“Symbols are not only a record of thoughts,
means of its image and fixation, -
no, they affect the very thought,
they... guide her, and that's enough
move them on paper... in order to
unmistakably reach new truths.
L. Carnot
Mathematical signs serve primarily for accurate (uniquely defined) recording of mathematical concepts and sentences. Their totality in the real conditions of their application by mathematicians constitutes what is called the mathematical language.
Mathematical signs allow you to write in a compact form sentences that are cumbersomely expressed 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 it.
But mathematicians cannot always say right away what this or that symbol that they have 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 were discovered only at the end of the 18th and at the beginning of the 19th 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 regular language:
function limit F (x) at some point X0 is called a constant number A, such that for an arbitrary number E>0 there is a positive d(E) such that from the condition |X - X 0 | Notation in quantifiers (in mathematical language) 2. Symbolism of mathematical signs and geometric figures. 1) Infinity is a concept used in mathematics, philosophy and the natural sciences. The infinity of some concept or attribute of some object means the impossibility of specifying boundaries or a quantitative measure for it. The term infinity corresponds to several different concepts, depending on the field of application, whether it be 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 various "infinities" are not interchangeable. For example, set theory implies different infinities, and one can be greater than the other. 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. In this case, there is no one "infinite" power. For example, the cardinality of the set of real numbers is greater than the cardinality of integers, because a 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 cardinality of the set) is "infinite" than the other. The founder of these concepts was the German mathematician Georg Cantor. In mathematical analysis, two symbols, plus and minus infinity, are added to the set of real numbers, which are 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 (as well as many others) were introduced to shorten the notation of longer expressions. Infinity is also inextricably linked with the designation of the infinitely small, for example, even 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) Circle - the locus of points in the 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 a locus of points in 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, directness, truth, justice, wisdom, honor. Symmetry is the idea through which a person tries to comprehend harmony and has long been considered a symbol of beauty. Symmetry is possessed by the so-called “curly” verses, the text of which has the shape of a rhombus. We - (E. Martov, 1894) 4) Rectangle. Of all geometric forms, this is the most rational, most reliable and regular figure; empirically this is explained by the fact that always and everywhere the rectangle was the favorite shape. With the help of it, a person adapted a space or any object for direct use in his life, for example: a house, a room, a table, a bed, etc. 5) The Pentagon is a regular pentagon in the form of a star, a symbol of eternity, perfection, the universe. Pentagon - an amulet of health, a sign on the door to drive away 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 position in society among the Japanese. 6) Regular hexagon, hexagon - a symbol of abundance, beauty, harmony, freedom, marriage, a symbol of the number 6, the image of a person (two arms, two legs, head and torso). 7) The cross is a symbol of the highest sacred values. The cross models the spiritual aspect, the ascent 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 straight 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 of the versions of the origin of the sign associates 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 the fire, as it were, in the center of Magen David. In Kabbalah, the two triangles symbolize the duality inherent in man: good versus evil, spiritual versus physical, and so on. The upward pointing triangle symbolizes our good deeds, which ascend to heaven and cause a stream of grace to descend back into this world (which symbolizes 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 distinguishing emblem of the Bolsheviks is the red five-pointed star, officially installed in the spring of 1918. Initially, Bolshevik propaganda called it the “Mars Star” (allegedly belonging to the ancient god of war - Mars), and then began to declare that “The five rays of the star means the union of the workers of all five continents in the struggle 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 (obviously of Middle Eastern origin) called the “pentagram” or “Star of Solomon”. It should be noted that the pentagram was often placed by the Bolsheviks on Red Army uniforms, in military equipment, various signs and all sorts of attributes of visual propaganda in a purely satanic way: with two “horns” up. 3. Masonic signs Masons Motto:"Freedom. Equality. Brotherhood". The social movement of free people who, on the basis of free choice, allow them to become better, to become closer to God, therefore, they are recognized to improve 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, the Masons asked God for blessings for any grandiose actions, for