Infinity.J. Wallis (1655).
First found in the treatise of the English mathematician John Valis "On Conic Sections".
The base of natural logarithms. L. Euler (1736).
Mathematical constant, transcendental number. This number is sometimes called non-feathered in honor of the Scottish scientist Napier, author of the work “Description of the Amazing Table of Logarithms” (1614). The constant first appears tacitly in an appendix to the English translation of Napier's above-mentioned work, published in 1618. The constant itself was first calculated by the Swiss mathematician Jacob Bernoulli while solving the problem of the limiting value of interest income.
2,71828182845904523...
The first known use of this constant, where it was denoted by the letter b, found in Leibniz's letters to Huygens, 1690-1691. Letter e Euler began using it in 1727, and the first publication with this letter was his work “Mechanics, or the Science of Motion, Explained Analytically” in 1736. Respectively, e usually called Euler number. Why was the letter chosen? e, exactly unknown. Perhaps this is due to the fact that the word begins with it exponential(“indicative”, “exponential”). Another assumption is that the letters a, b, c And d have already been used quite widely for other purposes, and e was the first "free" letter.
The ratio of the circumference to the diameter. W. Jones (1706), L. Euler (1736).
Mathematical constant, irrational number. The number "pi", the old name is Ludolph's number. Like any irrational number, π is represented as an infinite non-periodic decimal fraction:
π =3.141592653589793...
For the first time, the designation of this number by the Greek letter π was used by the British mathematician William Jones in the book “A New Introduction to Mathematics”, and it became generally accepted after the work of Leonhard Euler. This designation comes from the initial letter of the Greek words περιφερεια - circle, periphery and περιμετρος - perimeter. Johann Heinrich Lambert proved the irrationality of π in 1761, and Adrienne Marie Legendre proved the irrationality of π 2 in 1774. Legendre and Euler assumed that π could be transcendental, i.e. cannot satisfy any algebraic equation with integer coefficients, which was eventually proven in 1882 by Ferdinand von Lindemann.
Imaginary unit. L. Euler (1777, in print - 1794).
It is known that the equation x 2 =1 has two roots: 1 And -1 . The imaginary unit is one of the two roots of the equation x 2 = -1, denoted by a Latin letter i, another root: -i. This designation was proposed by Leonhard Euler, who took the first letter of the Latin word for this purpose imaginarius(imaginary). He also extended all standard functions to the complex domain, i.e. set of numbers representable as a+ib, Where a And b- real numbers. The term "complex number" was introduced into widespread use by the German mathematician Carl Gauss in 1831, although the term had previously been used in the same sense by the French mathematician Lazare Carnot in 1803.
Unit vectors. W. Hamilton (1853).
Unit vectors are often associated with the coordinate axes of a coordinate system (in particular, the axes of a Cartesian coordinate system). Unit vector directed along the axis X, denoted i, unit vector directed along the axis Y, denoted j, and the unit vector directed along the axis Z, denoted k. Vectors i, j, k are called unit vectors, they have unit modules. The term "ort" was introduced by the English mathematician and engineer Oliver Heaviside (1892), and the notation i, j, k- Irish mathematician William Hamilton.
Integer part of the number, antie. K.Gauss (1808).
The integer part of the number [x] of the number x is the largest integer not exceeding x. So, =5, [-3,6]=-4. The function [x] is also called "antier of x". The whole-part function symbol was introduced by Carl Gauss in 1808. Some mathematicians prefer to use instead the notation E(x), proposed in 1798 by Legendre.
Angle of parallelism. N.I. Lobachevsky (1835).
On the Lobachevsky plane - the angle between the straight lineb, passing through the pointABOUTparallel to the linea, not containing a pointABOUT, and perpendicular fromABOUT on a. α - the length of this perpendicular. As the point moves awayABOUT from the straight line athe angle of parallelism decreases from 90° to 0°. Lobachevsky gave a formula for the angle of parallelismP( α )=2arctg e - α /q , Where q— some constant associated with the curvature of Lobachevsky space.
Unknown or variable quantities. R. Descartes (1637).
In mathematics, a variable is a quantity characterized by the set of values it can take. This may mean both a real physical quantity, temporarily considered in isolation from its physical context, and some abstract quantity that has no analogues in the real world. The concept of a variable arose in the 17th century. initially under the influence of the demands of natural science, which brought to the fore the study of movement, processes, and not just states. This concept required new forms for its expression. Such new forms were the letter algebra and analytical geometry of Rene Descartes. For the first time, the rectangular coordinate system and the notation x, y were introduced by Rene Descartes in his work “Discourse on Method” in 1637. Pierre Fermat also contributed to the development of the coordinate method, but his works were first published after his death. Descartes and Fermat used the coordinate method only on the plane. The coordinate method for three-dimensional space was first used by Leonhard Euler already in the 18th century.
Vector. O. Cauchy (1853).
From the very beginning, a vector is understood as an object that has a magnitude, a direction and (optionally) a point of application. The beginnings of vector calculus appeared along with the geometric model of complex numbers in Gauss (1831). Hamilton published developed operations with vectors as part of his quaternion calculus (the vector was formed by the imaginary components of the quaternion). Hamilton proposed the term vector(from the Latin word vector, carrier) and described some operations of vector analysis. Maxwell used this formalism in his works on electromagnetism, thereby drawing the attention of scientists to the new calculus. Soon Gibbs's Elements of Vector Analysis came out (1880s), and then Heaviside (1903) gave vector analysis its modern look. The vector sign itself was introduced into use by the French mathematician Augustin Louis Cauchy in 1853.
Addition, subtraction. J. Widman (1489).
The plus and minus signs were apparently invented in the German mathematical school of “Kossists” (that is, algebraists). They are used in Jan (Johannes) Widmann's textbook A Quick and Pleasant Account for All Merchants, published in 1489. Previously, addition was denoted by the letter p(from Latin plus"more") or Latin word et(conjunction “and”), and subtraction - letter m(from Latin minus"less, less") For Widmann, the plus symbol replaces not only addition, but also the conjunction “and.” The origin of these symbols is unclear, but most likely they were previously used in trading as indicators of profit and loss. Both symbols soon became common in Europe - with the exception of Italy, which continued to use the old designations for about a century.
Multiplication. W. Outred (1631), G. Leibniz (1698).
The multiplication sign in the form of an oblique cross was introduced in 1631 by the Englishman William Oughtred. Before him, the letter was most often used M, although other notations were also proposed: the rectangle symbol (French mathematician Erigon, 1634), asterisk (Swiss mathematician Johann Rahn, 1659). Later, Gottfried Wilhelm Leibniz replaced the cross with a dot (late 17th century) so as not to confuse it with the letter x; before him, such symbolism was found among the German astronomer and mathematician Regiomontanus (15th century) and the English scientist Thomas Herriot (1560 -1621).
Division. I.Ran (1659), G.Leibniz (1684).
William Oughtred used a slash / as a division sign. Gottfried Leibniz began to denote division with a colon. Before them, the letter was also often used D. Starting with Fibonacci, the horizontal line of the fraction is also used, which was used by Heron, Diophantus and in Arabic works. In England and the USA, the symbol ÷ (obelus), which was proposed by Johann Rahn (possibly with the participation of John Pell) in 1659, became widespread. An attempt by the American National Committee on Mathematical Standards ( National Committee on Mathematical Requirements) to remove obelus from practice (1923) was unsuccessful.
Percent. M. de la Porte (1685).
A hundredth of a whole, taken as a unit. The word “percent” itself comes from the Latin “pro centum”, which means “per hundred”. In 1685, the book “Manual of Commercial Arithmetic” by Mathieu de la Porte was published in Paris. In one place they talked about percentages, which were then designated “cto” (short for cento). However, the typesetter mistook this "cto" for a fraction and printed "%". So, due to a typo, this sign came into use.
Degrees. R. Descartes (1637), I. Newton (1676).
The modern notation for the exponent was introduced by Rene Descartes in his “ Geometry"(1637), however, only for natural powers with exponents greater than 2. Later, Isaac Newton extended this form of notation to negative and fractional exponents (1676), the interpretation of which had already been proposed by this time: the Flemish mathematician and engineer Simon Stevin, the English mathematician John Wallis and French mathematician Albert Girard.
Arithmetic root n-th power of a real number A≥0, - non-negative number n-th degree of which is equal to A. The arithmetic root of the 2nd degree is called a square root and can be written without indicating the degree: √. An arithmetic root of the 3rd degree is called a cube root. Medieval mathematicians (for example, Cardano) denoted the square root with the symbol R x (from the Latin Radix, root). The modern notation was first used by the German mathematician Christoph Rudolf, from the Cossist school, in 1525. This symbol comes from the stylized first letter of the same word radix. At first there was no line above the radical expression; it was later introduced by Descartes (1637) for a different purpose (instead of parentheses), and this feature soon merged with the root sign. In the 16th century, the cube root was denoted as follows: R x .u.cu (from lat. Radix universalis cubica). Albert Girard (1629) began to use the familiar notation for a root of an arbitrary degree. This format was established thanks to Isaac Newton and Gottfried Leibniz.
