Mathematical notation(“language of mathematics”) is a complex graphic notation system used to present abstract mathematical ideas and judgments in a human-readable form. It constitutes (in its complexity and diversity) a significant proportion of non-speech sign systems used by humanity. This article describes the generally accepted international notation system, although various cultures of the past had their own, and some of them even have limited use to this day.
Note that mathematical notation, as a rule, is used in conjunction with the written form of some natural language.
In addition to fundamental and applied mathematics, mathematical notations are widely used in physics, as well as (to a limited extent) in engineering, computer science, economics, and indeed in all areas of human activity where mathematical models are used. The differences between the proper mathematical and applied style of notation will be discussed throughout the text.
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Hello! This video is not about mathematics, but rather about etymology and semiotics. But I'm sure you'll like it. Go! Are you aware that the search for solutions to cubic equations in general form took mathematicians several centuries? This is partly why? Because there were no clear symbols for clear thoughts, maybe it’s our time. There are so many symbols that you can get confused. But you and I can’t be fooled, let’s figure it out. This is the capital inverted letter A. This is actually an English letter, listed first in the words "all" and "any". In Russian, this symbol, depending on the context, can be read like this: for anyone, everyone, everyone, everything and so on. We will call such a hieroglyph a universal quantifier. And here is another quantifier, but already existence. The English letter e is reflected in Paint from left to right, thereby hinting at the overseas verb “exist”, in our way we will read: there is, there is, there is, and in other similar ways. An exclamation mark to such an existential quantifier will add uniqueness. If this is clear, let's move on. You probably came across indefinite integrals in the eleventh grade, I would like to remind you that this is not just some kind of antiderivative, but the totality of all the antiderivatives of the integrand. So don't forget about C - the constant of integration. By the way, the integral icon itself is just an elongated letter s, an echo of the Latin word sum. This is precisely the geometric meaning of a definite integral: finding the area of a figure under a graph by summing infinitesimal quantities. As for me, this is the most romantic activity in mathematical analysis. But school geometry is most useful because it teaches logical rigor. By the first year you should have a clear understanding of what a consequence is, what equivalence is. Well, you can’t get confused about necessity and sufficiency, you know? Let's even try to dig a little deeper. If you decide to take up higher mathematics, then I can imagine how bad your personal life is, but that is why you will probably agree to take on a small exercise. There are three points, each with a left and a right side, which you need to connect with one of the three drawn symbols. Please hit pause, try it for yourself, and then listen to what I have to say. If x=-2, then |x|=2, but from left to right you can construct the phrase this way. In the second paragraph, absolutely the same thing is written on the left and right sides. And the third point can be commented on as follows: every rectangle is a parallelogram, but not every parallelogram is a rectangle. Yes, I know that you are no longer little, but still my applause for those who completed this exercise. Well, okay, that's enough, let's remember numerical sets. Natural numbers are used when counting: 1, 2, 3, 4 and so on. In nature, -1 apple does not exist, but, by the way, integers allow us to talk about such things. The letter ℤ screams to us about the important role of zero; the set of rational numbers is denoted by the letter ℚ, and this is no coincidence. In English, the word "quotient" means "attitude". By the way, if somewhere in Brooklyn an African-American comes up to you and says: “Keep it real!”, you can be sure that this is a mathematician, an admirer of real numbers. Well, you should read something about complex numbers, it will be more useful. We will now make a rollback, return to the first grade of the most ordinary Greek school. In short, let's remember the ancient alphabet. The first letter is alpha, then betta, this hook is gamma, then delta, followed by epsilon and so on, until the last letter omega. You can be sure that the Greeks also have capital letters, but we won’t talk about sad things now. We are better about fun - about limits. But there are no mysteries here; it is immediately clear from which word the mathematical symbol appeared. Well, therefore, we can move on to the final part of the video. Please try to recite the definition of the limit of a number sequence that is now written in front of you. Click pause quickly and think, and may you have the happiness of a one-year-old child who recognizes the word “mother.” If for any epsilon greater than zero there is a positive integer N such that for all numbers of the numerical sequence greater than N, the inequality |xₙ-a|<Ɛ (эпсилон), то тогда предел числовой последовательности xₙ , при n, стремящемся к бесконечности, равен числу a. Такие вот дела, ребята. Не беда, если вам не удалось прочесть это определение, главное в свое время его понять. Напоследок отмечу: множество тех, кто посмотрел этот ролик, но до сих пор не подписан на канал, не является пустым. Это меня очень печалит, так что во время финальной музыки покажу, как это исправить. Ну а остальным желаю мыслить критически, заниматься математикой! Счастливо! [Музыка / аплодиминнты]
General information
The system evolved, like natural languages, historically (see the history of mathematical notations), and is organized like the writing of natural languages, borrowing from there also many symbols (primarily from the Latin and Greek alphabets). Symbols, as in ordinary writing, are depicted with contrasting lines on a uniform background (black on white paper, light on a dark board, contrasting on a monitor, etc.), and their meaning is determined primarily by their shape and relative position. Color is not taken into account and is usually not used, but when using letters, their characteristics such as style and even typeface, which do not affect the meaning in ordinary writing, can play a meaningful role in mathematical notation.
Structure
Ordinary mathematical notations (in particular, the so-called mathematical formulas) are generally written in a line from left to right, but do not necessarily form a sequential string of characters. Individual blocks of characters can appear in the top or bottom half of a line, even when the characters do not overlap verticals. Also, some parts are located entirely above or below the line. From the grammatical point of view, almost any “formula” can be considered a hierarchically organized tree-type structure.
Standardization
Mathematical notation represents a system in the sense of the interconnection of its components, but, in general, Not constitute a formal system (in the understanding of mathematics itself). In any complex case, they cannot even be parsed programmatically. Like any natural language, the “language of mathematics” is full of inconsistent notations, homographs, different (among its speakers) interpretations of what is considered correct, etc. There is not even any visible alphabet of mathematical symbols, and in particular because The question of whether to consider two designations as different symbols or different spellings of the same symbol is not always clearly resolved.
Some mathematical notation (mostly related to measurement) is standardized in ISO 31-11, but overall notation standardization is rather lacking.
Elements of mathematical notation
Numbers
If it is necessary to use a number system with a base less than ten, the base is written in the subscript: 20003 8. Number systems with bases greater than ten are not used in generally accepted mathematical notation (although, of course, they are studied by science itself), since there are not enough numbers for them. In connection with the development of computer science, the hexadecimal number system has become relevant, in which the numbers from 10 to 15 are denoted by the first six Latin letters from A to F. To designate such numbers, several different approaches are used in computer science, but they have not been transferred to mathematics.
Superscript and subscript characters
Parentheses, related symbols, and delimiters
Parentheses "()" are used:
Square brackets "" are often used in grouping meanings when many pairs of brackets must be used. In this case, they are placed on the outside and (with careful typography) have a greater height than the brackets on the inside.
Square "" and parentheses "()" are used to indicate closed and open spaces, respectively.
Curly braces "()" are generally used for , although the same caveat applies to them as for square brackets. The left "(" and right ")" brackets can be used separately; their purpose is described.
Angle bracket characters " ⟨ ⟩ (\displaystyle \langle \;\rangle ) With neat typography, they should have obtuse angles and thus differ from similar ones that have a right or acute angle. In practice, one should not hope for this (especially when writing formulas manually) and one has to distinguish between them using intuition.
Pairs of symmetrical (relative to the vertical axis) symbols, including those different from those listed, are often used to highlight a piece of the formula. The purpose of paired brackets is described.
Indexes
Depending on the location, upper and lower indices are distinguished. The superscript may (but does not necessarily mean) exponentiation, about other uses.
Variables
In the sciences there are sets of quantities, and any of them can take either a set of values and be called variable value (variant), or only one value and be called a constant. In mathematics, quantities are often abstracted from the physical meaning, and then the variable quantity turns into abstract(or numeric) variable, denoted by some symbol that is not occupied by the special notations mentioned above.
Variable X is considered given if the set of values it accepts is specified (x). It is convenient to consider a constant quantity as a variable whose corresponding set (x) consists of one element.
Functions and Operators
In mathematics there is no significant difference between operator(unary), display And function.
However, it is understood that if to write the value of a mapping from given arguments it is necessary to specify , then the symbol of this mapping denotes a function; in other cases, they rather speak of an operator. Symbols for some functions of one argument are used with or without parentheses. Many elementary functions, for example sin x (\displaystyle \sin x) or sin (x) (\displaystyle \sin(x)), but elementary functions are always called functions.
Operators and relations (unary and binary)
Functions
A function can be mentioned in two senses: as an expression of its value given given arguments (written f (x) , f (x , y) (\displaystyle f(x),\ f(x,y)) etc.) or as a function itself. In the latter case, only the function symbol is inserted, without parentheses (although they are often written haphazardly).
There are many notations for common functions used in mathematical work without further explanation. Otherwise, the function must be described somehow, and in fundamental mathematics it is not fundamentally different from and is also denoted by an arbitrary letter. The most popular letter for denoting variable functions is f, g and most Greek letters are also often used.
Predefined (reserved) designations
However, single-letter designations can, if desired, be given a different meaning. For example, the letter i is often used as an index symbol in contexts where complex numbers are not used, and the letter may be used as a variable in some combinatorics. Also, set theory symbols (such as " ⊂ (\displaystyle \subset )" And " ⊃ (\displaystyle \supset )") and propositional calculi (such as " ∧ (\displaystyle \wedge)" And " ∨ (\displaystyle \vee)") can be used in another sense, usually as order relations and binary operations, respectively.
Indexing
Indexing is represented graphically (usually by bottoms, sometimes by tops) and is, in a sense, a way to expand the information content of a variable. However, it is used in three slightly different (albeit overlapping) senses.
The actual numbers
It is possible to have several different variables by denoting them with the same letter, similar to using . For example: x 1 , x 2 , x 3 … (\displaystyle x_(1),\x_(2),\x_(3)\ldots ). Usually they are connected by some kind of commonality, but in general this is not necessary.
Moreover, not only numbers, but also any symbols can be used as “indices”. However, when another variable and expression are written as an index, this entry is interpreted as “a variable with a number determined by the value of the index expression.”
In tensor analysis
In linear algebra, tensor analysis, differential geometry with indices (in the form of variables) are written
“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 on any grandiose actions or 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. Each of us from school (or rather from the 1st grade of primary school) should be familiar with such simple mathematical symbols as greater 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. 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 3x 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 constant quantities 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. A little 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
A huge contribution to the creation of 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
“... 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.” 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.
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
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 
