“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 century. early XIX century.
1. Symbolism of mathematical quantifiers
Like ordinary language, the language of mathematical signs allows the exchange of established mathematical truths, but being only an auxiliary tool attached to ordinary language and cannot exist without it.
Mathematical definition:
In ordinary language:
Limit of the function F (x) at some point X0 is a constant number A such that for an arbitrary number E>0 there exists a positive d(E) such that from the condition |X - X 0 | Writing in quantifiers (in mathematical language) 2. Symbolism of mathematical signs and geometric figures. 1) Infinity is a concept used in mathematics, philosophy and science. The infinity of a concept or attribute of a certain object means that it is impossible to indicate boundaries or a quantitative measure for it. The term infinity corresponds to several different concepts, depending on the field of application, be it mathematics, physics, philosophy, theology or everyday life. In mathematics there is no single concept of infinity; it is endowed with special properties in each section. Moreover, these different "infinities" are not interchangeable. For example, set theory implies different infinities, and one may be greater than the other. Let's say the number of integers is infinitely large (it is called countable). To generalize the concept of the number of elements for infinite sets, the concept of cardinality of a set is introduced in mathematics. However, there is no one “infinite” power. For example, the power of the set of real numbers is greater than the power of integers, because one-to-one correspondence cannot be built between these sets, and integers are included in the real numbers. Thus, in this case, one cardinal number (equal to the power of the set) is “infinite” than the other. The founder of these concepts was the German mathematician Georg Cantor. In calculus, two symbols are added to the set of real numbers, plus and minus infinity, used to determine boundary values and convergence. It should be noted that in this case we are not talking about “tangible” infinity, since any statement containing this symbol can be written using only finite numbers and quantifiers. These symbols (and many others) were introduced to shorten longer expressions. Infinity is also inextricably linked with the designation of the infinitely small, for example, Aristotle said: Infinity in most cultures appeared as an abstract quantitative designation for something incomprehensibly large, applied to entities without spatial or temporal boundaries. 2) A circle is a geometric locus of points on a plane, the distance from which to a given point, called the center of the circle, does not exceed a given non-negative number, called the radius of this circle. If the radius is zero, then the circle degenerates into a point. A circle is the geometric locus of points on a plane that are equidistant from a given point, called the center, at a given non-zero distance, called its radius. 3) Square (rhombus) - is a symbol of the combination and ordering of four different elements, for example the four main elements or the four seasons. Symbol of the number 4, equality, simplicity, integrity, truth, justice, wisdom, honor. Symmetry is the idea through which a person tries to comprehend harmony and has been considered a symbol of beauty since ancient times. The so-called “figured” verses, the text of which has the outline of a rhombus, have symmetry. We - (E.Martov, 1894) 4) Rectangle. Of all geometric forms, this is the most rational, most reliable and correct figure; empirically this is explained by the fact that the rectangle has always and everywhere been the favorite shape. With its help, a person adapted space or any object for direct use in his everyday life, for example: a house, room, table, bed, etc. 5) The Pentagon is a regular pentagon in the shape of a star, a symbol of eternity, perfection, and the universe. Pentagon - an amulet of health, a sign on the doors to ward off witches, the emblem of Thoth, Mercury, Celtic Gawain, etc., a symbol of the five wounds of Jesus Christ, prosperity, good luck among the Jews, the legendary key of Solomon; a sign of high status in Japanese society. 6) Regular hexagon, hexagon - a symbol of abundance, beauty, harmony, freedom, marriage, a symbol of the number 6, an image of a person (two arms, two legs, a head and a torso). 7) The cross is a symbol of the highest sacred values. The cross models the spiritual aspect, the ascension of the spirit, the aspiration to God, to eternity. The cross is a universal symbol of the unity of life and death. 8) A triangle is a geometric figure that consists of three points that do not lie on the same line, and three segments connecting these three points. 9) Six-pointed Star (Star of David) - consists of two equilateral triangles superimposed on one another. One version of the origin of the sign connects its shape with the shape of the White Lily flower, which has six petals. The flower was traditionally placed under the temple lamp, in such a way that the priest lit a fire, as it were, in the center of the Magen David. In Kabbalah, two triangles symbolize the inherent duality of man: good versus evil, spiritual versus physical, and so on. The upward-pointing triangle symbolizes our good deeds, which rise to heaven and cause a stream of grace to descend back to this world (which is symbolized by the downward-pointing triangle). Sometimes the Star of David is called the Star of the Creator and each of its six ends is associated with one of the days of the week, and the center with Saturday. 10) Five-pointed Star - The main distinctive emblem of the Bolsheviks is the red five-pointed star, officially installed in the spring of 1918. Initially, Bolshevik propaganda called it the “Star of Mars” (supposedly belonging to the ancient god of war - Mars), and then began to declare that “The five rays of the star mean the union of the working people of all five continents in the fight against capitalism.” In reality, the five-pointed star has nothing to do with either the militant deity Mars or the international proletariat, it is an ancient occult sign (apparently of Middle Eastern origin) called the “pentagram” or “Star of Solomon”. Let us note that the pentagram was often placed by the Bolsheviks on Red Army uniforms, military equipment, various signs and all kinds of attributes of visual propaganda in a purely satanic way: with two “horns” up. 3. Masonic signs Masons Motto:"Freedom. Equality. Brotherhood". A social movement of free people who, on the basis of free choice, make it possible to become better, to become closer to God, and therefore, they are recognized as improving the world. Signs The radiant eye (delta) is an ancient, religious sign. He says that God oversees his creations. With the image of this sign, Freemasons asked God for blessings for any grandiose actions or for their labors. The Radiant Eye is located on the pediment of the Kazan Cathedral in St. Petersburg. The combination of a compass and a square in a Masonic sign. For the uninitiated, this is a tool of labor (mason), and for the initiated, these are ways of understanding the world and the relationship between divine wisdom and human reason. For divine wisdom nothing is impossible, it can take on both a human form (-) and a divine form (0), it can contain everything. Thus, the human mind comprehends divine wisdom and embraces it. In philosophy, this statement is a postulate about absolute and relative truth. Hexagonal Star (Bethlehem) The letter G is the designation of God (German - Got), the great geometer of the Universe. Conclusion Mathematical symbols serve primarily to accurately record mathematical concepts and sentences. Their totality constitutes what is called mathematical language. Each of us from school (or rather from the 1st grade of primary school) should be familiar with such simple mathematical symbols as more sign And less than sign, and also the equal sign. However, if it is quite difficult to confuse something with the latter, then about How and in which direction are greater and less than signs written? (less sign And over sign, as they are sometimes called) many immediately after the same school bench forget, because they are rarely used by us in everyday life. But almost everyone, sooner or later, still has to encounter them, and they can only “remember” in which direction the character they need is written by turning to their favorite search engine for help. So why not answer this question in detail, at the same time telling visitors to our site how to remember the correct spelling of these signs for the future? It is precisely how to correctly write the greater-than and less-than sign that we want to remind you in this short note. It would also not be amiss to tell you that how to type greater than or equal signs on the keyboard And less or equal, because This question also quite often causes difficulties for users who encounter such a task very rarely. Let's get straight to the point. If you are not very interested in remembering all this for the future and it’s easier to “Google” again next time, but now you just need an answer to the question “in which direction to write the sign,” then we have prepared a short answer for you - the signs for more and less are written like this: as shown in the image below. Now let’s tell you a little more about how to understand and remember this for the future. In general, the logic of understanding is very simple - whichever side (larger or smaller) the sign in the direction of writing faces to the left is the sign. Accordingly, the sign looks more to the left with its wide side - the larger one. An example of using the greater than sign: As you can see, everything is quite logical and simple, so now you should not have questions about which direction to write the greater sign and the less sign in the future. If you already remember how to write the sign you need, then it will not be difficult for you to add one line from below, this way you will get the sign "less or equal" or sign "more or equal". However, regarding these signs, some people have another question - how to type such an icon on a computer keyboard? As a result, most simply put two signs in a row, for example, “greater than or equal” denoting as ">="
, which, in principle, is often quite acceptable, but can be done more beautifully and correctly. In fact, in order to type these characters, there are special characters that can be entered on any keyboard. Agree, signs "≤"