Logarithm, decimal logarithm, natural logarithm. I. Kepler (1624), B. Cavalieri (1632), A. Prinsheim (1893).
The term "logarithm" belongs to the Scottish mathematician John Napier ( “Description of the amazing table of logarithms”, 1614); it arose from a combination of the Greek words λογος (word, relation) and αριθμος (number). J. Napier's logarithm is an auxiliary number for measuring the ratio of two numbers. The modern definition of logarithm was first given by the English mathematician William Gardiner (1742). By definition, the logarithm of a number b based on a (a ≠ 1, a > 0) - exponent m, to which the number should be raised a(called the logarithm base) to get b. Designated log a b. So, m = log a b, If a m = b.
The first tables of decimal logarithms were published in 1617 by Oxford mathematics professor Henry Briggs. Therefore, abroad, decimal logarithms are often called Briggs logarithms. The term “natural logarithm” was introduced by Pietro Mengoli (1659) and Nicholas Mercator (1668), although the London mathematics teacher John Spidell compiled a table of natural logarithms back in 1619.
Until the end of the 19th century, there was no generally accepted notation for the logarithm, the basis a indicated to the left and above the symbol log, then above it. Ultimately, mathematicians came to the conclusion that the most convenient place for the base is below the line, after the symbol log. The logarithm sign - the result of the abbreviation of the word "logarithm" - appears in various forms almost simultaneously with the appearance of the first tables of logarithms, e.g. Log- by I. Kepler (1624) and G. Briggs (1631), log- by B. Cavalieri (1632). Designation ln for the natural logarithm was introduced by the German mathematician Alfred Pringsheim (1893).
Sine, cosine, tangent, cotangent. W. Outred (mid-17th century), I. Bernoulli (18th century), L. Euler (1748, 1753).
The abbreviations for sine and cosine were introduced by William Oughtred in the mid-17th century. Abbreviations for tangent and cotangent: tg, ctg introduced by Johann Bernoulli in the 18th century, they became widespread in Germany and Russia. In other countries the names of these functions are used tan, cot proposed by Albert Girard even earlier, at the beginning of the 17th century. Leonhard Euler (1748, 1753) brought the theory of trigonometric functions into its modern form, and we owe it to him for the consolidation of real symbolism.The term "trigonometric functions" was introduced by the German mathematician and physicist Georg Simon Klügel in 1770.
Indian mathematicians originally called the sine line "arha-jiva"(“half-string”, that is, half a chord), then the word "archa" was discarded and the sine line began to be called simply "jiva". Arabic translators did not translate the word "jiva" Arabic word "vatar", denoting string and chord, and transcribed in Arabic letters and began to call the sine line "jiba". Since in Arabic short vowels are not marked, but long “i” in the word "jiba" denoted in the same way as the semivowel “th”, the Arabs began to pronounce the name of the sine line "jibe", which literally means “hollow”, “sinus”. When translating Arabic works into Latin, European translators translated the word "jibe" Latin word sinus, having the same meaning.The term "tangent" (from lat.tangents- touching) was introduced by the Danish mathematician Thomas Fincke in his book The Geometry of the Round (1583).
Arcsine. K. Scherfer (1772), J. Lagrange (1772).
Inverse trigonometric functions are mathematical functions that are the inverse of trigonometric functions. The name of the inverse trigonometric function is formed from the name of the corresponding trigonometric function by adding the prefix "arc" (from Lat. arc- arc).The inverse trigonometric functions usually include six functions: arcsine (arcsin), arccosine (arccos), arctangent (arctg), arccotangent (arcctg), arcsecant (arcsec) and arccosecant (arccosec). Special symbols for inverse trigonometric functions were first used by Daniel Bernoulli (1729, 1736).Manner of denoting inverse trigonometric functions using a prefix arc(from lat. arcus, arc) appeared with the Austrian mathematician Karl Scherfer and was consolidated thanks to the French mathematician, astronomer and mechanic Joseph Louis Lagrange. It was meant that, for example, an ordinary sine allows one to find a chord subtending it along an arc of a circle, and the inverse function solves the opposite problem. Until the end of the 19th century, the English and German mathematical schools proposed other notations: sin -1 and 1/sin, but they are not widely used.
Hyperbolic sine, hyperbolic cosine. V. Riccati (1757).
Historians discovered the first appearance of hyperbolic functions in the works of the English mathematician Abraham de Moivre (1707, 1722). A modern definition and a detailed study of them was carried out by the Italian Vincenzo Riccati in 1757 in his work “Opusculorum”, he also proposed their designations: sh,ch. Riccati started from considering the unit hyperbola. An independent discovery and further study of the properties of hyperbolic functions was carried out by the German mathematician, physicist and philosopher Johann Lambert (1768), who established the wide parallelism of the formulas of ordinary and hyperbolic trigonometry. N.I. Lobachevsky subsequently used this parallelism in an attempt to prove the consistency of non-Euclidean geometry, in which ordinary trigonometry is replaced by hyperbolic one.
Just as the trigonometric sine and cosine are the coordinates of a point on the coordinate circle, the hyperbolic sine and cosine are the coordinates of a point on a hyperbola. Hyperbolic functions are expressed in terms of an exponential and are closely related to trigonometric functions: sh(x)=0.5(e x -e -x) , ch(x)=0.5(e x +e -x). By analogy with trigonometric functions, hyperbolic tangent and cotangent are defined as the ratios of hyperbolic sine and cosine, cosine and sine, respectively.
Differential. G. Leibniz (1675, published 1684).
The main, linear part of the function increment.If the function y=f(x) one variable x has at x=x 0derivative, and incrementΔy=f(x 0 +?x)-f(x 0)functions f(x) can be represented in the formΔy=f"(x 0 )Δx+R(Δx) , where is the member R infinitesimal compared toΔx. First memberdy=f"(x 0 )Δxin this expansion and is called the differential of the function f(x) at the pointx 0. IN works of Gottfried Leibniz, Jacob and Johann Bernoulli the word"differentia"was used in the sense of “increment”, it was denoted by I. Bernoulli through Δ. G. Leibniz (1675, published 1684) used the notation for the “infinitesimal difference”d- the first letter of the word"differential", formed by him from"differentia".
Indefinite integral. G. Leibniz (1675, published 1686).
The word "integral" was first used in print by Jacob Bernoulli (1690). Perhaps the term is derived from the Latin integer- whole. According to another assumption, the basis was the Latin word integro- bring to its previous state, restore. The sign ∫ is used to represent an integral in mathematics and is a stylized representation of the first letter of the Latin word summa - sum. It was first used by the German mathematician and founder of differential and integral calculus, Gottfried Leibniz, at the end of the 17th century. Another of the founders of differential and integral calculus, Isaac Newton, did not propose an alternative symbolism for the integral in his works, although he tried various options: a vertical bar above the function or a square symbol that stands in front of the function or borders it. Indefinite integral for a function y=f(x) is the set of all antiderivatives of a given function.
Definite integral. J. Fourier (1819-1822).
Definite integral of a function f(x) with a lower limit a and upper limit b can be defined as the difference F(b) - F(a) = a ∫ b f(x)dx , Where F(x)- some antiderivative of a function f(x) . Definite integral a ∫ b f(x)dx numerically equal to the area of the figure bounded by the x-axis and straight lines x=a And x=b and the graph of the function f(x). The design of a definite integral in the form we are familiar with was proposed by the French mathematician and physicist Jean Baptiste Joseph Fourier at the beginning of the 19th century.
Derivative. G. Leibniz (1675), J. Lagrange (1770, 1779).
Derivative is the basic concept of differential calculus, characterizing the rate of change of a function f(x) when the argument changes x . It is defined as the limit of the ratio of the increment of a function to the increment of its argument as the increment of the argument tends to zero, if such a limit exists. A function that has a finite derivative at some point is called differentiable at that point. The process of calculating the derivative is called differentiation. The reverse process is integration. In classical differential calculus, the derivative is most often defined through the concepts of the theory of limits, but historically the theory of limits appeared later than differential calculus.
The term “derivative” was introduced by Joseph Louis Lagrange in 1797, the denotation of a derivative using a stroke is also used by him (1770, 1779), and dy/dx- Gottfried Leibniz in 1675. The manner of denoting the time derivative with a dot over a letter comes from Newton (1691).The Russian term “derivative of a function” was first used by a Russian mathematicianVasily Ivanovich Viskovatov (1779-1812).
Partial derivative. A. Legendre (1786), J. Lagrange (1797, 1801).
For functions of many variables, partial derivatives are defined - derivatives with respect to one of the arguments, calculated under the assumption that the remaining arguments are constant. Designations ∂f/ ∂ x, ∂ z/ ∂ y introduced by French mathematician Adrien Marie Legendre in 1786; fx",z x "- Joseph Louis Lagrange (1797, 1801); ∂ 2 z/ ∂ x 2, ∂ 2 z/ ∂ x ∂ y- partial derivatives of the second order - German mathematician Carl Gustav Jacob Jacobi (1837).
Difference, increment. I. Bernoulli (late 17th century - first half of the 18th century), L. Euler (1755).
The designation of increment by the letter Δ was first used by the Swiss mathematician Johann Bernoulli. The delta symbol came into general use after the work of Leonhard Euler in 1755.
Sum. L. Euler (1755).
Sum is the result of adding quantities (numbers, functions, vectors, matrices, etc.). To denote the sum of n numbers a 1, a 2, ..., a n, the Greek letter “sigma” Σ is used: a 1 + a 2 + ... + a n = Σ n i=1 a i = Σ n 1 a i. The Σ sign for the sum was introduced by Leonhard Euler in 1755.
Work. K.Gauss (1812).
A product is the result of multiplication. To denote the product of n numbers a 1, a 2, ..., a n, the Greek letter pi Π is used: a 1 · a 2 · ... · a n = Π n i=1 a i = Π n 1 a i. For example, 1 · 3 · 5 · ... · 97 · 99 = ? 50 1 (2i-1). The Π sign for a product was introduced by the German mathematician Carl Gauss in 1812. In Russian mathematical literature, the term “product” was first encountered by Leonty Filippovich Magnitsky in 1703.
Factorial. K. Crump (1808).
The factorial of a number n (denoted n!, pronounced "en factorial") is the product of all natural numbers up to n inclusive: n! = 1·2·3·...·n. For example, 5! = 1·2·3·4·5 = 120. By definition, 0 is assumed! = 1. Factorial is defined only for non-negative integers. The factorial of n is equal to the number of permutations of n elements. For example, 3! = 6, indeed,
♣ ♦
♣ ♦
♣ ♦
♦ ♣
♦ ♣
♦ ♣
All six and only six permutations of three elements.
The term "factorial" was introduced by the French mathematician and politician Louis Francois Antoine Arbogast (1800), the designation n! - French mathematician Christian Crump (1808).
Modulus, absolute value. K. Weierstrass (1841).
The absolute value of a real number x is a non-negative number defined as follows: |x| = x for x ≥ 0, and |x| = -x for x ≤ 0. For example, |7| = 7, |- 0.23| = -(-0.23) = 0.23. The modulus of a complex number z = a + ib is a real number equal to √(a 2 + b 2).
It is believed that the term “module” was proposed by the English mathematician and philosopher, Newton’s student, Roger Cotes. Gottfried Leibniz also used this function, which he called “modulus” and denoted: mol x. The generally accepted notation for absolute magnitude was introduced in 1841 by the German mathematician Karl Weierstrass. For complex numbers, this concept was introduced by French mathematicians Augustin Cauchy and Jean Robert Argan at the beginning of the 19th century. In 1903, the Austrian scientist Konrad Lorenz used the same symbolism for the length of a vector.
Norm. E. Schmidt (1908).
A norm is a functional defined on a vector space and generalizing the concept of the length of a vector or modulus of a number. The "norm" sign (from the Latin word "norma" - "rule", "pattern") was introduced by the German mathematician Erhard Schmidt in 1908.
Limit. S. Lhuillier (1786), W. Hamilton (1853), many mathematicians (until the beginning of the twentieth century)
Limit is one of the basic concepts of mathematical analysis, meaning that a certain variable value in the process of its change under consideration indefinitely approaches a certain constant value. The concept of a limit was used intuitively in the second half of the 17th century by Isaac Newton, as well as by 18th-century mathematicians such as Leonhard Euler and Joseph Louis Lagrange. The first rigorous definitions of the sequence limit were given by Bernard Bolzano in 1816 and Augustin Cauchy in 1821. The symbol lim (the first 3 letters from the Latin word limes - border) appeared in 1787 by the Swiss mathematician Simon Antoine Jean Lhuillier, but its use did not yet resemble modern ones. The expression lim in a more familiar form was first used by the Irish mathematician William Hamilton in 1853.Weierstrass introduced a designation close to the modern one, but instead of the familiar arrow, he used an equal sign. The arrow appeared at the beginning of the 20th century among several mathematicians at once - for example, the English mathematician Godfried Hardy in 1908.
Zeta function, d Riemann zeta function. B. Riemann (1857).
Analytical function of a complex variable s = σ + it, for σ > 1, determined absolutely and uniformly by a convergent Dirichlet series:
ζ(s) = 1 -s + 2 -s + 3 -s + ... .
For σ > 1, the representation in the form of the Euler product is valid:
ζ(s) = Π p (1-p -s) -s ,
where the product is taken over all prime p. The zeta function plays a big role in number theory.As a function of a real variable, the zeta function was introduced in 1737 (published in 1744) by L. Euler, who indicated its expansion into a product. This function was then considered by the German mathematician L. Dirichlet and, especially successfully, by the Russian mathematician and mechanic P.L. Chebyshev when studying the law of distribution of prime numbers. However, the most profound properties of the zeta function were discovered later, after the work of the German mathematician Georg Friedrich Bernhard Riemann (1859), where the zeta function was considered as a function of a complex variable; He also introduced the name “zeta function” and the designation ζ(s) in 1857.
Gamma function, Euler Γ function. A. Legendre (1814).
The Gamma function is a mathematical function that extends the concept of factorial to the field of complex numbers. Usually denoted by Γ(z). The G-function was first introduced by Leonhard Euler in 1729; it is determined by the formula:
Γ(z) = limn→∞ n!·n z /z(z+1)...(z+n).
A large number of integrals, infinite products and sums of series are expressed through the G-function. Widely used in analytical number theory. The name "Gamma function" and the notation Γ(z) were proposed by the French mathematician Adrien Marie Legendre in 1814.
Beta function, B function, Euler B function. J. Binet (1839).
A function of two variables p and q, defined for p>0, q>0 by the equality:
B(p, q) = 0 ∫ 1 x p-1 (1-x) q-1 dx.
The beta function can be expressed through the Γ-function: B(p, q) = Γ(p)Г(q)/Г(p+q).Just as the gamma function for integers is a generalization of factorial, the beta function is, in a sense, a generalization of binomial coefficients.
The beta function describes many propertieselementary particles participating in strong interaction. This feature was noticed by the Italian theoretical physicistGabriele Veneziano in 1968. This marked the beginning string theory.
The name "beta function" and the designation B(p, q) were introduced in 1839 by the French mathematician, mechanic and astronomer Jacques Philippe Marie Binet.
Laplace operator, Laplacian. R. Murphy (1833).
Linear differential operator Δ, which assigns functions φ(x 1, x 2, ..., x n) of n variables x 1, x 2, ..., x n:
Δφ = ∂ 2 φ/∂х 1 2 + ∂ 2 φ/∂х 2 2 + ... + ∂ 2 φ/∂х n 2.
In particular, for a function φ(x) of one variable, the Laplace operator coincides with the operator of the 2nd derivative: Δφ = d 2 φ/dx 2 . The equation Δφ = 0 is usually called Laplace's equation; This is where the names “Laplace operator” or “Laplacian” come from. The designation Δ was introduced by the English physicist and mathematician Robert Murphy in 1833.
Hamilton operator, nabla operator, Hamiltonian. O. Heaviside (1892).
Vector differential operator of the form
∇ = ∂/∂x i+ ∂/∂y · j+ ∂/∂z · k,
Where i, j, And k- coordinate unit vectors. The basic operations of vector analysis, as well as the Laplace operator, are expressed in a natural way through the Nabla operator.
In 1853, Irish mathematician William Rowan Hamilton introduced this operator and coined the symbol ∇ for it as an inverted Greek letter Δ (delta). In Hamilton, the tip of the symbol pointed to the left; later, in the works of the Scottish mathematician and physicist Peter Guthrie Tate, the symbol acquired its modern form. Hamilton called this symbol "atled" (the word "delta" read backwards). Later, English scholars, including Oliver Heaviside, began to call this symbol "nabla", after the name of the letter ∇ in the Phoenician alphabet, where it occurs. The origin of the letter is associated with a musical instrument such as the harp, ναβλα (nabla) in ancient Greek meaning “harp”. The operator was called the Hamilton operator, or nabla operator.
Function. I. Bernoulli (1718), L. Euler (1734).