their labors. The Radiant Eye is located on the pediment of the Kazan Cathedral in St. Petersburg. The combination of compass and square in the Masonic sign. For the uninitiated, this is a tool of labor (a bricklayer), and for the initiated, these are ways of knowing the world and the relationship between divine wisdom and human reason. For divine wisdom, there is nothing impossible, it can take on both the human form (-) and the divine form (0), it can accommodate everything. Thus, the human mind comprehends the divine wisdom, 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 signs serve primarily to accurately record mathematical concepts and sentences. Their totality constitutes what is called the mathematical language. The course uses geometric language, made up of notations and symbols adopted in the course of mathematics (in particular, in the new geometry course in high school). The whole variety of designations and symbols, as well as the connections between them, can be divided into two groups: group I - designations of geometric figures and relations between them; group II designations of logical operations, constituting the syntactic basis of the geometric language. The following is a complete list of math symbols used in this course. Particular attention is paid to the symbols that are used to designate the projections of geometric shapes. Group I SYMBOLS DESIGNATED GEOMETRIC FIGURES AND RELATIONSHIPS BETWEEN THEM A. Designation of geometric shapes 1. The geometric figure is denoted - F. 2. Points are indicated by capital letters of the Latin alphabet or Arabic numerals: A, B, C, D, ... , L, M, N, ... 1,2,3,4,...,12,13,14,... 3. Lines arbitrarily located in relation to the projection planes are indicated by lowercase letters of the Latin alphabet: a, b, c, d, ... , l, m, n, ... Level lines are indicated: h - horizontal; f- frontal. The following notation is also used for straight lines: (AB) - a straight line passing through the points A and B; [AB) - a ray with the beginning at point A; [AB] - a straight line segment bounded by points A and B. 4. Surfaces are denoted by lowercase letters of the Greek alphabet: α, β, γ, δ,...,ζ,η,ν,... To emphasize the way the surface is defined, you should specify the geometric elements by which it is defined, for example: α(a || b) - plane α is determined by parallel lines a and b; β(d 1 d 2 gα) - the surface β is determined by the guides d 1 and d 2 , the generatrix g and the plane of parallelism α. 5. Angles are indicated: ∠ABC - angle with apex at point B, as well as ∠α°, ∠β°, ... , ∠φ°, ... 6. Angular: the value (degree measure) is indicated by the sign, which is placed above the angle: The value of the angle ABC; The value of the angle φ. A right angle is marked with a square with a dot inside 7. Distances between geometric figures are indicated by two vertical segments - ||. For example: |AB| - distance between points A and B (length of segment AB); |Aa| - distance from point A to line a; |Aα| - distances from point A to surface α; |ab| - distance between lines a and b; |αβ| distance between surfaces α and β. 8. For projection planes, the following designations are accepted: π 1 and π 2, where π 1 is the horizontal projection plane; π 2 -fryuntal plane of projections. When replacing projection planes or introducing new planes, the latter denote π 3, π 4, etc. 9. Projection axes are denoted: x, y, z, where x is the x-axis; y is the y-axis; z - applicate axis. The constant line of the Monge diagram is denoted by k. 10. Projections of points, lines, surfaces, any geometric figure are indicated by the same letters (or numbers) as the original, with the addition of a superscript corresponding to the projection plane on which they were obtained: A", B", C", D", ... , L", M", N", horizontal projections of points; A", B", C", D", ... , L", M" , N", ... frontal projections of points; a" , b" , c" , d" , ... , l", m" , n" , - horizontal projections of lines; a" ,b" , c" , d" , ... , l" , m " , n" , ... frontal projections of lines; α", β", γ", δ",...,ζ",η",ν",... horizontal projections of surfaces; α", β", γ", δ",...,ζ" ,η",ν",... frontal projections of surfaces. 11. Traces of planes (surfaces) are indicated by the same letters as the horizontal or frontal, with the addition of a subscript 0α, emphasizing that these lines lie in the projection plane and belong to the plane (surface) α. So: h 0α - horizontal trace of the plane (surface) α; f 0α - frontal trace of the plane (surface) α. 12. Traces of straight lines (lines) are indicated by capital letters, which begin words that define the name (in Latin transcription) of the projection plane that the line crosses, with a subscript indicating belonging to the line. For example: H a - horizontal trace of a straight line (line) a; F a - frontal trace of a straight line (line) a. 13. The sequence of points, lines (of any figure) is marked with subscripts 1,2,3,..., n: A 1, A 2, A 3,..., A n; a 1 , a 2 , a 3 ,...,a n ; α 1 , α 2 , α 3 ,...,α n ; F 1 , F 2 , F 3 ,..., F n etc. The auxiliary projection of the point, obtained as a result of the transformation to obtain the actual value of the geometric figure, is denoted by the same letter with the subscript 0: A 0 , B 0 , C 0 , D 0 , ... Axonometric projections 14. Axonometric projections of points, lines, surfaces are indicated by the same letters as nature with the addition of the superscript 0: A 0, B 0, C 0, D 0, ... 1 0 , 2 0 , 3 0 , 4 0 , ... a 0 , b 0 , c 0 , d 0 , ... α 0 , β 0 , γ 0 , δ 0 , ... 