a leg is represented by c, b, assume a, b, and c are in the smallest possible terms. By the Pythagorean theorem, c2 = a2+b2 = b2+b2 = 2b2, since c2 = 2b2, c2 is divisible by 2, and therefore even. Since c2 is even, c must be even, since c is even, dividing c by 2 yields an integer. Squaring both sides of c = 2y yields c2 =2, or c2 = 4y2, substituting 4y2 for c2 in the first equation gives us 4y2= 2b2. Dividing by 2 yields 2y2 = b2, since y is an integer, and 2y2 = b2, b2 is divisible by 2, and therefore even. Since b2 is even, b must be even and we have just show that both b and c must be even. Hence they have a factor of 2. However this contradicts the assumption that they have no common factors and this contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers. Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. ”Another legend states that Hippasus was merely exiled for this revelation, the discovery of incommensurable ratios was indicative of another problem facing the Greeks, the relation of the discrete to the continuous. Brought into light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a number of units of a given size. ”However Zeno found that in fact “ in general are not discrete collections of units. That in fact, these divisions of quantity must necessarily be infinite, for example, consider a line segment, this segment can be split in half, that half split in half, the half of the half in half, and so on
8.
Transcendental number
–
In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with integer coefficients. The best-known transcendental numbers are π and e, though only a few classes of transcendental numbers are known, transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the numbers are countable while the sets of real. All real transcendental numbers are irrational, since all numbers are algebraic. Another irrational number that is not transcendental is the ratio, φ or ϕ. The name transcendental comes from the root trans meaning across and length of numbers, euler was probably the first person to define transcendental numbers in the modern sense. Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1768 paper proving the number π is irrational, and proposed a tentative sketch of a proof of πs transcendence. In other words, the nth digit of this number is 1 only if n is one of the numbers 1. Liouville showed that number is what we now call a Liouville number. Liouville showed that all Liouville numbers are transcendental, the first number to be proven transcendental without having been specifically constructed for the purpose was e, by Charles Hermite in 1873. In 1874, Georg Cantor proved that the numbers are countable. He also gave a new method for constructing transcendental numbers, in 1878, Cantor published a construction that proves there are as many transcendental numbers as there are real numbers. Cantors work established the ubiquity of transcendental numbers, in 1882, Ferdinand von Lindemann published a proof that the number π is transcendental. He first showed that ea is transcendental when a is algebraic, then, since eiπ = −1 is algebraic, iπ and therefore π must be transcendental. This approach was generalized by Karl Weierstrass to the Lindemann–Weierstrass theorem, the transcendence of π allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle. The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem and this work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms. The set of numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a number of zeroes
9.
Parentheses
–
A bracket is a tall punctuation mark typically used in matched pairs within text, to set apart or interject other text. The matched pair may be described as opening and closing, or left, forms include round, square, curly, and angle brackets, and various other pairs of symbols. Chevrons were the earliest type of bracket to appear in written English, desiderius Erasmus coined the term lunula to refer to the rounded parentheses, recalling the shape of the crescent moon. Some of the names are regional or contextual. Sometimes referred to as angle brackets, in cases as HTML markup. Occasionally known as broken brackets or brokets, ⸤ ⸥, ｢ ｣ – corner brackets ⟦ ⟧ – double square brackets, white square brackets Guillemets, ‹ › and « », are sometimes referred to as chevrons or angle brackets. The characters ‹ › and « », known as guillemets or angular quote brackets, are actually quotation mark glyphs used in several European languages, which one of each pair is the opening quote mark and which is the closing quote varies between languages. In English, typographers generally prefer to not set brackets in italics, however, in other languages like German, if brackets enclose text in italics, they are usually set in italics too. Parentheses /pəˈrɛnθᵻsiːz/ contain material that serves to clarify or is aside from the main point, a milder effect may be obtained by using a pair of commas as the delimiter, though if the sentence contains commas for other purposes, visual confusion may result. In American usage, parentheses are considered separate from other brackets. Parentheses may be used in writing to add supplementary information. They can also indicate shorthand for either singular or plural for nouns and it can also be used for gender neutral language, especially in languages with grammatical gender, e. g. he agreed with his physician. Parenthetical phrases have been used extensively in informal writing and stream of consciousness literature, examples include the southern American author William Faulkner as well as poet E. E. Cummings. Parentheses have historically been used where the dash is used in alternatives, such as parenthesis) is used to indicate an interval from a to c that is inclusive of a. That is, [5, 12) would be the set of all numbers between 5 and 12, including 5 but not 12. The numbers may come as close as they like to 12, including 11.999 and so forth, in some European countries, the notation [5, 12[ is also used for this. The endpoint adjoining the bracket is known as closed, whereas the endpoint adjoining the parenthesis is known as open, if both types of brackets are the same, the entire interval may be referred to as closed or open as appropriate. Whenever +∞ or −∞ is used as an endpoint, it is considered open
10.
Exponentiation
–
Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. The exponent is usually shown as a superscript to the right of the base, Some common exponents have their own names, the exponent 2 is called the square of b or b squared, the exponent 3 is called the cube of b or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b, when n is a positive integer and b is not zero, b−n is naturally defined as 1/bn, preserving the property bn × bm = bn + m. The definition of exponentiation can be extended to any real or complex exponent. Exponentiation by integer exponents can also be defined for a variety of algebraic structures. The term power was used by the Greek mathematician Euclid for the square of a line, archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10. In the late 16th century, Jost Bürgi used Roman numerals for exponents, early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie, there, the notation is introduced in Book I. Nicolas Chuquet used a form of notation in the 15th century. The word exponent was coined in 1544 by Michael Stifel, samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the square, cube, zenzizenzic, sursolid, zenzicube, second sursolid. Biquadrate has been used to refer to the power as well. Some mathematicians used exponents only for greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d, another historical synonym, involution, is now rare and should not be confused with its more common meaning. In 1748 Leonhard Euler wrote consider exponentials or powers in which the exponent itself is a variable and it is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant. With this introduction of transcendental functions, Euler laid the foundation for the introduction of natural logarithm as the inverse function for y = ex. The expression b2 = b ⋅ b is called the square of b because the area of a square with side-length b is b2, the expression b3 = b ⋅ b ⋅ b is called the cube of b because the volume of a cube with side-length b is b3. The exponent indicates how many copies of the base are multiplied together, for example,35 =3 ⋅3 ⋅3 ⋅3 ⋅3 =243. The base 3 appears 5 times in the multiplication, because the exponent is 5
11.
Real number
–
In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
12.
Integral
–
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two operations of calculus, with its inverse, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total, roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative. In this case, it is called an integral and is written. The integrals discussed in this article are those termed definite integrals, a rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space and this method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was developed in China around the 3rd century AD by Liu Hui. This method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi. The next significant advances in integral calculus did not begin to appear until the 17th century, further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the theorem of calculus. Wallis generalized Cavalieris method, computing integrals of x to a power, including negative powers. The major advance in integration came in the 17th century with the independent discovery of the theorem of calculus by Newton. The theorem demonstrates a connection between integration and differentiation and this connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the mathematical framework that both Newton and Leibniz developed
13.
Complex number
–
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
14.
Multi-valued function
–
In mathematics, a multivalued function is a left-total relation. In the strict sense, a well-defined function associates one, and only one, the term multivalued function is, therefore, a misnomer because functions are single-valued. Multivalued functions often arise as inverses of functions that are not injective, such functions do not have an inverse function, but they do have an inverse relation. The multivalued function corresponds to this inverse relation, every real number greater than zero has two real square roots. The square roots of 4 are in the set, the square root of 0 is 0. Each complex number except zero has two roots, three cube roots, and in general n nth roots. The complex logarithm function is multiple-valued, the values assumed by log for real numbers a and b are log a 2 + b 2 + i arg +2 π n i for all integers n. Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic and we have tan = tan = tan = tan = ⋯ =1. As a consequence, arctan is intuitively related to several values, π/4, 5π/4, −3π/4 and we can treat arctan as a single-valued function by restricting the domain of tan x to −π/2 < x < π/2 – a domain over which tan x is monotonically increasing. Thus, the range of arctan becomes −π/2 < y < π/2 and these values from a restricted domain are called principal values. The indefinite integral can be considered as a multivalued function, the indefinite integral of a function is the set of functions whose derivative is that function. The constant of integration follows from the fact that the derivative of a constant function is 0 and these are all examples of multivalued functions that come about from non-injective functions. Since the original functions do not preserve all the information of their inputs, often, the restriction of a multivalued function is a partial inverse of the original function. Multivalued functions of a variable have branch points. For example, for the nth root and logarithm functions,0 is a point, for the arctangent function. Using the branch points, these functions may be redefined to be single-valued functions, by restricting the range. A suitable interval may be found through use of a branch cut, as in the case with real functions, the restricted range may be called principal branch of the function. Set-valued analysis is the study of sets in the spirit of mathematical analysis, instead of considering collections of only points, set-valued analysis considers collections of sets
15.
Complex logarithm
–
In complex analysis, a complex logarithm function is an inverse of the complex exponential function, just as the real natural logarithm ln x is the inverse of the real exponential function ex. Thus, a logarithm of a number z is a complex number w such that ew = z. The notation for such a w is ln z or log z, since every nonzero complex number z has infinitely many logarithms, care is required to give such notation an unambiguous meaning. If z = reiθ with r >0, then w = ln r + iθ is one logarithm of z, for a function to have an inverse, it must map distinct values to distinct values, i. e. be injective. But the complex function is not injective, because ew+2πi = ew for any w. So the exponential function does not have a function in the standard sense. There are two solutions to this problem and this is analogous to the definition of arcsin x on as the inverse of the restriction of sin θ to the interval, there are infinitely many real numbers θ with sin θ = x, but one chooses the one in. Branches have the advantage that they can be evaluated at complex numbers, on the other hand, the function on the Riemann surface is elegant in that it packages together all branches of log z and does not require an arbitrary choice as part of its definition. For each nonzero complex number z = x + yi, the principal value Log z is the logarithm whose imaginary part lies in the interval (−π, the expression Log 0 is left undefined since there is no complex number w satisfying ew =0. The principal value can be described also in a few other ways, to give a formula for Log z, begin by expressing z in polar form, z = reiθ. For example, Log = ln 3 − πi/2, another way to describe Log z is as the inverse of a restriction of the complex exponential function, as in the previous section. In fact, the function maps S bijectively to the punctured complex plane C × = C ∖. The conformal mapping section below explains the properties of this map in more detail. When the notation log z appears without any particular logarithm having been specified, in particular, this gives a value consistent with the real value of ln z when z is a positive real number. The capitalization in the notation Log is used by authors to distinguish the principal value from other logarithms of z. Not all identities satisfied by ln extend to complex numbers and it is true that eLog z = z for all z ≠0, but the identity Log ez = z fails for z outside the strip S. For this reason, one cannot always apply Log to both sides of an identity ez = ew to deduce z = w, the function Log z is discontinuous at each negative real number, but continuous everywhere else in C ×. To explain the discontinuity, consider what happens to Arg z as z approaches a negative real number a, if z approaches a from above, then Arg z approaches π, which is also the value of Arg a itself
16.