A mathematical concept that reflects the relationship between elements of sets. We can say that a function is a “law”, a “rule” according to which each element of one set (called the domain of definition) is associated with some element of another set (called the domain of values). The mathematical concept of a function expresses the intuitive idea of how one quantity completely determines the value of another quantity. Often the term "function" refers to a numerical function; that is, a function that puts some numbers in correspondence with others. For a long time, mathematicians specified arguments without parentheses, for example, like this - φх. This notation was first used by the Swiss mathematician Johann Bernoulli in 1718.Parentheses were used only in the case of multiple arguments or if the argument was a complex expression. Echoes of those times are the recordings still in use todaysin x, log xetc. But gradually the use of parentheses, f(x) , became a general rule. And the main credit for this belongs to Leonhard Euler.
Equality. R. Record (1557).
The equals sign was proposed by the Welsh physician and mathematician Robert Record in 1557; the outline of the symbol was much longer than the current one, as it imitated the image of two parallel segments. The author explained that there is nothing more equal in the world than two parallel segments of the same length. Before this, in ancient and medieval mathematics equality was denoted verbally (for example est egale). In the 17th century, Rene Descartes began to use æ (from lat. aequalis), and he used the modern equal sign to indicate that the coefficient can be negative. François Viète used the equal sign to denote subtraction. The Record symbol did not become widespread immediately. The spread of the Record symbol was hampered by the fact that since ancient times the same symbol was used to indicate the parallelism of straight lines; In the end, it was decided to make the parallelism symbol vertical. In continental Europe, the "=" sign was introduced by Gottfried Leibniz only at the turn of the 17th-18th centuries, that is, more than 100 years after the death of Robert Record, who first used it for this purpose.
Approximately equal, approximately equal. A.Gunther (1882).
Sign " ≈ " was introduced into use as a symbol for the relation "approximately equal" by the German mathematician and physicist Adam Wilhelm Sigmund Günther in 1882.
More less. T. Harriot (1631).
These two signs were introduced into use by the English astronomer, mathematician, ethnographer and translator Thomas Harriot in 1631; before that, the words “more” and “less” were used.
Comparability. K.Gauss (1801).
Comparison is a relationship between two integers n and m, meaning that the difference n-m of these numbers is divided by a given integer a, called the comparison modulus; it is written: n≡m(mod а) and reads “the numbers n and m are comparable modulo a”. For example, 3≡11(mod 4), since 3-11 is divisible by 4; the numbers 3 and 11 are comparable modulo 4. Congruences have many properties similar to those of equalities. Thus, a term located in one part of the comparison can be transferred with the opposite sign to another part, and comparisons with the same module can be added, subtracted, multiplied, both parts of the comparison can be multiplied by the same number, etc. For example,
3≡9+2(mod 4) and 3-2≡9(mod 4)
At the same time true comparisons. And from a pair of correct comparisons 3≡11(mod 4) and 1≡5(mod 4) the following follows:
3+1≡11+5(mod 4)
3-1≡11-5(mod 4)
3·1≡11·5(mod 4)
3 2 ≡11 2 (mod 4)
3·23≡11·23(mod 4)
Number theory deals with methods for solving various comparisons, i.e. methods for finding integers that satisfy comparisons of one type or another. Modulo comparisons were first used by the German mathematician Carl Gauss in his 1801 book Arithmetic Studies. He also proposed symbolism for comparisons that was established in mathematics.
Identity. B. Riemann (1857).
Identity is the equality of two analytical expressions, valid for any permissible values of the letters included in it. The equality a+b = b+a is valid for all numerical values of a and b, and therefore is an identity. To record identities, in some cases, since 1857, the sign “≡” (read “identically equal”) has been used, the author of which in this use is the German mathematician Georg Friedrich Bernhard Riemann. You can write down a+b ≡ b+a.
Perpendicularity. P. Erigon (1634).
Perpendicularity is the relative position of two straight lines, planes, or a straight line and a plane, in which the indicated figures form a right angle. The sign ⊥ to denote perpendicularity was introduced in 1634 by the French mathematician and astronomer Pierre Erigon. The concept of perpendicularity has a number of generalizations, but all of them, as a rule, are accompanied by the sign ⊥.
Parallelism. W. Outred (posthumous edition 1677).
Parallelism is the relationship between certain geometric figures; for example, straight. Defined differently depending on different geometries; for example, in the geometry of Euclid and in the geometry of Lobachevsky. The sign of parallelism has been known since ancient times, it was used by Heron and Pappus of Alexandria. At first, the symbol was similar to the current equals sign (only more extended), but with the advent of the latter, to avoid confusion, the symbol was turned vertically ||. It appeared in this form for the first time in the posthumous edition of the works of the English mathematician William Oughtred in 1677.
Intersection, union. J. Peano (1888).
The intersection of sets is a set that contains those and only those elements that simultaneously belong to all given sets. A union of sets is a set that contains all the elements of the original sets. Intersection and union are also called operations on sets that assign new sets to certain ones according to the rules indicated above. Denoted by ∩ and ∪, respectively. For example, if
A= (♠ ♣ ) And B= (♣ ♦),
That
A∩B= {♣ }
A∪B= {♠ ♣ ♦ } .
Contains, contains. E. Schroeder (1890).
If A and B are two sets and there are no elements in A that do not belong to B, then they say that A is contained in B. They write A⊂B or B⊃A (B contains A). For example,
{♠}⊂{♠ ♣}⊂{♠ ♣ ♦ }
{♠ ♣ ♦ }⊃{ ♦ }⊃{♦ }
The symbols “contains” and “contains” appeared in 1890 by the German mathematician and logician Ernst Schroeder.
Affiliation. J. Peano (1895).
If a is an element of the set A, then write a∈A and read “a belongs to A.” If a is not an element of the set A, write a∉A and read “a does not belong to A.” At first, the relations “contained” and “belongs” (“is an element”) were not distinguished, but over time these concepts required differentiation. The symbol ∈ was first used by the Italian mathematician Giuseppe Peano in 1895. The symbol ∈ comes from the first letter of the Greek word εστι - to be.
Quantifier of universality, quantifier of existence. G. Gentzen (1935), C. Pierce (1885).
Quantifier is a general name for logical operations that indicate the domain of truth of a predicate (mathematical statement). Philosophers have long paid attention to logical operations that limit the domain of truth of a predicate, but have not identified them as a separate class of operations. Although quantifier-logical constructions are widely used in both scientific and everyday speech, their formalization occurred only in 1879, in the book of the German logician, mathematician and philosopher Friedrich Ludwig Gottlob Frege “The Calculus of Concepts”. Frege's notation looked like cumbersome graphic constructions and was not accepted. Subsequently, many more successful symbols were proposed, but the notations that became generally accepted were ∃ for the existential quantifier (read “exists”, “there is”), proposed by the American philosopher, logician and mathematician Charles Peirce in 1885, and ∀ for the universal quantifier (read “any” , “each”, “everyone”), formed by the German mathematician and logician Gerhard Karl Erich Gentzen in 1935 by analogy with the symbol of the quantifier of existence (inverted first letters of the English words Existence (existence) and Any (any)). For example, record
(∀ε>0) (∃δ>0) (∀x≠x 0 , |x-x 0 |<δ) (|f(x)-A|<ε)
reads like this: “for any ε>0 there is δ>0 such that for all x not equal to x 0 and satisfying the inequality |x-x 0 |<δ, выполняется неравенство |f(x)-A|<ε".
Empty set. N. Bourbaki (1939).
A set that does not contain a single element. The sign of the empty set was introduced in the books of Nicolas Bourbaki in 1939. Bourbaki is the collective pseudonym of a group of French mathematicians created in 1935. One of the members of the Bourbaki group was Andre Weil, the author of the Ø symbol.
Q.E.D. D. Knuth (1978).
In mathematics, proof is understood as a sequence of reasoning built on certain rules, showing that a certain statement is true. Since the Renaissance, the end of a proof has been denoted by mathematicians by the abbreviation "Q.E.D.", from the Latin expression "Quod Erat Demonstrandum" - "What was required to be proved." When creating the computer layout system ΤΕΧ in 1978, American computer science professor Donald Edwin Knuth used a symbol: a filled square, the so-called “Halmos symbol”, named after the Hungarian-born American mathematician Paul Richard Halmos. Today, the completion of a proof is usually indicated by the Halmos Symbol. As an alternative, other signs are used: an empty square, a right triangle, // (two forward slashes), as well as the Russian abbreviation “ch.t.d.”
“Symbols are not only recordings of thoughts,
a means of depicting and consolidating it, -
no, they influence the thought itself,
they... guide her, and that’s enough
move them on paper... in order to
to unerringly reach new truths.”
L.Carnot
Mathematical signs serve primarily for precise (unambiguously defined) recording of mathematical concepts and sentences. Their totality in real conditions of their application by mathematicians constitutes what is called mathematical language.
Mathematical symbols make it possible to write in a compact form sentences that are cumbersome to express in ordinary language. This makes them easier to remember.
Before using certain signs in reasoning, the mathematician tries to say what each of them means. Otherwise they may not understand him.
But mathematicians cannot always immediately say what this or that symbol they introduced for any mathematical theory reflects. For example, for hundreds of years mathematicians operated with negative and complex numbers, but the objective meaning of these numbers and the operation with them was discovered only at the end of the 18th and beginning of the 19th centuries.
1. Symbolism of mathematical quantifiers
Like ordinary language, the language of mathematical signs allows the exchange of established mathematical truths, but being only an auxiliary tool attached to ordinary language and cannot exist without it.
Mathematical definition:
In ordinary language:
Limit of the function F (x) at some point X0 is a constant number A such that for an arbitrary number E>0 there exists a positive d(E) such that from the condition |X - X 0 | Writing in quantifiers (in mathematical language) 2. Symbolism of mathematical signs and geometric figures. 1) Infinity is a concept used in mathematics, philosophy and science. The infinity of a concept or attribute of a certain object means that it is impossible to indicate boundaries or a quantitative measure for it. The term infinity corresponds to several different concepts, depending on the field of application, be it mathematics, physics, philosophy, theology or everyday life. In mathematics there is no single concept of infinity; it is endowed with special properties in each section. Moreover, these different "infinities" are not interchangeable. For example, set theory implies different infinities, and one may be greater than the other. Let's say the number of integers is infinitely large (it is called countable). To generalize the concept of the number of elements for infinite sets, the concept of cardinality of a set is introduced in mathematics. However, there is no one “infinite” power. For example, the power of the set of real numbers is greater than the power of integers, because one-to-one correspondence cannot be built between these sets, and integers are included in the real numbers. Thus, in this case, one cardinal number (equal to the power of the set) is “infinite” than the other. The founder of these concepts was the German mathematician Georg Cantor. In calculus, two symbols are added to the set of real numbers, plus and minus infinity, used to determine boundary values and convergence. It should be noted that in this case we are not talking about “tangible” infinity, since any statement containing this symbol can be written using only finite numbers and quantifiers. These symbols (and many others) were introduced to shorten longer expressions. Infinity is also inextricably linked with the designation of the infinitely small, for example, Aristotle said: Infinity in most cultures appeared as an abstract quantitative designation for something incomprehensibly large, applied to entities without spatial or temporal boundaries. 2) A circle is a geometric locus of points on a plane, the distance from which to a given point, called the center of the circle, does not exceed a given non-negative number, called the radius of this circle. If the radius is zero, then the circle degenerates into a point. A circle is the geometric locus of points on a plane that are equidistant from a given point, called the center, at a given non-zero distance, called its radius. 3) Square (rhombus) - is a symbol of the combination and ordering of four different elements, for example the four main elements or the four seasons. Symbol of the number 4, equality, simplicity, integrity, truth, justice, wisdom, honor. Symmetry is the idea through which a person tries to comprehend harmony and has been considered a symbol of beauty since ancient times. The so-called “figured” verses, the text of which has the outline of a rhombus, have symmetry. We - (E.Martov, 1894) 4) Rectangle. Of all geometric forms, this is the most rational, most reliable and correct figure; empirically this is explained by the fact that the rectangle has always and everywhere been the favorite shape. With its help, a person adapted space or any object for direct use in his everyday life, for example: a house, room, table, bed, etc. 5) The Pentagon is a regular pentagon in the shape of a star, a symbol of eternity, perfection, and the universe. Pentagon - an amulet of health, a sign on the doors to ward off witches, the emblem of Thoth, Mercury, Celtic Gawain, etc., a symbol of the five wounds of Jesus Christ, prosperity, good luck among the Jews, the legendary key of Solomon; a sign of high status in Japanese society. 6) Regular hexagon, hexagon - a symbol of abundance, beauty, harmony, freedom, marriage, a symbol of the number 6, an image of a person (two arms, two legs, a head and a torso). 7) The cross is a symbol of the highest sacred values. The cross models the spiritual aspect, the ascension of the spirit, the aspiration to God, to eternity. The cross is a universal symbol of the unity of life and death. 8) A triangle is a geometric figure that consists of three points that do not lie on the same line, and three segments connecting these three points. 9) Six-pointed Star (Star of David) - consists of two equilateral triangles superimposed on one another. One version of the origin of the sign connects its shape with the shape of the White Lily flower, which has six petals. The flower was traditionally placed under the temple lamp, in such a way that the priest lit a fire, as it were, in the center of the Magen David. In Kabbalah, two triangles symbolize the inherent duality of man: good versus evil, spiritual versus physical, and so on. The upward-pointing triangle symbolizes our good deeds, which rise to heaven and cause a stream of grace to descend back to this world (which is symbolized by the downward-pointing triangle). Sometimes the Star of David is called the Star of the Creator and each of its six ends is associated with one of the days of the week, and the center with Saturday. 10) Five-pointed Star - The main distinctive emblem of the Bolsheviks is the red five-pointed star, officially installed in the spring of 1918. Initially, Bolshevik propaganda called it the “Star of Mars” (supposedly belonging to the ancient god of war - Mars), and then began to declare that “The five rays of the star mean the union of the working people of all five continents in the fight against capitalism.” In reality, the five-pointed star has nothing to do with either the militant deity Mars or the international proletariat, it is an ancient occult sign (apparently of Middle Eastern origin) called the “pentagram” or “Star of Solomon”. Let us note that the pentagram was often placed by the Bolsheviks on Red Army uniforms, military equipment, various signs and all kinds of attributes of visual propaganda in a purely satanic way: with two “horns” up. 3. Masonic signs Masons Motto:"Freedom. Equality. Brotherhood". A social movement of free people who, on the basis of free choice, make it possible to become better, to become closer to God, and therefore, they are recognized as improving the world. Signs The radiant eye (delta) is an ancient, religious sign. He says that God oversees his creations. With the image of this sign, Freemasons asked God for blessings for any grandiose actions or for their labors. The Radiant Eye is located on the pediment of the Kazan Cathedral in St. Petersburg. The combination of a compass and a square in a Masonic sign. For the uninitiated, this is a tool of labor (mason), and for the initiated, these are ways of understanding the world and the relationship between divine wisdom and human reason. For divine wisdom nothing is impossible, it can take on both a human form (-) and a divine form (0), it can contain everything. Thus, the human mind comprehends divine wisdom and embraces it. In philosophy, this statement is a postulate about absolute and relative truth. Hexagonal Star (Bethlehem) The letter G is the designation of God (German - Got), the great geometer of the Universe. Conclusion Mathematical symbols serve primarily to accurately record mathematical concepts and sentences. Their totality constitutes what is called mathematical language. When people interact for a long time within a certain field of activity, they begin to look for a way to optimize the communication process. The system of mathematical signs and symbols is an artificial language that was developed to reduce the amount of graphically transmitted information while fully preserving the meaning of the message. Any language requires learning, and the language of mathematics in this regard is no exception. To understand the meaning of formulas, equations and graphs, you need to have certain information in advance, understand the terms, notation system, etc. In the absence of such knowledge, the text will be perceived as written in an unfamiliar foreign language. In accordance with the needs of society, graphic symbols for simpler mathematical operations (for example, notation for addition and subtraction) were developed earlier than for complex concepts like integral or differential. The more complex the concept, the more complex the sign it is usually denoted. In the early stages of the development of civilization, people connected the simplest mathematical operations with familiar concepts based on associations. For example, in Ancient Egypt, addition and subtraction were indicated by a pattern of walking feet: lines directed in the direction of reading they indicated “plus”, and in the opposite direction - “minus”. Numbers, perhaps in all cultures, were initially designated by the corresponding number of lines. Later, conventional notations began to be used for recording - this saved time, as well as space on physical media. Letters were often used as symbols: this strategy became widespread in Greek, Latin and many other languages of the world. The history of the emergence of mathematical symbols and signs knows two of the most productive ways of creating graphic elements. Initially, any mathematical concept is expressed by a certain word or phrase and does not have its own graphic representation (besides the lexical one). However, performing calculations and writing formulas in words is a lengthy procedure and takes up an unreasonably large amount of space on a physical medium. A common way to create mathematical symbols is to transform the lexical representation of a concept into a graphic element. In other words, the word denoting a concept is shortened or transformed in some other way over time. For example, the main hypothesis for the origin of the plus sign is its abbreviation from the Latin et, the analogue of which in Russian is the conjunction “and”. Gradually, the first letter in cursive writing stopped being written, and t reduced to a cross. Another example is the "x" sign for the unknown, which was originally an abbreviation of the Arabic word for "something". In a similar way, signs for denoting the square root, percentage, integral, logarithm, etc. appeared. In the table of mathematical symbols and signs you can find more than a dozen graphic elements that appeared in this way. The second common option for the formation of mathematical signs and symbols is to assign the symbol in an arbitrary manner. In this case, the word and graphic designation are not related to each other - the sign is usually approved as a result of the recommendation of one of the members of the scientific community. For example, the signs for multiplication, division, and equality were proposed by mathematicians William Oughtred, Johann Rahn and Robert Record. In some cases, several mathematical symbols may have been introduced into science by one scientist. In particular, Gottfried Wilhelm Leibniz proposed a number of symbols, including integral, differential, and derivative. Every schoolchild knows signs such as “plus” and “minus”, as well as symbols for multiplication and division, despite the fact that there are several possible graphic signs for the last two mentioned operations. It is safe to say that people knew how to add and subtract many millennia before our era, but standardized mathematical signs and symbols denoting these actions and known to us today appeared only by the 14th-15th centuries. However, despite the establishment of a certain agreement in the scientific community, multiplication in our time can be represented by three different signs (a diagonal cross, a dot, an asterisk), and division by two (a horizontal line with dots above and below or a slash). For many centuries, the scientific community exclusively used Latin to communicate information, and many mathematical terms and symbols find their origins in this language. In some cases, graphic elements were the result of shortening words, less often - their intentional or accidental transformation (for example, due to a typo). The percentage designation (“%”) most likely comes from a misspelling of the abbreviation who(cento, i.e. “hundredth part”). In a similar way, the plus sign came about, the history of which is described above. Much more was formed by deliberate shortening of the word, although this is not always obvious. Not every person recognizes the letter in the square root sign R, i.e. the first character in the word Radix (“root”). The integral symbol also represents the first letter of the word Summa, but intuitively it looks like a capital letter f without a horizontal line. By the way, in the first publication the publishers made just such a mistake by printing f instead of this symbol. Not only Latin ones are used as graphic notations for various concepts, but also in the table of mathematical symbols you can find a number of examples of such names. The number Pi, which is the ratio of the circumference of a circle to its diameter, comes from the first letter of the Greek word for circle. There are several other lesser-known irrational numbers, denoted by letters of the Greek alphabet. An extremely common sign in mathematics is “delta,” which reflects the amount of change in the value of variables. Another commonly used sign is “sigma”, which functions as a sum sign. Moreover, almost all Greek letters are used in mathematics in one way or another. However, these mathematical signs and symbols and their meaning are known only to people who are engaged in science professionally. A person does not need this knowledge in everyday life. Oddly enough, many intuitive symbols were invented quite recently. In particular, the horizontal arrow replacing the word “therefore” was proposed only in 1922. Quantifiers of existence and universality, i.e. signs read as: “there is ...” and “for any ...”, were introduced in 1897 and 1935 respectively. Symbols from the field of set theory were invented in 1888-1889. And the crossed out circle, which is known to any high school student today as the sign of an empty set, appeared in 1939. Thus, symbols for such complex concepts as integral or logarithm were invented centuries earlier than some intuitive symbols that are easily perceived and learned even without prior preparation. Due to the fact that a significant part of the concepts was described in scientific works in Latin, a number of names of mathematical signs and symbols in English and Russian are the same. For example: Plus, Integral, Delta function, Perpendicular, Parallel, Null. Some concepts in the two languages are called differently: for example, division is Division, multiplication is Multiplication. In rare cases, the English name for a mathematical sign becomes somewhat widespread in the Russian language: for example, the slash in recent years is often called “slash”. The easiest and most convenient way to familiarize yourself with the list of mathematical signs is to look at a special table that contains operation signs, symbols of mathematical logic, set theory, geometry, combinatorics, mathematical analysis, and linear algebra. This table presents the basic mathematical symbols in English. When performing various types of work, it is often necessary to use formulas that use characters that are not on the computer keyboard. Like graphic elements from almost any field of knowledge, mathematical signs and symbols in Word can be found in the “Insert” tab. In the 2003 or 2007 versions of the program, there is an “Insert Symbol” option: when you click on the button on the right side of the panel, the user will see a table that presents all the necessary mathematical symbols, Greek lowercase and uppercase letters, different types of brackets and much more. In program versions released after 2010, a more convenient option has been developed. When you click on the “Formula” button, you go to the formula constructor, which provides for the use of fractions, entering data under the root, changing the register (to indicate powers or serial numbers of variables). All the signs from the table presented above can also be found here. The mathematical notation system is an artificial language that only simplifies the writing process, but cannot bring an understanding of the subject to an outside observer. Thus, memorizing signs without studying terms, rules, and logical connections between concepts will not lead to mastery of this area of knowledge. The human brain easily learns signs, letters and abbreviations - mathematical symbols are remembered by themselves when studying the subject. Understanding the meaning of each specific action creates such strong signs that the signs denoting the terms, and often the formulas associated with them, remain in memory for many years and even decades. Since any language, including an artificial one, is open to changes and additions, the number of mathematical signs and symbols will certainly grow over time. It is possible that some elements will be replaced or adjusted, while others will be standardized in the only possible form, which is relevant, for example, for multiplication or division signs. The ability to use mathematical symbols at the level of a full school course is practically necessary in the modern world. In the context of the rapid development of information technology and science, widespread algorithmization and automation, mastery of the mathematical apparatus should be taken for granted, and the mastery of mathematical symbols as an integral part of it. Since calculations are used in the humanities, economics, natural sciences, and, of course, in the field of engineering and high technology, understanding mathematical concepts and knowledge of symbols will be useful for any specialist. The development of mathematical symbolism was closely related to the general development of concepts and methods of mathematics. First Mathematical signs there were signs to depict numbers - numbers,
the emergence of which, apparently, preceded writing. The most ancient numbering systems - Babylonian and Egyptian - appeared as early as 3 1/2 millennium BC. e. First Mathematical signs for arbitrary quantities appeared much later (starting from the 5th-4th centuries BC) in Greece. Quantities (areas, volumes, angles) were depicted in the form of segments, and the product of two arbitrary homogeneous quantities was depicted in the form of a rectangle built on the corresponding segments. In "Principles" Euclid
(3rd century BC) quantities are denoted by two letters - the initial and final letters of the corresponding segment, and sometimes just one. U Archimedes
(3rd century BC) the latter method becomes common. Such a designation contained possibilities for the development of letter calculus. However, in classical ancient mathematics, letter calculus was not created. The beginnings of letter representation and calculus appeared in the late Hellenistic era as a result of the liberation of algebra from geometric form. Diophantus
(probably 3rd century) recorded unknown ( X) and its degree with the following signs: [ - from the Greek term dunamiV (dynamis - force), denoting the square of the unknown, - from the Greek cuboV (k_ybos) - cube]. To the right of the unknown or its powers, Diophantus wrote coefficients, for example 3 x 5 was depicted (where = 3). When adding, Diophantus attributed the terms to each other, and used a special sign for subtraction; Diophantus denoted equality with the letter i [from the Greek isoV (isos) - equal]. For example, the equation (x 3 + 8x) - (5x 2 + 1) =X Diophantus would have written it like this:
(Here means that the unit does not have a multiplier in the form of a power of the unknown). Several centuries later, the Indians introduced various Mathematical signs for several unknowns (abbreviations for the names of colors denoting unknowns), square, square root, subtrahend. So, the equation 3X 2 + 10x - 8 = x 2 + 1 In recording Brahmagupta
(7th century) would look like: Ya va 3 ya 10 ru 8 Ya va 1 ya 0 ru 1 (ya - from yawat - tawat - unknown, va - from varga - square number, ru - from rupa - rupee coin - free term, a dot over the number means the subtracted number). The creation of modern algebraic symbolism dates back to the 14th-17th centuries; it was determined by the successes of practical arithmetic and the study of equations. In various countries they spontaneously appear Mathematical signs for some actions and for powers of unknown magnitude. Many decades and even centuries pass before one or another convenient symbol is developed. So, at the end of 15 and. N. Shuke
and L. Pacioli
used addition and subtraction signs (from Latin plus and minus), German mathematicians introduced modern + (probably an abbreviation of Latin et) and -. Back in the 17th century. you can count about a dozen Mathematical signs for the multiplication action. There were also different Mathematical signs unknown and its degrees. In the 16th - early 17th centuries. more than ten notations competed for the square of the unknown alone, e.g. se(from census - a Latin term that served as a translation of the Greek dunamiV, Q(from quadratum), , A (2), , Aii, aa, a 2 etc. Thus, the equation x 3 + 5 x = 12 the Italian mathematician G. Cardano (1545) would have the form: from the German mathematician M. Stiefel (1544): from the Italian mathematician R. Bombelli (1572): French mathematician F. Vieta (1591): from the English mathematician T. Harriot (1631): In the 16th and early 17th centuries. equal signs and brackets are used: square (R. Bombelli
, 1550), round (N. Tartaglia,
1556), figured (F. Viet,
1593). In the 16th century the modern form takes on the notation of fractions. A significant step forward in the development of mathematical symbolism was the introduction by Viet (1591) Mathematical signs for arbitrary constants in the form of capital consonant letters of the Latin alphabet B, D, which gave him the opportunity for the first time to write down algebraic equations with arbitrary coefficients and operate with them. Viet depicted unknowns with vowels in capital letters A, E,... For example, Viet's recording In our symbols it looks like this: x 3 + 3bx = d.
Viet was the creator of algebraic formulas. R. Descartes
(1637) gave the signs of algebra a modern look, denoting unknowns with the last letters of Lat. alphabet x, y, z, and arbitrary data values - with initial letters a, b, c. The current record of the degree belongs to him. Descartes' notations had a great advantage over all previous ones. Therefore, they soon received universal recognition. Further development Mathematical signs was closely connected with the creation of infinitesimal analysis, for the development of the symbolism of which the basis was already largely prepared in algebra. Dates of origin of some mathematical symbols L. Euler limit many mathematicians early 20th century and for an infinitesimal increment o. Somewhat earlier J. Wallis
(1655) proposed the infinity sign ¥. The creator of modern symbolism of differential and integral calculus is G. Leibniz.
In particular, he owns the currently used Mathematical signs differentials dx,d 2 x,d 3 x
and integral
Enormous credit for creating the symbolism of modern mathematics belongs to L. Euler.
He introduced (1734) into general use the first sign of a variable operation, namely the sign of the function f(x)
(from Latin functio). After Euler's work, the signs for many individual functions, such as trigonometric functions, became standard. Euler is the author of the notation for the constants e(base of natural logarithms, 1736), p [probably from Greek perijereia (periphereia) - circle, periphery, 1736], imaginary unit (from the French imaginaire - imaginary, 1777, published 1794). In the 19th century the role of symbolism is increasing. At this time, the signs of the absolute value |x| appear. (TO. Weierstrass,
1841), vector (O. Cauchy,
1853), determinant (A. Cayley,
1841), etc. Many theories that arose in the 19th century, for example tensor calculus, could not be developed without suitable symbolism. Along with the specified standardization process Mathematical signs in modern literature one can often find Mathematical signs, used by individual authors only within the scope of this study. From the point of view of mathematical logic, among Mathematical signs The following main groups can be outlined: A) signs of objects, B) signs of operations, C) signs of relations. For example, the signs 1, 2, 3, 4 represent numbers, that is, objects studied by arithmetic. The addition sign + by itself does not represent any object; it receives subject content when it is indicated which numbers add up: the notation 1 + 3 represents the number 4. The sign > (greater than) is a sign of the relationship between numbers. The relation sign receives a completely definite content when it is indicated between which objects the relation is considered. To the listed three main groups Mathematical signs adjacent to the fourth: D) auxiliary signs that establish the order of combination of the main signs. A sufficient idea of such signs is given by brackets indicating the order of actions. The signs of each of the three groups A), B) and C) are of two kinds: 1) individual signs of well-defined objects, operations and relations, 2) general signs of “unvariable” or “unknown” objects, operations and relations. Examples of signs of the first kind can serve (see also table): A 1) Designations of natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9; transcendental numbers e and p; imaginary unit i.
B 1) Signs of arithmetic operations +, -, ·, ´,:; root extraction, differentiation signs of the sum (union) È and the product (intersection) Ç of sets; this also includes the signs of individual functions sin, tg, log, etc. 1) Equal and inequality signs =, >,<, ¹, знаки параллельности || и перпендикулярности ^, знаки принадлежности Î элемента некоторому множеству и включения Ì одного множества в другое и т.п. Signs of the second kind depict arbitrary objects, operations and relations of a certain class or objects, operations and relations that are subject to some pre-agreed conditions. For example, when writing the identity ( a + b)(a - b) = a 2 -b 2 letters A And b represent arbitrary numbers; when studying functional dependence at = X 2 letters X And y - arbitrary numbers connected by a given relationship; when solving the equation X denotes any number that satisfies this equation (as a result of solving this equation, we learn that only two possible values +1 and -1 correspond to this condition). From a logical point of view, it is legitimate to call such general signs signs of variables, as is customary in mathematical logic, without being afraid of the fact that the “domain of change” of a variable may turn out to consist of one single object or even “empty” (for example, in the case of equations , without a solution). Further examples of this type of signs can be: A 2) Designations of points, lines, planes and more complex geometric figures with letters in geometry. B 2) Designations f, , j for functions and operator calculus notation, when with one letter L represent, for example, an arbitrary operator of the form:
Notations for “variable relations” are less common; they are used only in mathematical logic (see. Algebra of logic
) and in relatively abstract, mostly axiomatic, mathematical studies. Lit.: Cajori., A history of mathematical notations, v. 1-2, Chi., 1928-29. Article about the word " Mathematical signs" in the Great Soviet Encyclopedia was read 39,764 times Each of us from school (or rather from the 1st grade of primary school) should be familiar with such simple mathematical symbols as more sign And less than sign, and also the equal sign. However, if it is quite difficult to confuse something with the latter, then about How and in which direction are greater and less than signs written? (less sign And over sign, as they are sometimes called) many immediately after the same school bench forget, because they are rarely used by us in everyday life. But almost everyone, sooner or later, still has to encounter them, and they can only “remember” in which direction the character they need is written by turning to their favorite search engine for help. So why not answer this question in detail, at the same time telling visitors to our site how to remember the correct spelling of these signs for the future? It is precisely how to correctly write the greater-than and less-than sign that we want to remind you in this short note. It would also not be amiss to tell you that how to type greater than or equal signs on the keyboard And less or equal, because This question also quite often causes difficulties for users who encounter such a task very rarely. Let's get straight to the point. If you are not very interested in remembering all this for the future and it’s easier to “Google” again next time, but now you just need an answer to the question “in which direction to write the sign,” then we have prepared a short answer for you - the signs for more and less are written like this: as shown in the image below. Now let’s tell you a little more about how to understand and remember this for the future. In general, the logic of understanding is very simple - whichever side (larger or smaller) the sign in the direction of writing faces to the left is the sign. Accordingly, the sign looks more to the left with its wide side - the larger one. An example of using the greater than sign: As you can see, everything is quite logical and simple, so now you should not have questions about which direction to write the greater sign and the less sign in the future. If you already remember how to write the sign you need, then it will not be difficult for you to add one line from below, this way you will get the sign "less or equal" or sign "more or equal". However, regarding these signs, some people have another question - how to type such an icon on a computer keyboard? As a result, most simply put two signs in a row, for example, “greater than or equal” denoting as ">="
, which, in principle, is often quite acceptable, but can be done more beautifully and correctly. In fact, in order to type these characters, there are special characters that can be entered on any keyboard. Agree, signs "≤"
And "≥"
look much better. In order to write “greater than or equal to” on the keyboard with one sign, you don’t even need to go into the table of special characters - just write the greater than sign while holding down the key "alt". Thus, the key combination (entered in the English layout) will be as follows. Or you can just copy the icon from this article if you only need to use it once. Here it is, please. ≥
As you probably already guessed, you can write “less than or equal to” on the keyboard by analogy with the greater than sign - just write the less than sign while holding down the key "alt". The keyboard shortcut you need to enter in the English keyboard will be as follows. Or just copy it from this page if that makes it easier for you, here it is. ≤
As you can see, the rule for writing greater than and less than signs is quite simple to remember, and in order to type the greater than or equal to and less than or equal to symbols on the keyboard, you just need to press an additional key - it’s simple.
“... it is always possible to come up with a larger number, because the number of parts into which a segment can be divided has no limit; therefore, infinity is potential, never actual, and no matter what number of divisions is given, it is always potentially possible to divide this segment into an even larger number.” Let us note that Aristotle made a great contribution to the awareness of infinity, dividing it into potential and actual, and from this side came closely to the foundations of mathematical analysis, also pointing to five sources of ideas about it:
Further, infinity was developed in philosophy and theology along with the exact sciences. For example, in theology, the infinity of God does not so much give a quantitative definition as it means unlimited and incomprehensible. In philosophy, this is an attribute of space and time.
Modern physics comes close to the relevance of infinity denied by Aristotle - that is, accessibility in the real world, and not just in the abstract. For example, there is the concept of a singularity, closely related to black holes and the big bang theory: it is a point in spacetime at which mass in an infinitesimal volume is concentrated with infinite density. There is already solid indirect evidence for the existence of black holes, although the big bang theory is still under development.
The circle is a symbol of the Sun, Moon. One of the most common symbols. It is also a symbol of infinity, eternity, and perfection.
The poem is a rhombus.
Among the darkness.
The eye is resting.
The darkness of the night is alive.
The heart sighs greedily,
The whispers of the stars sometimes reach us.
And the azure feelings are crowded.
Everything was forgotten in the dewy brilliance.
Let's give you a fragrant kiss!
Shine quickly!
Whisper again
As then:
"Yes!"
Of course, you may not agree with these statements.
However, no one will deny that any image evokes associations in a person. But the problem is that some objects, plots or graphic elements evoke the same associations in all people (or rather, many), while others evoke completely different ones.
Properties of a triangle as a figure: strength, immutability.
Axiom A1 of stereometry says: “Through 3 points of space that do not lie on the same straight line, a plane passes, and only one!”
To test the depth of understanding of this statement, a task is usually asked: “There are three flies sitting on the table, at three ends of the table. At a certain moment, they fly apart in three mutually perpendicular directions at the same speed. When will they be on the same plane again?” The answer is the fact that three points always, at any moment, define a single plane. And it is precisely 3 points that define the triangle, so this figure in geometry is considered the most stable and durable.
The triangle is usually referred to as a sharp, “offensive” figure associated with the masculine principle. The equilateral triangle is a masculine and solar sign representing divinity, fire, life, heart, mountain and ascension, well-being, harmony and royalty. An inverted triangle is a feminine and lunar symbol, representing water, fertility, rain, and divine mercy.
State symbols of the United States also contain the Six-Pointed Star in different forms, in particular it is on the Great Seal of the United States and on banknotes. The Star of David is depicted on the coats of arms of the German cities of Cher and Gerbstedt, as well as the Ukrainian Ternopil and Konotop. Three six-pointed stars are depicted on the flag of Burundi and represent the national motto: “Unity. Job. Progress".
In Christianity, a six-pointed star is a symbol of Christ, namely the union of the divine and human nature in Christ. That is why this sign is inscribed in the Orthodox Cross.
Government”, which is under the complete control of Freemasonry.
Very often, Satanists draw a pentagram with both ends up so that it is easy to fit the devil’s head “Pentagram of Baphomet” there. The portrait of the “Fiery Revolutionary” is placed inside the “Pentagram of Baphomet”, which is the central part of the composition of the special Chekist order “Felix Dzerzhinsky” designed in 1932 (the project was later rejected by Stalin, who deeply hated “Iron Felix”).
The Marxist plans for a “world proletarian revolution” were clearly of Masonic origin; a number of the most prominent Marxists were members of Freemasonry. L. Trotsky was one of them, and it was he who proposed making the Masonic pentagram the identifying emblem of Bolshevism.
International Masonic lodges secretly provided the Bolsheviks with full support, especially financial.
Freemasons are comrades of the Creator, supporters of social progress, against inertia, inertia and ignorance. Outstanding representatives of Freemasonry are Nikolai Mikhailovich Karamzin, Alexander Vasilievich Suvorov, Mikhail Illarionovich Kutuzov, Alexander Sergeevich Pushkin, Joseph Goebbels.
The square, as a rule, from below is human knowledge of the world. From the point of view of Freemasonry, a person comes into the world to understand the divine plan. And for knowledge you need tools. The most effective science in understanding the world is mathematics.
The square is the oldest mathematical instrument, known since time immemorial. Graduation of the square is already a big step forward in the mathematical tools of cognition. A person understands the world with the help of sciences; mathematics is the first of them, but not the only one.
However, the square is wooden, and it holds what it can hold. It cannot be moved apart. If you try to expand it to accommodate more, you will break it.
So people who try to understand the entire infinity of the divine plan either die or go crazy. “Know your boundaries!” - this is what this sign tells the World. Even if you were Einstein, Newton, Sakharov - the greatest minds of mankind! - understand that you are limited by the time in which you were born; in understanding the world, language, brain capacity, a variety of human limitations, the life of your body. Therefore, yes, learn, but understand that you will never fully understand!
What about the compass? The compass is divine wisdom. You can use a compass to describe a circle, but if you spread its legs, it will be a straight line. And in symbolic systems, a circle and a straight line are two opposites. The straight line denotes a person, his beginning and end (like a dash between two dates - birth and death). The circle is a symbol of deity because it is a perfect figure. They oppose each other - divine and human figures. Man is not perfect. God is perfect in everything.
People always know the truth, but always relative truth. And absolute truth is known only to God.
Learn more and more, realizing that you will not be able to fully understand the truth - what depths we find in an ordinary compass with a square! Who would have thought!
This is the beauty and charm of Masonic symbolism, its enormous intellectual depth.
Since the Middle Ages, the compass, as a tool for drawing perfect circles, has become a symbol of geometry, cosmic order and planned actions. At this time, the God of Hosts was often depicted in the image of the creator and architect of the Universe with a compass in his hands (William Blake “The Great Architect”, 1794).
The Hexagonal Star meant Unity and the Struggle of Opposites, the struggle of Man and Woman, Good and Evil, Light and Darkness. One cannot exist without the other. The tension that arises between these opposites creates the world as we know it.
The upward triangle means “Man strives for God.” Triangle down - “Divinity descends to Man.” In their connection our world exists, which is the union of the Human and the Divine. The letter G here means that God lives in our world. He is truly present in everything he created.
The decisive force in the development of mathematical symbolism is not the “free will” of mathematicians, but the requirements of practice and mathematical research. It is real mathematical research that helps to find out which system of signs best reflects the structure of quantitative and qualitative relationships, which is why they can be an effective tool for their further use in symbols and emblems.Models for the formation of graphic symbols
Converting a Verbal Representation
Custom character assignment
Simplest operations
Letters
Greek letters
Signs of logic
Mathematical symbols in English
symbol table
Mathematical symbols in a text editor
Is it worth learning math symbols?
Finally
sign
meaning
Who entered
When entered Signs of individual objects
¥
infinity
J. Wallis
1655
e
base of natural logarithms
L. Euler
1736
p
ratio of circumference to diameter
W. Jones
1706
i
square root of -1
L. Euler
1777 (printed 1794)
i j k
unit vectors, unit vectors
W. Hamilton
1853
P(a)
angle of parallelism
N.I. Lobachevsky
1835Signs of variable objects
x,y,z
unknown or variable quantities
R. Descartes
1637
r
vector
O. Cauchy
1853Individual Operations Signs
+
addition
German mathematicians
Late 15th century
–
subtraction
´
multiplication
W. Outred
1631
×
multiplication
G. Leibniz
1698
:
division
G. Leibniz
1684
a 2 , a 3 ,…, a n
degrees
R. Descartes
1637
I. Newton
1676
roots
K. Rudolph
1525
A. Girard
1629
Log
logarithm
I. Kepler
1624
log
B. Cavalieri
1632
sin
sinus
L. Euler
1748
cos
cosine
tg
tangent
L. Euler
1753
arc.sin
arcsine
J. Lagrange
1772 Sh
hyperbolic sine V. Riccati
1757 Ch
hyperbolic cosine
dx, ddx, …
differential
G. Leibniz
1675 (printed 1684)
d 2 x, d 3 x,…
integral
G. Leibniz
1675 (printed 1686)
derivative
G. Leibniz
1675
¦¢x
derivative
J. Lagrange
1770, 1779
y'
¦¢(x)
Dx
difference
L. Euler
1755
partial derivative
A. Legendre
1786
definite integral
J. Fourier
1819-22
sum
L. Euler
1755
P
work
K. Gauss
1812
!
factorial
K. Crump
1808
|x|
module
K. Weierstrass
1841
lim
W. Hamilton,
1853,
lim
n = ¥
lim
n ® ¥
x
zeta function
B. Riemann
1857
G
gamma function
A. Legendre
1808
IN
beta function
J. Binet
1839
D
delta (Laplace operator)
R. Murphy
1833
Ñ
nabla (Hamilton cameraman)
W. Hamilton
1853Signs of variable operations
jx
function
I. Bernouli
1718
f(x)
L. Euler
1734Signs of individual relationships
=
equality
R. Record
1557
>
more
T. Garriott
1631
<
less
º
comparability
K. Gauss
1801
parallelism
W. Outred
1677
^
perpendicularity
P. Erigon
1634
AND. Newton
in his method of fluxions and fluents (1666 and subsequent years) he introduced signs for successive fluxions (derivatives) of a quantity (in the form 
How to write the less sign is probably not worth explaining again. Exactly the same as the greater sign. If the sign faces to the left with its narrow side - the smaller one, then the sign in front of you is smaller.
An example of using the less than sign:Greater than or equal to/less than or equal to sign
Greater than or equal sign on keyboard
Less than or equal sign on keyboard