And "≥"
look much better. In order to write “greater than or equal to” on the keyboard with one sign, you don’t even need to go into the table of special characters - just write the greater than sign while holding down the key "alt". Thus, the key combination (entered in the English layout) will be as follows. Or you can just copy the icon from this article if you only need to use it once. Here it is, please. ≥
As you probably already guessed, you can write “less than or equal to” on the keyboard by analogy with the greater than sign - just write the less than sign while holding down the key "alt". The keyboard shortcut you need to enter in the English keyboard will be as follows. Or just copy it from this page if that makes it easier for you, here it is. ≤
As you can see, the rule for writing greater than and less than signs is quite simple to remember, and in order to type the greater than or equal to and less than or equal to symbols on the keyboard, you just need to press an additional key - it’s simple. The development of mathematical symbolism was closely related to general development 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 alphabetic representation and calculus appeared in the late Hellenistic era as a result of the liberation of algebra from geometric shape. Diophantus
(probably 3rd century) recorded unknown ( X) and its degree with the following signs: [ - from the Greek term dunamiV (dynamis - force), denoting the square of the unknown, - from the Greek cuboV (k_ybos) - cube]. To the right of the unknown or its powers, Diophantus wrote coefficients, for example 3 x 5 was depicted (where = 3). When adding, Diophantus attributed the terms to each other, and used a special sign for subtraction; Diophantus denoted equality with the letter i [from the Greek isoV (isos) - equal]. For example, the equation (x 3 + 8x) - (5x 2 + 1) =X Diophantus would have written it like this:
(Here means that the unit does not have a multiplier in the form of a power of the unknown). Several centuries later, the Indians introduced various Mathematical signs for several unknowns (abbreviations for the names of colors denoting unknowns), a square, square root, the number to be subtracted. 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 - tavat - unknown, va - from varga - square number, ru - from rupa - rupee coin - free term, the dot over the number means the number being 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 appear spontaneously 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 modern look accepts 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. Somewhat earlier J. Wallis
(1655) proposed the infinity sign ¥. The creator of modern symbolism of differential and integral calculus is G. Leibniz.
In particular, he owns the currently used Mathematical signs differentials dx,d 2 x,d 3 x
and integral
Enormous credit for creating the symbolism of modern mathematics belongs to L. Euler.
He introduced (1734) into general use the first sign of a variable operation, namely the sign of the function f(x)
(from Latin functio). After Euler's work, the signs for many individual functions, such as trigonometric functions, became standard. Euler is the author of the notation for the constants e(base of natural logarithms, 1736), p [probably from Greek perijereia (periphereia) - circle, periphery, 1736], imaginary unit (from the French imaginaire - imaginary, 1777, published 1794). In the 19th century the role of symbolism is increasing. At this time, the signs of the absolute value |x| appear. (TO. Weierstrass,
1841), vector (O. Cauchy,
1853), determinant (A. Cayley,
1841), etc. Many theories that arose in the 19th century, for example tensor calculus, could not be developed without suitable symbolism. Along with the specified standardization process Mathematical signs in modern literature one can often find Mathematical signs, used by individual authors only within the scope of this study. From the point of view of mathematical logic, among Mathematical signs The following main groups can be outlined: A) signs of objects, B) signs of operations, C) signs of relations. For example, the signs 1, 2, 3, 4 represent numbers, that is, objects studied by arithmetic. The addition sign + by itself does not represent any object; it receives subject content when it is indicated which numbers add up: the notation 1 + 3 represents the number 4. The sign > (greater than) is a sign of the relationship between numbers. The relation sign receives a completely definite content when it is indicated between which objects the relation is considered. To the listed three main groups Mathematical signs adjacent to the fourth: D) auxiliary signs that establish the order of combination of the main signs. A sufficient idea of such signs is given by brackets indicating the order of actions. The signs of each of the three groups A), B) and C) are of two kinds: 1) individual signs of well-defined objects, operations and relationships, 2) common signs“non-variable” or “unknown” objects, operations and relationships. Examples of signs of the first kind can serve (see also table): A 1) Designation natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9; transcendental numbers e and p; imaginary unit i.
B 1) Signs 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 39,765 times As you know, mathematics loves precision and brevity - it’s not without reason that a single formula can, in verbal form, take up a paragraph, and sometimes even a whole page of text. Thus, graphical elements used throughout the world in science are designed to increase the speed of writing and the compactness of data presentation. In addition, standardized graphic images can be recognized by a native speaker of any language who has basic knowledge in the relevant field. The history of mathematical signs and symbols goes back many centuries - some of them were invented randomly and were intended to indicate other phenomena; others became the product of the activities of scientists who purposefully form an artificial language and are guided exclusively by practical considerations. The history of the origin of symbols denoting the simplest arithmetic operations is not known for certain. However, there is a fairly plausible hypothesis for the origin of the plus sign, which looks like crossed horizontal and vertical lines. In accordance with it, the addition symbol originates in the Latin union et, which is translated into Russian as “and”. Gradually, in order to speed up the writing process, the word was shortened to a vertically oriented cross, resembling the letter t. The earliest reliable example of such a reduction dates back to the 14th century. The generally accepted minus sign appeared, apparently, later. In the 14th and even 15th centuries, a number of symbols were used in scientific literature to denote the operation of subtraction, and only by the 16th century did “plus” and “minus” in their modern form begin to appear together in mathematical works. Oddly enough, the mathematical signs and symbols for these two arithmetic operations are not completely standardized today. A popular symbol for multiplication is the diagonal cross proposed by the mathematician Oughtred in the 17th century, which can be seen, for example, on calculators. In mathematics lessons at school, the same operation is usually represented as a point - this method was proposed by Leibniz in the same century. Another representation method is an asterisk, which is most often used in computer representation of various calculations. It was proposed to use it in the same 17th century by Johann Rahn. For the division operation, a slash sign (proposed by Oughtred) and a horizontal line with dots above and below are provided (the symbol was introduced by Johann Rahn). The first designation option is more popular, but the second is also quite common. Mathematical signs and symbols and their meanings sometimes change over time. However, all three methods of graphically representing multiplication, as well as both methods for division, are to one degree or another valid and relevant today. As with many other mathematical signs and symbols, the designation of equality was originally verbal. For quite a long time, the generally accepted designation was the abbreviation ae from the Latin aequalis (“equal”). However, in the 16th century, a Welsh mathematician named Robert Record proposed two horizontal lines located one below the other as a symbol. As the scientist argued, it is impossible to think of anything more equal to each other than two parallel segments. Despite the fact that a similar sign was used to indicate the parallelism of lines, the new equality symbol gradually became widespread. By the way, such signs as “more” and “less”, depicting ticks turned in different directions, appeared only in the 17th-18th centuries. Today they seem intuitive to any schoolchild. Slightly more complex signs of equivalence (two wavy lines) and identity (three horizontal parallel lines) came into use only in the second half of the 19th century. The history of the emergence of mathematical signs and symbols also contains very interesting cases of rethinking graphics as science develops. The sign for the unknown, called today “X,” originates in the Middle East at the dawn of the last millennium. Back in the 10th century in the Arab world, famous at that historical period for its scientists, the concept of the unknown was denoted by a word literally translated as “something” and beginning with the sound “Ш”. In order to save materials and time, the word in treatises began to be shortened to the first letter. Many decades later, the written works of Arab scientists ended up in the cities of the Iberian Peninsula, in the territory of modern Spain. Scientific treatises began to be translated into the national language, but a difficulty arose - in Spanish there is no phoneme “Ш”. Borrowed Arabic words starting with it were written according to a special rule and were preceded by the letter X. The scientific language of that time was Latin, in which the corresponding sign is called “X”. Thus, the sign, which at first glance is just a randomly chosen symbol, has a deep history and was originally an abbreviation of the Arabic word for “something.” Unlike “X,” Y and Z, familiar to us from school, as well as a, b, c, have a much more prosaic origin story. In the 17th century, Descartes published a book called Geometry. In this book, the author proposed standardizing symbols in equations: in accordance with his idea, the last three letters of the Latin alphabet (starting from “X”) began to denote unknown values, and the first three - known values. The history of such a word as “sine” is truly unusual. The corresponding trigonometric functions were originally named in India. The word corresponding to the concept of sine literally meant “string”. During the heyday of Arabic science, Indian treatises were translated, and the concept, which had no analogue in the Arabic language, was transcribed. By coincidence, what came out in the letter resembled the real-life word “hollow”, the semantics of which had nothing to do with the original term. As a result, when Arabic texts were translated into Latin in the 12th century, the word "sine" emerged, meaning "hollow" and established as a new mathematical concept. But the mathematical signs and symbols for tangent and cotangent have not yet been standardized - in some countries they are usually written as tg, and in others - as tan. As can be seen from the examples described above, the emergence of mathematical signs and symbols largely occurred in the 16th-17th centuries. The same period saw the emergence of today's familiar forms of recording such concepts as percentage, square root, degree. Percentage, i.e. one hundredth, has long been designated as cto (short for Latin cento). It is believed that the sign that is generally accepted today appeared as a result of a typo about four hundred years ago. The resulting image was perceived as a successful way to shorten it and caught on. The root sign was originally a stylized letter R (short for the Latin word radix, “root”). The upper bar, under which the expression is written today, served as parentheses and was a separate symbol, separate from the root. Parentheses were invented later - they came into widespread use thanks to the work of Leibniz (1646-1716). Thanks to his work, the integral symbol was introduced into science, looking like an elongated letter S - short for the word “sum”. Finally, the sign for the operation of exponentiation was invented by Descartes and modified by Newton in the second half of the 17th century. Considering that the familiar graphic images of “plus” and “minus” were introduced into circulation only a few centuries ago, it does not seem surprising that mathematical signs and symbols denoting complex phenomena began to be used only in the century before last. Thus, the factorial, which looks like an exclamation mark after a number or variable, appeared only at the beginning of the 19th century. Around the same time, the capital “P” to denote work and the limit symbol appeared. It is somewhat strange that the signs for Pi and the algebraic sum appeared only in the 18th century - later than, for example, the integral symbol, although intuitively it seems that they are more commonly used. The graphical representation of the ratio of circumference to diameter comes from the first letter of the Greek words meaning "circumference" and "perimeter". And the “sigma” sign for an algebraic sum was proposed by Euler in the last quarter of the 18th century. As you know, the language of science in Europe for many centuries was Latin. Physical, medical and many other terms were often borrowed in the form of transcriptions, much less often - in the form of tracing paper. Thus, many mathematical signs and symbols in English are called almost the same as in Russian, French or German. The more complex the essence of a phenomenon, the higher the likelihood that it will have the same name in different languages. The simplest mathematical signs and symbols in Word are indicated by the usual key combination Shift+number from 0 to 9 in the Russian or English layout. Separate keys are reserved for some commonly used signs: plus, minus, equal, slash. If you want to use graphic images of an integral, an algebraic sum or product, Pi, etc., you need to open the “Insert” tab in Word and find one of two buttons: “Formula” or “Symbol”. In the first case, a constructor will open, allowing you to build an entire formula within one field, and in the second, a table of symbols will open, where you can find any mathematical symbols. Unlike chemistry and physics, where the number of symbols to remember can exceed a hundred units, mathematics operates with a relatively small number of symbols. We learn the simplest of them in early childhood, learning to add and subtract, and only at the university in certain specialties do we become familiar with a few complex mathematical signs and symbols. Pictures for children help in a matter of weeks to achieve instant recognition of the graphic image of the required operation; much more time may be needed to master the skill of performing these operations and understanding their essence. Thus, the process of memorizing signs occurs automatically and does not require much effort. The value of mathematical signs and symbols lies in the fact that they are easily understood by people who speak different languages and are native speakers of different cultures. For this reason, it is extremely useful to understand and be able to reproduce graphical representations of various phenomena and operations. The high level of standardization of these signs determines their use in a wide variety of fields: in the field of finance, information technology, engineering, etc. For anyone who wants to do business related to numbers and calculations, knowledge of mathematical signs and symbols and their meanings becomes a vital necessity .
“... 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 deity, 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
1853Signs individual transactions
+
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 
Plus and minus
Multiplication and division
Equality, identity, equivalence
Sign of the unknown - “X”
Designation of other unknowns
Trigonometric terms
Some other signs
Later designations
Names of symbols in different languages
Computer notation of mathematical symbols
How to Remember Math Symbols
Finally