Real-valued function
–
In mathematics, a real-valued function or real function is a function whose values are real numbers. In other words, it is a function that assigns a number to each member of its domain. Many important function spaces are defined to consist of real functions, let X be an arbitrary set. Let F denote the set of all functions from X to real numbers R. F is an ordered ring. The σ-algebra of Borel sets is an important structure on real numbers, if X has its σ-algebra and a function f is such that the preimage f −1 of any Borel set B belongs to that σ-algebra, then f is said to be measurable. Measurable functions also form a space and an algebra as explained above. Moreover, a set of real-valued functions on X can actually define a σ-algebra on X generated by all preimages of all Borel sets and this is the way how σ-algebras arise in probability theory, where real-valued functions on the sample space Ω are real-valued random variables. Real numbers form a space and a complete metric space. Continuous real-valued functions are important in theories of topological spaces and of metric spaces, the extreme value theorem states that for any real continuous function on a compact space its global maximum and minimum exist. The concept of space itself is defined with a real-valued function of two variables, the metric, which is continuous. The space of functions on a compact Hausdorff space has a particular importance. Convergent sequences also can be considered as real-valued continuous functions on a topological space. Continuous functions also form a space and an algebra as explained above. Real numbers are used as the codomain to define smooth functions, a domain of a real smooth function can be the real coordinate space, a topological vector space, an open subset of them, or a smooth manifold. Spaces of smooth functions also are vector spaces and algebras as explained above, a measure on a set is a non-negative real-valued functional on a σ-algebra of subsets. Lp spaces on sets with a measure are defined from aforementioned real-valued measurable functions, though, real-valued Lp spaces still have some of the structure explicated above. For example, pointwise product of two L2 functions belongs to L1, Real analysis Partial differential equations, a major user of real-valued functions Norm Scalar Weisstein, Eric W. Real Function
17.
Inverse function
–
I. e. f = y if and only if g = x. As a simple example, consider the function of a real variable given by f = 5x −7. Thinking of this as a procedure, to reverse this and get x back from some output value, say y. In this case means that we should add 7 to y. In functional notation this inverse function would be given by, g = y +75, with y = 5x −7 we have that f = y and g = x. Not all functions have inverse functions, in order for a function f, X → Y to have an inverse, it must have the property that for every y in Y there must be one, and only one x in X so that f = y. This property ensures that a function g, Y → X will exist having the necessary relationship with f, let f be a function whose domain is the set X, and whose image is the set Y. Then f is invertible if there exists a g with domain Y and image X, with the property. If f is invertible, the g is unique, which means that there is exactly one function g satisfying this property. That function g is called the inverse of f, and is usually denoted as f −1. Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y, not all functions have an inverse. For a function to have an inverse, each element y ∈ Y must correspond to no more than one x ∈ X, a function f with this property is called one-to-one or an injection. If f −1 is to be a function on Y, then each element y ∈ Y must correspond to some x ∈ X. Functions with this property are called surjections. This property is satisfied by definition if Y is the image of f, to be invertible a function must be both an injection and a surjection. If a function f is invertible, then both it and its inverse function f−1 are bijections, there is another convention used in the definition of functions. This can be referred to as the set-theoretic or graph definition using ordered pairs in which a codomain is never referred to, under this convention all functions are surjections, and so, being a bijection simply means being an injection. Authors using this convention may use the phrasing that a function is invertible if, the two conventions need not cause confusion as long as it is remembered that in this alternate convention the codomain of a function is always taken to be the range of the function. With this type of function it is impossible to deduce an input from its output, such a function is called non-injective or, in some applications, information-losing
18.
Exponential function
–
In mathematics, an exponential function is a function of the form in which the input variable x occurs as an exponent. A function of the form f = b x + c, as functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the logarithm of the base b. The argument of the function can be any real or complex number or even an entirely different kind of mathematical object. Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the function is the most important function in mathematics. In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same change in the dependent variable. The graph of y = e x is upward-sloping, and increases faster as x increases, the graph always lies above the x -axis but can get arbitrarily close to it for negative x, thus, the x -axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y -coordinate at that point, as implied by its derivative function. Its inverse function is the logarithm, denoted log, ln, or log e, because of this. The exponential function exp, C → C can be characterized in a variety of equivalent ways, the constant e is then defined as e = exp = ∑ k =0 ∞. The exponential function arises whenever a quantity grows or decays at a proportional to its current value. One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683 to the number lim n → ∞ n now known as e, later, in 1697, Johann Bernoulli studied the calculus of the exponential function. If instead interest is compounded daily, this becomes 365, letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, exp = lim n → ∞ n first given by Euler. This is one of a number of characterizations of the exponential function, from any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity, exp = exp ⋅ exp which is why it can be written as ex. The derivative of the function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself is expressible in terms of the exponential function and this function property leads to exponential growth and exponential decay. The exponential function extends to a function on the complex plane. Eulers formula relates its values at purely imaginary arguments to trigonometric functions, the exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra
19.
Group (mathematics)
–
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
20.
Function (mathematics)
–
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that each real number x to its square x2. The output of a function f corresponding to a x is denoted by f. In this example, if the input is −3, then the output is 9, likewise, if the input is 3, then the output is also 9, and we may write f =9. The input variable are sometimes referred to as the argument of the function, Functions of various kinds are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function, some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function, in science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, sometimes the codomain is called the functions range, but more commonly the word range is used to mean, instead, specifically the set of outputs. For example, we could define a function using the rule f = x2 by saying that the domain and codomain are the numbers. The image of this function is the set of real numbers. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, another important operation defined on functions is function composition, where the output from one function becomes the input to another function. Linking each shape to its color is a function from X to Y, each shape is linked to a color, there is no shape that lacks a color and no shape that has more than one color. This function will be referred to as the color-of-the-shape function, the input to a function is called the argument and the output is called the value. The set of all permitted inputs to a function is called the domain of the function. Thus, the domain of the function is the set of the four shapes. The concept of a function does not require that every possible output is the value of some argument, a second example of a function is the following, the domain is chosen to be the set of natural numbers, and the codomain is the set of integers. The function associates to any number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6, a third example of a function has the set of polygons as domain and the set of natural numbers as codomain
21.
Binary logarithm
–
In mathematics, the binary logarithm is the power to which the number 2 must be raised to obtain the value n. That is, for any number x, x = log 2 n ⟺2 x = n. For example, the logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2. The binary logarithm is the logarithm to the base 2, the binary logarithm function is the inverse function of the power of two function. As well as log2, alternative notations for the binary logarithm include lg, ld, lb, and log. Binary logarithms can be used to calculate the length of the representation of a number in the numeral system. In computer science, they count the number of steps needed for binary search, other areas in which the binary logarithm is frequently used include combinatorics, bioinformatics, the design of sports tournaments, and photography. Binary logarithms are included in the standard C mathematical functions and other software packages. The integer part of a binary logarithm can be using the find first set operation on an integer value. The fractional part of the logarithm can be calculated efficiently, the powers of two have been known since antiquity, for instance they appear in Euclids Elements, Props. And the binary logarithm of a power of two is just its position in the sequence of powers of two. On this basis, Michael Stifel has been credited with publishing the first known table of binary logarithms in 1544 and his book Arthmetica Integra contains several tables that show the integers with their corresponding powers of two. Reversing the rows of these allow them to be interpreted as tables of binary logarithms. Earlier than Stifel, the 8th century Jain mathematician Virasena is credited with a precursor to the binary logarithm, virasenas concept of ardhacheda has been defined as the number of times a given number can be divided evenly by two. This definition gives rise to a function that coincides with the logarithm on the powers of two, but it is different for other integers, giving the 2-adic order rather than the logarithm. The modern form of a logarithm, applying to any number was considered explicitly by Leonhard Euler in 1739. Euler established the application of binary logarithms to music theory, long before their more significant applications in information theory, as part of his work in this area, Euler published a table of binary logarithms of the integers from 1 to 8, to seven decimal digits of accuracy. Alternatively, it may be defined as ln n/ln 2, where ln is the natural logarithm, using the complex logarithm in this definition allows the binary logarithm to be extended to the complex numbers
22.
Half-life
–
Half-life is the time required for a quantity to reduce to half its initial value. The term is used in nuclear physics to describe how quickly unstable atoms undergo. The term is used more generally to characterize any type of exponential or non-exponential decay. For example, the medical sciences refer to the biological half-life of drugs, the converse of half-life is doubling time. The original term, half-life period, dating to Ernest Rutherfords discovery of the principle in 1907, was shortened to half-life in the early 1950s. Rutherford applied the principle of a radioactive elements half-life to studies of age determination of rocks by measuring the period of radium to lead-206. Half-life is constant over the lifetime of an exponentially decaying quantity, the accompanying table shows the reduction of a quantity as a function of the number of half-lives elapsed. A half-life usually describes the decay of discrete entities, such as radioactive atoms, in that case, it does not work to use the definition that states half-life is the time required for exactly half of the entities to decay. For example, if there are 3 radioactive atoms with a half-life of one second, instead, the half-life is defined in terms of probability, Half-life is the time required for exactly half of the entities to decay on average. In other words, the probability of a radioactive atom decaying within its half-life is 50%, for example, the image on the right is a simulation of many identical atoms undergoing radioactive decay. Note that after one half-life there are not exactly one-half of the remaining, only approximately. Nevertheless, when there are many identical atoms decaying, the law of large numbers suggests that it is a good approximation to say that half of the atoms remain after one half-life. There are various simple exercises that demonstrate probabilistic decay, for example involving flipping coins or running a computer program. The three parameters t1⁄2, τ, and λ are all related in the following way. Amount approaches zero as t approaches infinity as expected, some quantities decay by two exponential-decay processes simultaneously. There is a half-life describing any exponential-decay process, for example, The current flowing through an RC circuit or RL circuit decays with a half-life of RCln or lnL/R, respectively. For this example, the half time might be used instead of half life. In a first-order chemical reaction, the half-life of the reactant is ln/λ, in radioactive decay, the half-life is the length of time after which there is a 50% chance that an atom will have undergone nuclear decay
23.
Exponential decay
–
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the differential equation. The solution to this equation is, N = N0 e − λ t, where N is the quantity at time t, and N0 = N is the initial quantity, i. e. the quantity at time t =0. If the decaying quantity, N, is the number of elements in a certain set. This is called the lifetime, τ, and it can be shown that it relates to the decay rate, λ, in the following way. For example, if the population of the assembly, N, is 1000. A very similar equation will be seen below, which arises when the base of the exponential is chosen to be 2, in that case the scaling time is the half-life. A more intuitive characteristic of exponential decay for many people is the time required for the quantity to fall to one half of its initial value. This time is called the half-life, and often denoted by the symbol t1/2, the half-life can be written in terms of the decay constant, or the mean lifetime, as, t 1 /2 = ln λ = τ ln . When this expression is inserted for τ in the equation above, and ln 2 is absorbed into the base. Thus, the amount of left is 2−1 = 1/2 raised to the number of half-lives that have passed. Thus, after 3 half-lives there will be 1/23 = 1/8 of the material left. Therefore, the mean lifetime τ is equal to the half-life divided by the log of 2, or. E. g. polonium-210 has a half-life of 138 days, the equation that describes exponential decay is d N d t = − λ N or, by rearranging, d N N = − λ d t. This is the form of the equation that is most commonly used to describe exponential decay, any one of decay constant, mean lifetime, or half-life is sufficient to characterise the decay. The notation λ for the constant is a remnant of the usual notation for an eigenvalue. In this case, λ is the eigenvalue of the negative of the operator with N as the corresponding eigenfunction. The units of the constant are s−1
24.
Compound interest
–
Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. It is the result of reinvesting interest, rather than paying it out, Compound interest is standard in finance and economics. Compound interest may be contrasted with simple interest, where interest is not added to the principal, the simple annual interest rate is the interest amount per period, multiplied by the number of periods per year. The simple annual interest rate is known as the nominal interest rate. The compounding frequency is the number of times per year the accumulated interest is paid out, or capitalized, the frequency could be yearly, half-yearly, quarterly, monthly, weekly, daily. For example, monthly capitalization with annual rate of interest means that the frequency is 12. The effect of compounding depends on, The nominal interest rate which is applied, the nominal rate cannot be directly compared between loans with different compounding frequencies. Both the nominal interest rate and the frequency are required in order to compare interest-bearing financial instruments. The effective annual rate is the accumulated interest that would be payable up to the end of one year. There are usually two aspects to the rules defining these rates, The rate is the compound interest rate. The effect of fees or taxes which the customer is charged, exactly which fees and taxes are included or excluded varies by country. May or may not be comparable between different jurisdictions, because the use of terms may be inconsistent, and vary according to local practice. 1,000 Brazilian real is deposited into a Brazilian savings account paying 20% per annum, at the end of one year,1,000 x 20% =200 BRL interest is credited to the account. The account then earns 1,200 x 20% =240 BRL in the second year. A rate of 1% per month is equivalent to an annual interest rate of 12%, but allowing for the effect of compounding. The interest on bonds and government bonds is usually payable twice yearly. The amount of interest paid is the disclosed interest rate divided by two and multiplied by the principal, the yearly compounded rate is higher than the disclosed rate. Canadian mortgage loans are generally compounded semi-annually with monthly payments, U. S. mortgages use an amortizing loan, not compound interest
25.
Algebraic number
–
An algebraic number is any complex number that is a root of a non-zero polynomial in one variable with rational coefficients. All integers and rational numbers are algebraic, as are all roots of integers, the same is not true for all real and complex numbers because they also include transcendental numbers such as π and e. Almost all real and complex numbers are transcendental, the rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the above definition because x = a/b is the root of bx − a. The quadratic surds are algebraic numbers, if the quadratic polynomial is monic then the roots are quadratic integers. The constructible numbers are numbers that can be constructed from a given unit length using straightedge. These include all quadratic surds, all numbers, and all numbers that can be formed from these using the basic arithmetic operations. Any expression formed from algebraic numbers using any combination of the arithmetic operations. Polynomial roots that cannot be expressed in terms of the arithmetic operations. This happens with many, but not all, polynomials of degree 5 or higher, gaussian integers, those complex numbers a + bi where both a and b are integers are also quadratic integers. Trigonometric functions of rational multiples of π, that is, the trigonometric numbers, for example, each of cos π/7, cos 3π/7, cos 5π/7 satisfies 8x3 − 4x2 − 4x +1 =0. This polynomial is irreducible over the rationals, and so these three cosines are conjugate algebraic numbers. Likewise, tan 3π/16, tan 7π/16, tan 11π/16, tan 15π/16 all satisfy the irreducible polynomial x4 − 4x3 − 6x2 + 4x +1, and so are conjugate algebraic integers. Some irrational numbers are algebraic and some are not, The numbers √2 and 3√3/2 are algebraic since they are roots of polynomials x2 −2 and 8x3 −3, the golden ratio φ is algebraic since it is a root of the polynomial x2 − x −1. The numbers π and e are not algebraic numbers, hence they are transcendental, the set of algebraic numbers is countable. Hence, the set of numbers has Lebesgue measure zero. Given an algebraic number, there is a monic polynomial of least degree that has the number as a root. This polynomial is called its minimal polynomial, if its minimal polynomial has degree n, then the algebraic number is said to be of degree n. An algebraic number of degree 1 is a rational number, a real algebraic number of degree 2 is a quadratic irrational
26.
Euler's identity
–
Eulers identity is named after the Swiss mathematician Leonhard Euler. It is considered to be an example of mathematical beauty, in particular, when x = π, or one half-turn around a circle, e i π = cos π + i sin π. Since cos π = −1 and sin π =0, it follows that e i π = −1 +0 i, Eulers identity is often cited as an example of deep mathematical beauty. Three of the arithmetic operations occur exactly once each, addition, multiplication. The identity also links five fundamental mathematical constants, The number 0, the number 1, the multiplicative identity. The number π, which is ubiquitous in the geometry of Euclidean space and analytical mathematics The number e, the base of natural logarithms, which occurs widely in mathematical analysis. Furthermore, the equation is given in the form of an expression set equal to zero, the mathematics writer Constance Reid has opined that Eulers identity is the most famous formula in all mathematics. A poll of readers conducted by The Mathematical Intelligencer in 1990 named Eulers identity as the most beautiful theorem in mathematics, in another poll of readers that was conducted by Physics World in 2004, Eulers identity tied with Maxwells equations as the greatest equation ever. A study of the brains of sixteen mathematicians found that the emotional brain lit up more consistently for Eulers identity than for any other formula. Eulers identity is also a case of the more general identity that the nth roots of unity, for n >1. Eulers identity is the case where n =2, in another field of mathematics, by using quaternion exponentiation, one can show that a similar identity also applies to quaternions. Let be the elements, then, e 13 π +1 =0. In general, given real a1, a2, and a3 such that a12 + a22 + a32 =1, then, for octonions, with real an such that a12 + a22 +. + a72 =1 and the basis elements, then. It has been claimed that Eulers identity appears in his work of mathematical analysis published in 1748. However, it is whether this particular concept can be attributed to Euler himself. De Moivres formula Exponential function Gelfonds constant Conway, John Horton, the greatest equations ever, PhysicsWeb, October 2004. Equations as icons, PhysicsWeb, March 2007, prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
27.
Exponential growth
–
Exponential decay occurs in the same way when the growth rate is negative. In the case of a domain of definition with equal intervals, it is also called geometric growth or geometric decay. In either exponential growth or exponential decay, the ratio of the rate of change of the quantity to its current size remains constant over time. The formula for growth of a variable x at the growth rate r. This formula is transparent when the exponents are converted to multiplication, in this way, each increase in the exponent by a full interval can be seen to increase the previous total by another five percent. Since the time variable, which is the input to function, occurs as the exponent. Biology The number of microorganisms in a culture will increase exponentially until an essential nutrient is exhausted, typically the first organism splits into two daughter organisms, who then each split to form four, who split to form eight, and so on. Because exponential growth indicates constant growth rate, it is assumed that exponentially growing cells are at a steady-state. However, cells can grow exponentially at a constant rate while remodelling their metabolism, a virus typically will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people, human population, if the number of births and deaths per person per year were to remain at current levels. This means that the time of the American population is approximately 50 years. Physics Avalanche breakdown within a dielectric material, a free electron becomes sufficiently accelerated by an externally applied electrical field that it frees up additional electrons as it collides with atoms or molecules of the dielectric media. These secondary electrons also are accelerated, creating larger numbers of free electrons, the resulting exponential growth of electrons and ions may rapidly lead to complete dielectric breakdown of the material. Each uranium nucleus that undergoes fission produces multiple neutrons, each of which can be absorbed by adjacent uranium atoms, due to the exponential rate of increase, at any point in the chain reaction 99% of the energy will have been released in the last 4.6 generations. It is an approximation to think of the first 53 generations as a latency period leading up to the actual explosion. Economics Economic growth is expressed in terms, implying exponential growth. For example, U. S. GDP per capita has grown at a rate of approximately two percent since World War 2. Finance Compound interest at a constant interest rate provides exponential growth of the capital, pyramid schemes or Ponzi schemes also show this type of growth resulting in high profits for a few initial investors and losses among great numbers of investors
28.
John Napier
–
John Napier of Merchiston, also signed as Neper, Nepair, nicknamed Marvellous Merchiston) was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston and his Latinized name was Ioannes Neper. John Napier is best known as the discoverer of logarithms and he also invented the so-called Napiers bones and made common the use of the decimal point in arithmetic and mathematics. Napiers birthplace, Merchiston Tower in Edinburgh, is now part of the facilities of Edinburgh Napier University, Napier died from the effects of gout at home at Merchiston Castle and his remains were buried in the kirkyard of St Giles. Following the loss of the kirkyard there to build Parliament House, archibald Napier was 16 years old when John Napier was born. As was the practice for members of the nobility at that time, he was privately tutored and did not have formal education until he was 13. He did not stay in very long. It is believed that he dropped out of school in Scotland, in 1571, Napier, aged 21, returned to Scotland, and bought a castle at Gartness in 1574. On the death of his father in 1608, Napier and his family moved into Merchiston Castle in Edinburgh and he died at the age of 67. In such conditions, it is surprising that many mathematicians were acutely aware of the issues of computation and were dedicated to relieving practitioners of the calculation burden. In particular, the Scottish mathematician John Napier was famous for his devices to assist with computation and he invented a well-known mathematical artifact, the ingenious numbering rods more quaintly known as “Napiers bones, ” that offered mechanical means for facilitating computation. He appreciated that, for the most part, practitioners who had laborious computations generally did them in the context of trigonometry, therefore, as well as developing the logarithmic relation, Napier set it in a trigonometric context so it would be even more relevant. His work, Mirifici Logarithmorum Canonis Descriptio contained fifty-seven pages of explanatory matter, the book also has an excellent discussion of theorems in spherical trigonometry, usually known as Napiers Rules of Circular Parts. Modern English translations of both Napiers books on logarithms, and their description can be found on the web, as well as a discussion of Napiers Bones and his invention of logarithms was quickly taken up at Gresham College, and prominent English mathematician Henry Briggs visited Napier in 1615. Among the matters discussed were a re-scaling of Napiers logarithms. Napier delegated to Briggs the computation of a revised table, the computational advance available via logarithms, the converse of powered numbers or exponential notation, was such that it made calculations by hand much quicker. The way was opened to later scientific advances, in astronomy, dynamics and he improved Simon Stevins decimal notation. Lattice multiplication, used by Fibonacci, was more convenient by his introduction of Napiers bones
29.
Leonhard Euler
–
He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
30.
History of logarithms
–
The Napierian logarithms were published first in 1614. Henry Briggs introduced common logarithms, which were easier to use, tables of logarithms were published in many forms over four centuries. The idea of logarithms was also used to construct the slide rule, the Babylonians sometime in 2000–1600 BC may have invented the quarter square multiplication algorithm to multiply two numbers using only addition, subtraction and a table of quarter squares. Thus, such a table served a purpose to tables of logarithms. However, the method could not be used for division without an additional table of reciprocals. Large tables of squares were used to simplify the accurate multiplication of large numbers from 1817 onwards until this was superseded by the use of computers. The Indian mathematician Virasena worked with the concept of ardhaccheda, the number of times a number of the form 2n could be halved, for exact powers of 2, this equals the binary logarithm, but it differs from the logarithm for other numbers. He described a product formula for this concept and also introduced analogous concepts for base 3, michael Stifel published Arithmetica integra in Nuremberg in 1544, which contains a table of integers and powers of 2 that has been considered an early version of a table of binary logarithms. In the 16th and early 17th centuries an algorithm called prosthaphaeresis was used to approximate multiplication and division and this used the trigonometric identity cos α cos β =12 or similar to convert the multiplications to additions and table lookups. However, logarithms are more straightforward and require less work and it can be shown using Eulers formula that the two techniques are related. This procedure achieves the same as the logarithms will a few years later, the method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio. Johannes Kepler, who used logarithm tables extensively to compile his Ephemeris and therefore dedicated it to Napier, the accent in calculation led Justus Byrgius on the way to these very logarithms many years before Napiers system appeared, but. Instead of rearing up his child for the benefit he deserted it in the birth. By repeated subtractions Napier calculated L for L ranging from 1 to 100, the result for L=100 is approximately 0.99999 =1 − 10−5. Napier then calculated the products of these numbers with 107L for L from 1 to 50, the invention was quickly and widely met with acclaim. The works of Bonaventura Cavalieri, Edmund Wingate, Xue Fengzuo, as the common log of ten is one, of a hundred is two, and a thousand is three, the concept of common logarithms is very close to the decimal-positional number system. The common log is said to have base 10, but base 10,000 is ancient, in his book The Sand Reckoner, Archimedes used the myriad as the base of a number system designed to count the grains of sand in the universe. As was noted in 2000, In antiquity Archimedes gave a recipe for reducing multiplication to addition by making use of geometric progression of numbers, in 1616 Henry Briggs visited Napier at Edinburgh in order to discuss the suggested change to Napiers logarithms
31.
Gregoire de Saint-Vincent
–
Grégoire de Saint-Vincent was a Flemish Jesuit and mathematician. He is remembered for his work on quadrature of the hyperbola, grégoire gave the clearest early account of the summation of geometric series. He also resolved Zenos paradox by showing that the time intervals involved formed a geometric progression, for example, the ungula is formed by cutting a right circular cylinder by means of an oblique plane through a diameter of the circular base. And also the ’double ungula formed from cylinders with axes at right angles, ungula was changed to onglet in French by Blaise Pascal when he wrote Traité des trilignes rectangles et leurs onglets. Grégoire wrote his manuscript in the 1620s but it waited until 1647 before publication, then it attracted a great deal of attention. because of the systematic approach to volumetric integration developed under the name ductus plani in planum. Saint-Vincent found that the area under a hyperbola is the same over as over when a/b = c/d. This observation led to the natural logarithm, the stated property allows one to define a function A which is the area under said curve from 1 to x, which has the property that A = A + A. This functional property characterizes logarithms, and it was mathematical fashion to call such a function A a logarithm, in particular when we choose the rectangular hyperbola xy =1, one recovers the natural logarithm. To a large extent, recognition of de Saint-Vincents achievement in quadrature of the hyperbola is due to his student and co-worker A. A. de Sarasa, a modern approach to his theorem uses squeeze mapping in linear algebra. The following estimation was given by a historian of the calculus, As a consequence of the work of Gregory St. David Eugene Smith History of Mathematics, hans Wussing 6000 Jahre Mathematik, eine kulturgeschichtliche Zeitreise, S.433, Springer, ISBN9783540771920. Gregory Saint Vincent, and his polar coordinates from Jesuit History, Tradition, oConnor, John J. Robertson, Edmund F. Gregorius Saint-Vincent, MacTutor History of Mathematics archive, University of St Andrews
32.
Alphonse Antonio de Sarasa
–
Alphonse Antonio de Sarasa was a Jesuit mathematician who contributed to the understanding of logarithms, particularly as areas under a hyperbola. Alphonse de Sarasa was born in 1618, in Nieuwpoort in Flanders, in 1632 he was admitted as a novice in Ghent. It was there that he worked alongside Gregoire de Saint-Vincent whose ideas he developed, exploited, according to Sommervogel, Alphonse de Sarasa also held academic positions in Antwerp and Brussels. In 1649 Alphonse de Sarasa published Solutio problematis a R. P. Marino Mersenne Minimo propositi, burn explains that the term logarithm was used differently in the seventeenth century. Logarithms were any arithmetic progression which corresponded to a geometric progression, burn quotes de Sarasa on this point, …the foundation of the teaching embracing logarithms are contained in Saint-Vincent’s Opus Geometricum, part 4 of Book 6, de Hyperbola. Alphonse Antonio de Sarasa died in Brussels in 1667
33.
Quadrature (mathematics)
–
In mathematics, quadrature is a historical term which means determining area. Quadrature problems served as one of the sources of problems in the development of calculus. By Greek tradition, these constructions had to be performed using only a compass, for a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side x = a b. A similar geometrical construction solves the problems of quadrature of a parallelogram, problems of quadrature for curvilinear figures are much more difficult. The quadrature of the circle with compass and straightedge was proved in the 19th century to be impossible, nevertheless, for some figures a quadrature can be performed. The quadratures of the surface of a sphere and a parabola segment discovered by Archimedes became the highest achievement of analysis in antiquity, the area of the surface of a sphere is equal to four times the area of the circle formed by a great circle of this sphere. The area of a segment of a parabola determined by a line cutting it is 4/3 the area of a triangle inscribed in this segment. For the proof of these results, Archimedes used the method of exhaustion of Eudoxus, in medieval Europe, quadrature meant the calculation of area by any method. Most often the method of indivisibles was used, it was less rigorous than the geometric constructions of the Greeks, but it was simpler and more powerful. John Wallis algebrised this method, he wrote in his Arithmetica Infinitorum some series which are equivalent to what is now called the definite integral, isaac Barrow and James Gregory made further progress, quadratures for some algebraic curves and spirals. Christiaan Huygens successfully performed a quadrature of the area of some solids of revolution. The quadrature of the hyperbola by Saint-Vincent and de Sarasa provided a new function, with the invention of integral calculus came a universal method for area calculation. Gaussian quadrature Hyperbolic angle Numerical integration Quadrance Quadratrix Tanh-sinh quadrature Boyer, a History of Mathematics, 2nd ed. rev. by Uta C. Babin translator, William Alexander Myers editor, link from HathiTrust, christoph Scriba Gregorys Converging Double Sequence, a new look at the controversy between Huygens and Gregory over the analytical quadrature of the circle, Historia Mathematica 10, 274–85
34.
Hyperbola
–
In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other, the hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, each branch of the hyperbola has two arms which become straighter further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the point about which each branch reflects to form the other branch. In the case of the curve f =1 / x the asymptotes are the two coordinate axes, hyperbolas share many of the ellipses analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term, many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids, hyperboloids, hyperbolic geometry, hyperbolic functions, and gyrovector spaces. The word hyperbola derives from the Greek ὑπερβολή, meaning over-thrown or excessive, hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones. The term hyperbola is believed to have coined by Apollonius of Perga in his definitive work on the conic sections. The rectangle could be applied to the segment, be shorter than the segment or exceed the segment, the midpoint M of the line segment joining the foci is called the center of the hyperbola. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, C2 is called the director circle of the hyperbola. In order to get the branch of the hyperbola, one has to use the director circle related to F1. This property should not be confused with the definition of a hyperbola with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the hyperbola if the condition is fulfilled 2 + y 2 −2 + y 2 = ±2 a. Remove the square roots by suitable squarings and use the relation b 2 = c 2 − a 2 to obtain the equation of the hyperbola, the shape parameters a, b are called the semi major axis and semi minor axis or conjugate axis. As opposed to an ellipse, a hyperbola has two vertices
35.
Hyperbolic sector
–
A hyperbolic sector is a region of the Cartesian plane bounded by rays from the origin to two points and and by the rectangular hyperbola xy =1. A hyperbolic sector in standard position has a =1 and b >1, hyperbolic sectors are the basis for the hyperbolic functions. The area of a sector in standard position is ln b. Proof, Integrate under 1/x from 1 to b, add triangle, when in standard position, a hyperbolic sector corresponds to a positive hyperbolic angle at the origin, with the measure of the latter being defined as the area of the former. The length of the base of this triangle is 2 cosh u, and the altitude is 2 sinh u, the analogy between circular and hyperbolic functions was described by Augustus De Morgan in his Trigonometry and Double Algebra. William Burnside used such triangles, projecting from a point on the hyperbola xy =1 onto the main diagonal, students of integral calculus know that f = xp has an algebraic antiderivative except in the case p = –1 corresponding to the quadrature of the hyperbola. The other cases are given by Cavalieris quadrature formula and his findings led to the natural logarithm function, once called the hyperbolic logarithm since it is obtained by integrating, or finding the area, under the hyperbola. Before 1748 and the publication of Introduction to the Analysis of the Infinite, leonhard Euler changed that when he introduced transcendental functions such as 10x. Euler identified e as the value of b producing a unit of area, then the natural logarithm could be recognized as the inverse function to the transcendental function ex. When Felix Klein wrote his book on geometry in 1928. To establish hyperbolic measure on a line, he noted that the area of a hyperbolic sector provided visual illustration of the concept, hyperbolic sectors can also be drawn to the hyperbola y =1 + x 2. The area of hyperbolic sectors has been used to define hyperbolic distance in a geometry textbook. Squeeze mapping Mellen W. Haskell On the introduction of the notion of hyperbolic functions Bulletin of the American Mathematical Society 1, 155–9
36.
Programming language
–
A programming language is a formal computer language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to programs to control the behavior of a machine or to express algorithms. From the early 1800s, programs were used to direct the behavior of such as Jacquard looms. Thousands of different programming languages have created, mainly in the computer field. Many programming languages require computation to be specified in an imperative form while other languages use forms of program specification such as the declarative form. The description of a language is usually split into the two components of syntax and semantics. Some languages are defined by a document while other languages have a dominant implementation that is treated as a reference. Some languages have both, with the language defined by a standard and extensions taken from the dominant implementation being common. A programming language is a notation for writing programs, which are specifications of a computation or algorithm, some, but not all, authors restrict the term programming language to those languages that can express all possible algorithms. For example, PostScript programs are created by another program to control a computer printer or display. More generally, a language may describe computation on some, possibly abstract. It is generally accepted that a specification for a programming language includes a description, possibly idealized. In most practical contexts, a programming language involves a computer, consequently, abstractions Programming languages usually contain abstractions for defining and manipulating data structures or controlling the flow of execution. Expressive power The theory of computation classifies languages by the computations they are capable of expressing, all Turing complete languages can implement the same set of algorithms. ANSI/ISO SQL-92 and Charity are examples of languages that are not Turing complete, markup languages like XML, HTML, or troff, which define structured data, are not usually considered programming languages. Programming languages may, however, share the syntax with markup languages if a computational semantics is defined, XSLT, for example, is a Turing complete XML dialect. Moreover, LaTeX, which is used for structuring documents. The term computer language is used interchangeably with programming language
37.
Common logarithm
–
In mathematics, the common logarithm is the logarithm with base 10. It is indicated by log10, or sometimes Log with a capital L, on calculators it is usually log, but mathematicians usually mean natural logarithm rather than common logarithm when they write log. To mitigate this ambiguity the ISO80000 specification recommends that log10 should be written lg, before the early 1970s, handheld electronic calculators were not available and mechanical calculators capable of multiplication were bulky, expensive and not widely available. Instead, tables of logarithms were used in science, engineering. Use of logarithms avoided laborious and error prone paper and pencil multiplications and divisions, because logarithms were so useful, tables of base-10 logarithms were given in appendices of many text books. Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as well, see log table for the history of such tables. The fractional part is known as the mantissa, thus log tables need only show the fractional part. Tables of common logarithms typically listed the mantissa, to 4 or 5 decimal places or more, of number in a range. Such a range would cover all possible values of the mantissa, the integer part, called the characteristic, can be computed by simply counting how many places the decimal point must be moved so that it is just to the right of the first significant digit. For example, the logarithm of 120 is given by, log 10 120 = log 10 =2 + log 10 1.2 ≈2 +0.07918. The last number —the fractional part or the mantissa of the logarithm of 120—can be found in the table shown. The location of the point in 120 tells us that the integer part of the common logarithm of 120. Numbers greater than 0 and less than 1 have negative logarithms, when reading a number in bar notation out loud, the symbol n ¯ is read as bar n, so that 2 ¯.07918 is read as bar 2 point 07918. The following table shows how the same mantissa can be used for a range of numbers differing by powers of ten and this holds for any positive real number x because, log 10 = log 10 + log 10 = log 10 + i. Since i is always an integer the mantissa comes from log 10 which is constant for given x and this allows a table of logarithms to include only one entry for each mantissa. In the example of 5×10i,0.698970 will be listed once indexed by 5, or 0.5, common logarithms are sometimes also called Briggsian logarithms after Henry Briggs, a 17th-century British mathematician. In 1616 and 1617 Briggs visited John Napier, the inventor of what are now called natural logarithms at Edinburgh in order to suggest a change to Napiers logarithms. During these conferences the alteration proposed by Briggs was agreed upon, because base 10 logarithms were most useful for computations, engineers generally simply wrote log when they meant log10
38.
Area
–
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
39.
Circular sector
–
A circular sector or circle sector, is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the angle in radians, r the radius of the circle. A sector with the angle of 180° is called a half-disk and is bounded by a diameter. The angle formed by connecting the endpoints of the arc to any point on the circumference that is not in the sector is equal to half the central angle, the total area of a circle is πr2. Circular segment – the part of the sector which remains after removing the triangle formed by the center of the circle, conic section Gerard, L. J. V. The Elements of Geometry, in Eight Books, or, First Step in Applied Logic, London, Longmans Green, Reader & Dyer,1874