15. Secondary projections are indicated by adding a superscript 1: A 1 0 , B 1 0 , C 1 0 , D 1 0 , ... 1 1 0 , 2 1 0 , 3 1 0 , 4 1 0 , ... a 1 0 , b 1 0 , c 1 0 , d 1 0 , ... α 1 0 , β 1 0 , γ 1 0 , δ 1 0 , ... To facilitate reading the drawings in the textbook, several colors were used in the design of the illustrative material, each of which has a certain semantic meaning: black lines (dots) indicate the initial data; green color is used for lines of auxiliary graphic constructions; red lines (dots) show the results of constructions or those geometric elements to which special attention should be paid. Mathematical notation("language of mathematics") - a complex graphical notation that serves to present abstract mathematical ideas and judgments in a human-readable form. It makes up (in its complexity and diversity) a significant proportion of non-speech sign systems used by mankind. This article describes the generally accepted international notation, although different cultures of the past had their own, and some of them even have limited use until now. Note that mathematical notation, as a rule, is used in conjunction with the written form of some of the natural languages. In addition to fundamental and applied mathematics, mathematical notation is widely used in physics, as well as (in its incomplete scope) in engineering, computer science, economics, and indeed in all areas of human activity where mathematical models are used. Differences between the proper mathematical and applied style of notation will be discussed in the course of the text. 1
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5 ✪ Sign / in math ✪ Mathematics Grade 3. Table of digits of multi-digit numbers ✪ Sets in math ✪ Mathematics 19. Math fun - Shishkin school 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 a solution to cubic equations in a general form took mathematicians several centuries? This is partly why? Because there were no clear symbols for clear thoughts, whether it's our time. There are so many characters that you can get confused. But you can't fool us, let's figure it out. This is an inverted capital 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, everyone, and so on. Such a hieroglyph will be called a universal quantifier. And here is another quantifier, but already existence. The English letter e was reflected in Paint from left to right, thus hinting at the overseas verb "exist", in our opinion we will read: exists, there is, there is another similar way. An exclamation mark would add uniqueness to such an existential quantifier. If this is clear, we move on. You probably came across indefinite integrals in the eleventh class, so I would like to remind you that this is not just some kind of antiderivative, but the collection of all 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: the search for the area of \u200b\u200bthe figure under the graph by summing infinitesimal values. To me, this is the most romantic activity in calculus. But school geometry is most useful because it teaches logical rigor. By the first course, you should have a clear understanding of what a consequence is, what an equivalence is. Well, you can't get confused between necessity and sufficiency, you understand? Let's even try to dig a little deeper. If you decide to take up higher mathematics, then I imagine how bad things are with your personal life, but that is why you will surely agree to overcome a small exercise. There are three points here, each has a left and right side, which you need to connect with one of the three drawn symbols. Please pause, try it out for yourself, and then listen to what I have to say. If x=-2, then |x|=2, but from left to right, so the phrase is already built. In the second paragraph, absolutely the same thing is written on the left and right sides. And the third point can be commented as follows: every rectangle is a parallelogram, but not every parallelogram is a rectangle. Yes, I know that you are no longer small, but still my applause to those who have coped with this exercise. Well, okay, enough, let's remember the number sets. Natural numbers are used in counting: 1, 2, 3, 4 and so on. In nature, -1 apple does not exist, but, by the way, integers allow you 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 approaches you and says: "Keep it real!", you can be sure that you are a mathematician, an admirer of real numbers. Well, you should read something about complex numbers, it will be more useful. We will now roll back, 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, up to the last letter omega. You can be sure that the Greeks also have capital letters, but we will not talk about sad things now. We are better about cheerful - about limits. But here there are just no riddles, 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 sound out the definition of the limit of the number sequence, which is now written in front of you. Click rather pause and think, and may you have the happiness of a one-year-old child who has learned the word "mother." If for any epsilon greater than zero there is a natural number 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] The system evolved like natural languages, historically (see the history of mathematical notation), and is organized like the writing of natural languages, borrowing many symbols from there as well (primarily from the Latin and Greek alphabets). Symbols, as well 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 the 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 semantic role in mathematical notation. Ordinary mathematical notation (in particular, the so-called mathematical formulas) are written in general in a string from left to right, but do not necessarily constitute a consecutive string of characters. Separate blocks of characters can be located in the upper or lower half of the line, even in the case when the characters do not overlap vertically. Also, some parts are located entirely above or below the line. On the grammatical side, almost any "formula" can be considered a hierarchically organized tree-type structure. Mathematical notation represents a system in terms of the relationship of its components, but, in general, not constitute a formal system (in the understanding of mathematics itself). They, in any complicated case, cannot even be disassembled programmatically. Like any natural language, the “language of mathematics” is full of inconsistent designations, homographs, different (among its speakers) interpretations of what is considered correct, etc. There is not even any foreseeable alphabet of mathematical symbols, and in particular because the question is not always unambiguously resolved whether to consider two designations as different characters or as different spellings of one character. Some of the mathematical notation (mainly related to measurements) is standardized in ISO 31 -11, but in general, there is rather no standardization of notation. If necessary, apply a number system with a base less than ten, the base is written in a subscript: 20003 8 . Number systems with bases greater than ten are not used in the 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 indicated by the first six Latin letters from A to F. Several different approaches are used to designate such numbers in computer science, but they are not transferred to mathematics. Parentheses "()" are used: Square brackets "" are often used in grouping meanings when you have to use many pairs of brackets. In this case, they are placed on the outside and (with neat typography) have a greater height than the brackets that are inside. Square "" and round "()" brackets are used to denote closed and open spaces, respectively. Curly braces "()" are usually used for , although the same caveat applies to them as for square brackets. Left "(" and right ")" brackets can be used separately; their purpose is described. Angle bracket symbols " ⟨ ⟩ (\displaystyle \langle \;\rangle )» with neat typography 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 manually writing formulas) and one has to distinguish between them with the help of intuition. Pairs of symmetric (with respect to the vertical axis) symbols, including those other than those listed, are often used to highlight a piece of a formula. The purpose of paired brackets is described. Depending on the location, superscripts and subscripts are distinguished. The superscript can mean (but does not necessarily mean) exponentiation to , about other uses of . 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 diverted from the physical meaning, and then the variable turns into abstract(or numeric) variable, denoted by some symbol not occupied by the special notation mentioned above. Variable X is considered given if the set of values it takes is specified (x). It is convenient to consider a constant value as a variable for which the corresponding set (x) consists of one element. Mathematically, there is no significant difference between operator(unary), mapping and function. However, it is understood that if to record the value of the mapping from the given arguments, it is necessary to specify , then the symbol of this mapping denotes a function, in other cases it is more likely to speak of an operator. Symbols of some functions of one argument are used with and without brackets. Many elementary functions, for example sin x (\displaystyle \sin x) or sin (x) (\displaystyle \sin(x)), but elementary functions are always called functions. A function can be referred to in two senses: as an expression of its value with given arguments (written f (x) , f (x , y) (\displaystyle f(x),\ f(x,y)) etc.) or actually as a function. In the latter case, only the function symbol is put, without brackets (although they often write it randomly). 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 does not fundamentally differ from and is also denoted by an arbitrary letter in the same way. The letter f is the most popular for variable functions, g and most Greek are also often used. However, single-letter designations can, if desired, be given a different meaning. For example, the letter i is often used as an index in a context where complex numbers are not used, and the letter can be used as a variable in some combinatorics. Also, set theory symbols (such as " ⊂ (\displaystyle \subset )" and " ⊃ (\displaystyle \supset )”) and propositional calculus (such as “ ∧ (\displaystyle \wedge )" and " ∨ (\displaystyle\vee )”) can be used in another sense, usually as an order relation and a binary operation, respectively. Indexing is plotted (usually bottom, sometimes top) and is, in a sense, a way to expand the content of a variable. However, it is used in three slightly different (though overlapping) senses. You can have multiple 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 commonality, but in general this is not necessary. Moreover, as "indexes" you can use not only numbers, but also any characters. However, when another variable and expression is written as an index, this entry is interpreted as "a variable with a number determined by the value of the index expression." In linear algebra, tensor analysis, differential geometry with indices (in the form of variables) are written Abstract algebra makes extensive use of symbols to simplify and shorten text, as well as standard notation for some groups. The following is a list of the most common algebraic notation, the corresponding commands in ... Wikipedia Mathematical notations are symbols used to write mathematical equations and formulas in a compact way. In addition to numbers and letters of various alphabets (Latin, including Gothic, Greek and Hebrew), ... ... Wikipedia The article contains a list of commonly used abbreviations for mathematical functions, operators, and other mathematical terms. Contents 1 Abbreviations 1.1 Latin 1.2 Greek alphabet ... Wikipedia Unicode, or Unicode (eng. Unicode) is a character encoding standard that allows you to represent the signs of almost all written languages. The standard was proposed in 1991 by the non-profit organization Unicode Consortium (Eng. Unicode Consortium, ... ... Wikipedia A list of specific symbols used in mathematics can be seen in the article Table of mathematical symbols Mathematical notation ("language of mathematics") is a complex graphic notation system used to present abstract ... ... Wikipedia This term has other meanings, see Plus minus (meanings). ± ∓ Plus minus sign (±) is a mathematical symbol that is placed in front of some expression and means that the value of this expression can be both positive and ... Wikipedia It is necessary to check the quality of the translation and bring the article in line with the stylistic rules of Wikipedia. You can help ... Wikipedia Or mathematical symbols are signs that symbolize certain mathematical operations with their arguments. The most common ones are: Plus: + Minus:, - Multiplication sign: ×, ∙ Division sign::, ∕, ÷ Exposition sign to ... ... Wikipedia Operation signs or mathematical symbols are signs that symbolize certain mathematical operations with their arguments. The most common ones are: Plus: + Minus:, - Multiplication sign: ×, ∙ Division sign::, ∕, ÷ Construction sign ... ... Wikipedia
“... 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 real, and no matter how many divisions are given, it is always potentially possible to divide this segment into an even greater number. Note that Aristotle made a great contribution to the understanding of infinity, dividing it into potential and actual, and came close from this side to the foundations of mathematical analysis, also pointing out 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 unlimitedness and incomprehensibility. In philosophy, it is an attribute of space and time.
Modern physics comes close to the actuality 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 space-time at which mass in an infinitely small volume is concentrated with infinite density. There is already solid circumstantial evidence for the existence of black holes, although the big bang theory is still under development.
The circle is a symbol of the Sun, the Moon. One of the most common characters. It is also a symbol of infinity, eternity, perfection.
The poem is a rhombus.
In the midst of darkness.
The eye is resting.
The darkness of the night is alive.
The heart sighs eagerly
The whisper of the stars flies at times.
And azure feelings are crowded by the crowd.
Everything was forgotten in the dewy brilliance.
Fragrant kiss!
Shine fast!
Whisper again
As then:
"Yes!"
Of course, one can disagree 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, in many), while others are completely different.
Properties of a triangle as a figure: strength, immutability.
Axiom A1 of stereometry says: “Through 3 points of space that do not lie on one straight line, a plane passes, and moreover, only one!”
To check the depth of understanding of this statement, they usually set the backfill problem: “Three flies are sitting on the table, at three ends of the table. At a certain moment, they scatter in three mutually perpendicular directions with 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 3 points that define a 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 ascent, prosperity, harmony and royalty. The inverted triangle is a female and lunar symbol, personifies water, fertility, rain, divine mercy.
US state symbols also contain the Six-pointed Star in various 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, the six-pointed star is a symbol of Christ, namely the union in Christ of divine and human nature. That is why this sign is inscribed in the Orthodox Cross.
Government”, which is under the complete control of Freemasonry.
Quite often, Satanists draw a pentagram with two ends up, so that it is easy to enter 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 hates the “Iron Felix”).
The Marxist plans for a "world proletarian revolution" were clearly of Masonic origin, and a number of the most prominent Marxists were members of Freemasonry. L. Trotsky belonged to them, it was he who proposed to make the Masonic pentagram the identification emblem of Bolshevism.
International Masonic lodges secretly provided the Bolsheviks with comprehensive support, especially financial.
Freemasons are associates of the Creator, associates of social progress, against inertia, inertia and ignorance. Outstanding representatives of freemasonry - Karamzin Nikolai Mikhailovich, Suvorov Alexander Vasilyevich, Kutuzov Mikhail Illarionovich, Pushkin Alexander Sergeevich, Goebbels Joseph.
The square, as a rule, from below is a human knowledge of the world. From the point of view of Freemasonry, a person comes into the world to know the divine plan. And knowledge requires tools. The most effective science in the knowledge of the world is mathematics.
The square is the oldest mathematical tool known from time immemorial. The graduation of a square is already a big step forward in the mathematical tools of knowledge. Man cognizes the world with the help of the sciences of mathematics, 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. If you try to push it apart to fit more, you will break it.
So people who try to know the whole infinity of the divine plan either die or go crazy. "Know your limits!" - that's what this sign tells the World. Even if you are Einstein, Newton, Sakharov - the greatest minds of mankind! - understand that you are limited by the time in which you were born; in the knowledge of the world, language, brain size, a variety of human limitations, the life of your body. Therefore - yes, learn, but understand that you will never fully know!
And the circle? The compass is divine wisdom. A compass can describe a circle, and if you push its legs apart, it will be a straight line. And in symbolic systems, a circle and a straight line are two opposites. A straight line denotes a person, his beginning and end (like a dash between two dates - birth and death). The circle is a symbol of the deity, since it is a perfect figure. They oppose each other - the divine and human figures. Man is not perfect. God is perfect in everything.
People always know the truth, but always relative truth. And the absolute truth is known only to God.
Learn more and more, realizing that you will not be able to know the truth to the end - what depths we find in an ordinary compass with a square! Who would have thought!
This is the beauty and charm of Masonic symbolism, in its great 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 painted 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 the Unity and Struggle of Opposites, the fight 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 triangle up means - "A person strives for God." Triangle down - "The Deity descends to Man." In their combination, our world exists, which is the combination of the Human and the Divine. The letter G here means that God lives in our world. He is really 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, mathematical research. It is real mathematical research that helps to find out which sign system best reflects the structure of quantitative and qualitative relations, which can be an effective tool for their further use in symbols and emblems.B. Symbols Denoting Relations Between Geometric Figures
no.
Designation
Content
Symbolic notation example
1
≡
Match (AB) ≡ (CD) - a straight line passing through points A and B,
coincides with the line passing through points C and D2
≅
Congruent ∠ABC≅∠MNK - angle ABC is congruent to angle MNK
3
∼
Similar ΔABS∼ΔMNK - triangles ABC and MNK are similar
4
||
Parallel α||β - plane α is parallel to plane β
5
⊥
Perpendicular a⊥b - lines a and b are perpendicular
6
interbreed with d - lines c and d intersect
7
Tangents t l - line t is tangent to line l.
βα - plane β tangent to surface α8
→
Are displayed F 1 → F 2 - the figure F 1 is mapped onto the figure F 2
9
S projection center.
If the projection center is not a proper point,
its position is indicated by an arrow,
indicating the direction of projection -
10
s Projection direction -
11
P Parallel projection p s α Parallel projection - parallel projection
to the plane α in the direction sB. Set-theoretic notation
no.
Designation
Content
Symbolic notation example
An example of symbolic notation in geometry
1
M,N Sets -
-
2
A,B,C,... Set elements -
-
3
{ ... }
Comprises... F(A, B, C,... ) Ф(A, B, C,...) - figure Ф consists of points A, B, C, ...
4
∅
Empty set L - ∅ - the set L is empty (contains no elements) -
5
∈
Belongs to, is an element 2∈N (where N is the set of natural numbers) -
the number 2 belongs to the set NA ∈ a - point A belongs to the line a
(point A lies on line a)6
⊂
Includes, contains N⊂M - the set N is a part (subset) of the set
M of all rational numbersa⊂α - line a belongs to the plane α (understood in the sense:
the set of points of the line a is a subset of the points of the plane α)7
∪
An association C \u003d A U B - set C is a union of sets
A and B; (1, 2. 3, 4.5) = (1.2.3)∪(4.5)ABCD = ∪ [BC] ∪ - broken line, ABCD is
union of segments [AB], [BC],8
∩
Intersection of many М=К∩L - the set М is the intersection of the sets К and L
(contains elements belonging to both the set K and the set L).
M ∩ N = ∅- intersection of sets M and N is the empty set
(sets M and N do not have common elements)a = α ∩ β - line a is the intersection
planes α and β
and ