Double less sign. Remember the greater than and less than signs! The simplest way

Along with arithmetic operations, there is an acquaintance with such abstract concepts as "greater than", "less than" and "equal to". It will not be difficult for a child to determine which side has more objects and which one has less. But here the setting of signs sometimes causes difficulties. Game methods will help to learn the signs.

"Hungry Bird"

To play, you will need a sign - an open beak (a "more" sign). It can be cut out of cardboard or made into a large model from a disposable plate. To interest the baby, you can glue or draw eyes, feathers, and make the mouth open .

The explanation starts with some background: “This bird is small, loves to eat well. And she always chooses the pile in which there is more food.

After that, it is clearly shown that the bird opens its beak to the side where there are more objects.

Further, the information received is fixed: heaps with grains are laid out on the table, and the child determines in which direction the bird will turn its beak . If it is not possible to correctly position it the first time, you need to help by saying again that the mouth is open towards more food. Then you can offer several more similar tasks: the numbers are written on the sheet, you need to glue the beak correctly.

Examples can be diversified by replacing the bird with a pike, a crocodile, or any other predator that also opens its mouth towards a larger number.

There may be unusual situations where the number of items in both piles will be equal. If the child notices this, it means that he is attentive.

For this you must be commended , and then show 2 identical strips and explain that they are the same as the number of objects in piles, and since the number of objects is equal, then the sign is called “equal”.

Arrows

A small schoolchild can be explained signs based on comparing them with arrows pointing in different directions.

Difficulties may arise when reading expressions. But this difficulty can also be overcome: by putting the sign correctly, he will be able to correctly read the expression . After completing a few exercises, the child will remember that the arrow pointing to the left means the sign "less". If she points to the right, then the sign reads: "more."

Strengthening exercises

After explaining the rules for setting the sign, you need to practice in performing similar tasks.

For this purpose, tasks of this type are suitable:

1. "Put a sign" (4 and 5 - need a "less than" sign).
2. "More less" - the child shows signs with the thumb and forefinger of both hands, comparing the sizes of various objects or their number (the plane is larger than the dragonfly, the strawberry is smaller than the watermelon).
3. "What number" - there are signs, a number is written on one side, you need to guess what number will be on the other side (in the expression "_<5» на месте пропуска могут стоять числа 0 – 4).
4. "Fill in the numbers" - you need to correctly put the numbers to the left and right of the specified sign (the number 8 will be to the left of the "greater than" sign, and the number 2 to the right).

To develop logic and thinking, you can supplement the exercises with the following tasks:

• "From which direction did the object escape?" - 3 triangles are drawn on the left, 2 squares on the right, and there is a “=” sign between them. The child must guess that there is not enough square on the right for the equality to be true. If you can’t do this right away, you can solve the problem practically by adding a triangle first on the left, and then a square on the right.
• “What needs to be done to make inequality right?” - taking into account the situation, the child determines which side to remove or add objects so that the sign stands correctly.

Video tutorial will tell you about the signs: greater than, less than and equal

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The development of mathematical symbolism was closely connected with the general development of the concepts and methods of mathematics. First Mathematical signs there were signs for depicting numbers - numbers, the emergence of which, apparently, preceded writing. The most ancient numbering systems - Babylonian and Egyptian - appeared as early as 3 1/2 millennia BC. e.

First Mathematical signs for arbitrary values ​​appeared much later (starting from the 5th-4th centuries BC) in Greece. Quantities (area, volumes, angles) were shown as segments, and the product of two arbitrary homogeneous quantities - as a rectangle built on the corresponding segments. In "Beginnings" Euclid (3rd century BC) quantities are indicated by two letters - the initial and final letters of the corresponding segment, and sometimes even one. At Archimedes (3rd century BC) the latter method becomes common. Such a designation contained the possibilities for the development of literal calculus. However, in classical ancient mathematics, literal calculus was not created.

The beginnings of letter representation and calculus arise in the late Hellenistic era as a result of the liberation of algebra from geometric form. Diophantus (probably 3rd century) wrote down an unknown ( X) and its degrees with the following signs:

[ - from the Greek term dunamiV (dynamis - strength), denoting the square of the unknown, - from the Greek cuboV (k_ybos) - cube]. To the right of the unknown or its degrees, Diophantus wrote the coefficients, for example, 3x5 was depicted

(where = 3). When adding, Diophantus attributed terms to each other, for subtraction he used a special sign; 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 write it like this:

(Here

means that the unit does not have a multiplier in the form of a power of the unknown).

A few centuries later, the Indians introduced various Mathematical signs for several unknowns (abbreviations for the names of colors denoting unknowns), square, square root, subtracted number. 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 yavat - tawat - unknown, va - from varga - square number, ru - from rupa - rupee coin - a free member, a dot above the number means the number to be subtracted).

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 spontaneously appear Mathematical signs for some actions and for powers of an unknown quantity. 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 lat. plus and minus), German mathematicians introduced modern + (probably an abbreviation of lat. et) and -. Back in the 17th century can count about ten Mathematical signs for the multiplication operation.

were different and Mathematical signs unknown and its degrees. In the 16th - early 17th centuries. more than ten notations competed for the square of the unknown alone, for example 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 come into use: square (R. Bombelli , 1550), round (N. Tartaglia, 1556), curly (F. viet, 1593). In the 16th century modern look accepts fractions.

A significant step forward in the development of mathematical symbolism was the introduction by Vieta (1591) Mathematical signs for arbitrary constants in the form of capital consonants of the Latin alphabet B, D, which made it possible for him for the first time to write down algebraic equations with arbitrary coefficients and operate with them. Unknown Viet depicted vowels in capital letters A, E, ... For example, the record Vieta

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 given quantities - in initial letters a, b, c. He also owns the current record of the degree. Descartes' notation had a great advantage over all the 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 prepared to a large extent in algebra.

Dates of occurrence of some mathematical signs

and for an infinitesimal increment o. Somewhat earlier, J. Wallis (1655) proposed the infinity sign ¥.

The creator of the modern symbolism of differential and integral calculus is G. Leibniz. He, in particular, belongs to the currently used Mathematical signs differentials

dx, d 2 x, d 3 x

and integral

A huge merit in creating the symbolism of modern mathematics belongs to L. Euler. He introduced (1734) into general use the first sign of the variable operation, namely the sign of the function f(x) (from lat. functio). After Euler's work, the signs for many individual functions, such as trigonometric functions, acquired a standard character. Euler owns the notation for constants e(base of natural logarithms, 1736), p [probably from Greek perijereia (periphereia) - circumference, periphery, 1736], imaginary unit

(from the French imaginaire - imaginary, 1777, published in 1794).

In the 19th century the role of symbolism is growing. At this time, signs of the absolute value |x| (TO. Weierstrass, 1841), vector (O. Cauchy, 1853), determiner

(A. Cayley, 1841) and others. Many theories that arose in the 19th century, such as 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 depict 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 are added: the notation 1 + 3 depicts the number 4. The sign > (greater than) is the sign of the relationship between numbers. The sign of the relation receives a quite definite content when it is indicated between which objects the relation is considered. To the above three main groups Mathematical signs adjoins 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 in which actions are performed.

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 "non-repetitive" or "unknown" objects, operations and relations.

Examples of signs of the first kind can serve (see also the table):

A 1) Legend 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 sum (union) È and product (intersection) Ç of sets; this also includes the signs of the individual functions sin, tg, log, etc.

1) Equals 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 subject to some predetermined conditions. For example, when writing the identity ( a + b)(a - b) = a 2 -b 2 letters A And b denote arbitrary numbers; when studying functional dependence at = X 2 letters X And y - arbitrary numbers related by a given ratio; when solving the equation

X denotes any number that satisfies the given equation (as a result of solving this equation, we learn that only two possible values ​​\u200b\u200b+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 circumstance that the “region of change” of a variable may turn out to consist of a single object or even “empty” (for example, in the case of equations with no solution). Further examples of such signs are:

A 2) Designation of points, lines, planes and more complex geometric shapes with letters in geometry.

B 2) Notation f, , j for functions and notation of operator calculus, when one letter L depict, for example, an arbitrary operator of the form:

The notation for "variable ratios" is less common, and is used only in mathematical logic (cf. 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 has been read 39931 times

Infinity.J. Wallis (1655).

For the first time it is found in the treatise of the English mathematician John Valis "On Conic Sections".

Base of natural logarithms. L. Euler (1736).

Mathematical constant, transcendental number. This number is sometimes called non-Perov in honor of the Scottish scientist Napier, author of the work "Description of the amazing table of logarithms" (1614). For the first time, the constant is tacitly present in the appendix to the English translation of the aforementioned work by Napier, published in 1618. The very same constant was first calculated by the Swiss mathematician Jacob Bernoulli in the course of 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 started using Euler in 1727, and the first publication with this letter was his Mechanics, or the Science of Motion, Stated Analytically, 1736. Respectively, e commonly called Euler number. Why was the letter chosen? e, is not exactly known. Perhaps this is due to the fact that the word begins with it exponential("exponential", "exponential"). Another assumption is that the letters a, b, c And d already widely used for other purposes, and e was the first "free" letter.

The ratio of the circumference of a circle to its diameter. W. Jones (1706), L. Euler (1736).

Mathematical constant, irrational number. The number "pi", the old name is Ludolf's number. Like any irrational number, π is represented by an infinite non-periodic decimal fraction:

π=3.141592653589793...

For the first time, the designation of this number with 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 Adrien Marie Legendre in 1774 proved the irrationality of π 2 . 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 press - 1794).

It is known that the equation x 2 \u003d 1 has two roots: 1 And -1 . The imaginary unit is one of the two roots of the equation x 2 \u003d -1, denoted by the Latin letter i, another root: -i. This designation was proposed by Leonhard Euler, who took the first letter of the Latin word for this imaginarius(imaginary). He also extended all the standard functions to the complex domain, i.e. set of numbers representable in the form a+ib, Where a And b are real numbers. The term "complex number" was introduced into wide use by the German mathematician Carl Gauss in 1831, although the term had previously been used in the same sense by the French mathematician Lazar Carnot in 1803.

Unit vectors. W. Hamilton (1853).

Unit vectors are often associated with the coordinate axes of the coordinate system (in particular, with the axes of the Cartesian coordinate system). Unit vector directed along the axis X, denoted i, a 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 orts, they have identity 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.

The integer part of a 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 integer part function symbol was introduced by Carl Gauss in 1808. Some mathematicians prefer to use the notation E(x) proposed in 1798 by Legendre instead.

Angle of parallelism. N.I. Lobachevsky (1835).

On the Lobachevsky plane - the angle between the linebpassing through the pointABOUTparallel to a straight linea, not containing a dotABOUT, and perpendicular fromABOUT on a. α is the length of this perpendicular. As the point is removedABOUT from straight 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 is some constant related to the curvature of the Lobachevsky space.

Unknown or variable quantities. R. Descartes (1637).

In mathematics, a variable is a quantity characterized by the set of values ​​that it can take. This can 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. The literal algebra and analytic geometry of René Descartes were such new forms. For the first time, the rectangular coordinate system and the notation x, y were introduced by Rene Descartes in his work "Discourse on the method" in 1637. Pierre Fermat also contributed to the development of the coordinate method, but his work was 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 applied by Leonhard Euler already in the 18th century.

Vector. O.Koshi (1853).

From the very beginning, a vector is understood as an object having a magnitude, a direction, and (optionally) an application point. The beginnings of vector calculus appeared along with the geometric model of complex numbers in Gauss (1831). Advanced operations on vectors were published by Hamilton as part of his quaternion calculus (the imaginary components of a quaternion formed a vector). Hamilton coined the term vector(from the Latin word vector, carrier) and described some vector analysis operations. This formalism was used by Maxwell in his works on electromagnetism, thereby drawing the attention of scientists to the new calculus. Gibbs' Elements of Vector Analysis (1880s) soon followed, and then Heaviside (1903) gave vector analysis its modern look. The vector sign itself was introduced 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 Count for All Merchants, published in 1489. Prior to this, addition was denoted by the letter p(from Latin plus"more") or the Latin word et(conjunction "and"), and subtraction - by letter m(from Latin minus"less, less"). In Widman, the plus symbol replaces not only addition, but also the union "and". The origin of these symbols is unclear, but most likely they were previously used in trading as signs of profit and loss. Both symbols soon became common in Europe - with the exception of Italy, which used 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 Outred. Before him, the most commonly used letter M, although other designations were also proposed: the symbol of a rectangle (French mathematician Erigon, 1634), an asterisk (Swiss mathematician Johann Rahn, 1659). Later, Gottfried Wilhelm Leibniz replaced the cross with a dot (end of the 17th century), so as not to be confused with the letter x; before him, such symbolism was found by the German astronomer and mathematician Regiomontanus (XV century) and the English scientist Thomas Harriot (1560 -1621).

Division. I.Ran (1659), G.Leibniz (1684).

William Outred used the slash / as the division sign. Colon division began to denote Gottfried Leibniz. Before them, the letter was also often used D. Starting from Fibonacci, the horizontal line of the fraction is also used, which was used by Heron, Diophantus and in Arabic writings. In England and the United States, the ÷ (obelus) symbol, 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 the obelus from practice (1923) was inconclusive.

Percent. M. de la Porte (1685).

One hundredth of a whole, taken as a unit. The word "percent" itself comes from the Latin "pro centum", which means "one hundred". In 1685, the book Manual of Commercial Arithmetic by Mathieu de la Porte was published in Paris. In one place, it was about percentages, which then meant "cto" (short for cento). However, the typesetter mistook that "cto" for a fraction and typed "%". So because of a typo, this sign came into use.

Degrees. R. Descartes (1637), I. Newton (1676).

The modern notation for the exponent was introduced by René Descartes in his " geometries"(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 Vallis 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 the square root and can be written without indicating the degree: √. The arithmetic root of the 3rd degree is called the cube root. Medieval mathematicians (for example, Cardano) denoted the square root with the symbol R x (from the Latin Radix, root). The modern designation 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. The line above the radical expression was absent at first; it was later introduced by Descartes (1637) for a different purpose (instead of brackets), and this feature soon merged with the sign of the root. The cube root in the 16th century was designated as follows: R x .u.cu (from lat. Radix universalis cubica). Albert Girard (1629) began to use the usual notation for the 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 the logarithm was first given by the English mathematician William Gardiner (1742). By definition, the logarithm of a number b by reason a (a ≠ 1, a > 0) - exponent m, to which the number should be raised a(called the base of the logarithm) to get b. Denoted 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 brigs. 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 as early as 1619.

Before late XIX century there was no generally accepted notation for the logarithm, the base a indicated to the left and above the symbol log, then over it. Ultimately, mathematicians came to the conclusion that the most convenient place for the base is below the line, after the symbol log. The sign of the logarithm - the result of the reduction of the word "logarithm" - occurs in various types almost simultaneously with the appearance of the first tables of logarithms, for example Log- I. Kepler (1624) and G. Briggs (1631), log- B. Cavalieri (1632). Designation ln for the natural logarithm was introduced by the German mathematician Alfred Pringsheim (1893).

Sine, cosine, tangent, cotangent. W. Outred (middle of the 17th century), I. Bernoulli (18th century), L. Euler (1748, 1753).

Shorthand notation for sine and cosine was introduced by William Outred in the middle of the 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. Leonard Euler (1748, 1753) brought the theory of trigonometric functions into its modern form, and we also owe him the consolidation of real symbolism.The term "trigonometric functions" was introduced by the German mathematician and physicist Georg Simon Klugel in 1770.

The sine line of Indian mathematicians was originally called "arha jiva"("semi-string", that is, half of the 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 the bowstring and chord, and transcribed in Arabic letters and began to call the sine line "jiba". Since short vowels are not indicated in Arabic, and long "and" in the word "jiba" denoted in the same way as the semivowel "y", the Arabs began to pronounce the name of the sine line "jibe", which literally means "hollow", "bosom". 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 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).Inverse trigonometric functions usually include six functions: arcsine (arcsin), arccosine (arccos), arctangent (arctg), arccotangent (arcctg), arcsecant (arcsec) and arccosecant (arccosec). For the first time, special symbols for inverse trigonometric functions were used by Daniel Bernoulli (1729, 1736).Manner of notating inverse trigonometric functions with a prefix arc(from lat. arcus, arc) appeared at the Austrian mathematician Karl Scherfer and gained a foothold thanks to the French mathematician, astronomer and mechanic Joseph Louis Lagrange. It was meant that, for example, the usual sine allows you to find the chord subtending it along the 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 offered other notation: sin -1 and 1/sin, but they are not widely used.

Hyperbolic sine, hyperbolic cosine. W. Riccati (1757).

Historians discovered the first appearance of hyperbolic functions in the writings of the English mathematician Abraham de Moivre (1707, 1722). The modern definition and detailed study of them was carried out by the Italian Vincenzo Riccati in 1757 in the work "Opusculorum", he also proposed their designations: sh,ch. Riccati proceeded from the consideration of a single 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 a wide parallelism between the formulas of ordinary and hyperbolic trigonometry. N.I. Lobachevsky subsequently used this parallelism, trying to prove the consistency of non-Euclidean geometry, in which ordinary trigonometry is replaced by hyperbolic.

Just as the trigonometric sine and cosine are the coordinates of a point on a coordinate circle, the hyperbolic sine and cosine are the coordinates of a point on a hyperbola. Hyperbolic functions are expressed in terms of an exponent 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 ratios of hyperbolic sine and cosine, cosine and sine, respectively.

Differential. G. Leibniz (1675, in press 1684).

The main, linear part of the function increment.If the function y=f(x) one variable x has at x=x0derivative, and incrementΔy \u003d f (x 0 +? x) -f (x 0)functions f(x) can be represented asΔy \u003d f "(x 0) Δx + R (Δx) , where member R infinitely small compared toΔx. First Memberdy=f"(x 0 )Δxin this expansion is called the differential of the function f(x) at the pointx0. IN works of Gottfried Leibniz, Jacob and Johann Bernoulli word"differentia"was used in the sense of "increment", I. Bernoulli denoted it through Δ. G. Leibniz (1675, published in 1684) used the notation for "infinitely small difference"d- the first letter of the word"differential", formed by him from"differentia".

Indefinite integral. G. Leibniz (1675, in press 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- restore, restore. The sign ∫ is used to denote an integral in mathematics and is a stylized image of the first letter of a Latin word summa- sum. It was first used by the German mathematician Gottfried Leibniz, the founder of differential and integral calculus, at the end of the 17th century. Another of the founders of differential and integral calculus, Isaac Newton, did not offer an alternative symbolism of the integral in his works, although he tried various options: a vertical bar above a function or a square symbol that stands in front of a function or borders it. Indefinite integral for a function y=f(x) is the collection of all antiderivatives of the given function.

Definite integral. J. Fourier (1819-1822).

Definite integral of a function f(x) with 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 function f(x) . Definite integral a ∫ b f(x)dx numerically equal to the area of ​​\u200b\u200bthe figure bounded by the x-axis, straight lines x=a And x=b and function graph f(x). The design of a definite integral in the form familiar to us was proposed by the French mathematician and physicist Jean Baptiste Joseph Fourier in early XIX century.

Derivative. G. Leibniz (1675), J. Lagrange (1770, 1779).

Derivative - 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, however, historically, the theory of limits appeared later than differential calculus.

The term "derivative" was introduced by Joseph Louis Lagrange in 1797; dy/dx— Gottfried Leibniz in 1675. The manner of designating the derivative with respect to time with a dot above the letter comes from Newton (1691).The Russian term "derivative of a function" was first used by a Russian mathematicianVasily Ivanovich Viskovatov (1779-1812).

Private 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. Notation ∂f/ ∂ x, ∂ z/ ∂ y introduced by the French mathematician Adrien Marie Legendre in 1786; fx",zx"- Joseph Louis Lagrange (1797, 1801); ∂ 2z/ ∂ x2, ∂ 2z/ ∂ x ∂ y- second-order partial derivatives - 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 the increment by the letter Δ was first used by the Swiss mathematician Johann Bernoulli. IN general practice The use of the delta symbol came in after the work of Leonhard Euler in 1755.

Sum. L. Euler (1755).

The sum is the result of adding values ​​(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 sign Σ for the sum was introduced by Leonhard Euler in 1755.

Work. K. Gauss (1812).

The 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 symbol Π for the product was introduced by the German mathematician Carl Gauss in 1812. In Russian mathematical literature, the term "work" was first encountered by Leonty Filippovich Magnitsky in 1703.

Factorial. K.Krump (1808).

The factorial of a number n (denoted n!, pronounced "en factorial") is the product of all natural numbers up to and including n: n! = 1 2 3 ... n. For example, 5! = 1 2 3 4 5 = 120. By definition, 0! = 1. The factorial is defined only for non-negative integers. The factorial of a number 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 Kramp (1808).

Module, absolute value. K. Weierstrass (1841).

Module, the absolute value of the real number x - 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. 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 to be used by the English mathematician and philosopher, a student of Newton, Roger Cotes. Gottfried Leibniz also used this function, which he called "module" and denoted: mol x. The generally accepted notation for the absolute value was introduced in 1841 by the German mathematician Karl Weierstrass. For complex numbers, this concept was introduced by the 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 the modulus of a number. The sign "norm" (from the Latin word "norma" - "rule", "sample") was introduced by the German mathematician Erhard Schmidt in 1908.

Limit. S. Luillier (1786), W. Hamilton (1853), many mathematicians (until the beginning of the 20th century)

Limit - one of the basic concepts of mathematical analysis, meaning that some variable value in the process of its change under consideration approaches a certain constant value indefinitely. The concept of a limit was used intuitively as early as the second half of the 17th century by Isaac Newton, as well as by mathematicians of the 18th century, such as Leonhard Euler and Joseph Louis Lagrange. The first rigorous definitions of the limit of a sequence 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 with the Swiss mathematician Simon Antoine Jean Lhuillier, but its use did not yet resemble the modern one. The expression lim in a more familiar form for us was first used by the Irish mathematician William Hamilton in 1853.Weierstrass introduced a designation close to the modern one, but instead of the usual arrow, he used the equal sign. The arrow appeared at the beginning of the 20th century with several mathematicians at once - for example, with the English mathematician Godfried Hardy in 1908.

Zeta function, d Riemann zeta function. B. Riemann (1857).

Analytic function of the complex variable s = σ + it, for σ > 1, determined by the absolutely and uniformly 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 primes 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 decomposition into a product. Then this function was considered by the German mathematician L. Dirichlet and, especially successfully, by the Russian mathematician and mechanic P.L. Chebyshev in the study of 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 notation ζ(s) in 1857.

Gamma function, Euler Γ-function. A. Legendre (1814).

The gamma function is a mathematical function that extends the notion of factorial to the field of complex numbers. Usually denoted by Γ(z). The z-function was first introduced by Leonhard Euler in 1729; it is defined 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 analytic 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 in terms of the Γ-function: В(p, q) = Γ(p)Г(q)/Г(p+q).Just as the gamma function for integers is a generalization of the factorial, the beta function is, in a sense, a generalization of the binomial coefficients.

Many properties are described using the beta function.elementary particles participating in strong interaction. This feature was noticed by the Italian theoretical physicistGabriele Veneziano in 1968. It started string theory.

The name "beta function" and the notation 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 functions φ (x 1, x 2, ..., x n) from n variables x 1, x 2, ..., x n associates the function:

Δφ \u003d ∂ 2 φ / ∂x 1 2 + ∂ 2 φ / ∂x 2 2 + ... + ∂ 2 φ / ∂x 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 the Laplace equation; this is where the names "Laplace operator" or "Laplacian" come from. The notation Δ was introduced by the English physicist and mathematician Robert Murphy in 1833.

Hamiltonian 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 vectors. Through the nabla operator, the basic operations of vector analysis, as well as the Laplace operator, are expressed in a natural way.

In 1853, the Irish mathematician William Rowan Hamilton introduced this operator and coined the symbol ∇ for it in the form of an inverted Greek letter Δ (delta). At Hamilton, the point of the symbol pointed to the left; later, in the works of the Scottish mathematician and physicist Peter Guthrie Tate, the symbol acquired a modern look. Hamilton called this symbol the word "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 means "harp". The operator was called the Hamilton operator, or the 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 an intuitive idea of ​​how one quantity completely determines the value of another quantity. Often the term "function" means a numerical function; that is, a function that puts some numbers in line with others. For a long time, mathematicians gave arguments without brackets, for example, like this - φх. This notation was first used by the Swiss mathematician Johann Bernoulli in 1718.Parentheses were only used if there were many arguments, or if the argument was a complex expression. Echoes of those times are common and now recordssin x, lg xetc. But gradually the use of parentheses, f(x) , became general rule. And the main merit in this belongs to Leonhard Euler.

Equality. R. Record (1557).

The equal sign was proposed by the Welsh physician and mathematician Robert Record in 1557; the character's outline 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 that, in ancient and medieval mathematics, equality was denoted verbally (for example, est egale). Rene Descartes in the 17th century began to use æ (from lat. aequalis), and he used the modern equals sign to indicate that the coefficient could be negative. François Viète denoted subtraction with an equals sign. The symbol of the Record did not spread immediately. The spread of the Record symbol was hindered by the fact that since ancient times the same symbol has been used to indicate the parallelism of lines; in the end, it was decided to make the symbol of parallelism 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.

About the same, about the same. A. Günther (1882).

Sign " ≈" was introduced by German mathematician and physicist Adam Wilhelm Sigmund Günther in 1882 as a symbol for the relationship "about equal".

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 - the ratio between two integers n and m, meaning that the difference n-m of these numbers is divided by a given integer a, called the modulus of comparison; it is written: n≡m(mod a) and reads "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 congruent modulo 4. Comparisons have many properties similar to those of equalities. So, the term 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 true comparisons 3≡11(mod 4) and 1≡5(mod 4) the correctness of 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)

In number theory, methods for solving various comparisons are considered, i.e. methods for finding integers that satisfy comparisons of one kind or another. Modulo comparisons were first used by the German mathematician Carl Gauss in his 1801 book Arithmetical Investigations. He also proposed the symbolism established in mathematics for comparison.

Identity. B. Riemann (1857).

Identity - the equality of two analytical expressions, valid for any admissible 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. Can be written a+b ≡ b+a.

Perpendicularity. P.Erigon (1634).

Perpendicularity - the mutual arrangement of two straight lines, planes or a straight line and a plane, in which these figures make 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 (1677 posthumous edition).

Parallelism - the relationship between some geometric shapes; for example, straight lines. 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 a posthumous edition of the works of the English mathematician William Outred in 1677.

Intersection, union. J. Peano (1888).

An intersection of sets is a set that contains those and only those elements that simultaneously belong to all given sets. The 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 sets according to the above rules. Denoted ∩ 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 with 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 A, write a∉A and read "a does not belong to A". Initially, the relations "contained" and "belongs" ("is an element") were not distinguished, but over time, these concepts required a distinction. The membership sign ∈ 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.

The universal quantifier, the existential quantifier. G. Gentzen (1935), C. Pierce (1885).

A quantifier is a general name for logical operations that indicate the area of ​​truth of a predicate (mathematical statement). Philosophers have long paid attention to logical operations that limit the scope of the truth of a predicate, but did not single them out as a separate class of operations. Although quantifier-logical constructions are widely used both in scientific and everyday speech, their formalization took place 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 notation ∃ for the existential quantifier (read "exists", "there is"), proposed by the American philosopher, logician and mathematician Charles Pierce in 1885, and ∀ for the universal quantifier (read "any" , "every", "every"), formed by the German mathematician and logician Gerhard Karl Erich Gentzen in 1935 by analogy with the existential quantifier symbol (the reversed first letters of the English words Existence (existence) and Any (any)). For example, the entry

(∀ε>0) (∃δ>0) (∀x≠x 0 , |x-x 0 |<δ) (|f(x)-A|<ε)

reads as follows: "for any ε>0 there exists 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 any element. The empty set sign was introduced in the books of Nicolas Bourbaki in 1939. Bourbaki is the collective pseudonym of a group of French mathematicians formed 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, a proof is understood as a sequence of reasoning based on certain rules, showing that a certain statement is true. Since the Renaissance, the end of a proof has been denoted by mathematicians as "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, the American computer science professor Donald Edwin Knuth used a symbol: a filled square, the so-called "Halmos symbol", named after the American mathematician of Hungarian origin Paul Richard Halmos. Today, the completion of a proof is usually denoted by the Halmos Symbol. As an alternative, other signs are used: an empty square, a right triangle, // (two slashes), as well as the Russian abbreviation "ch.t.d.".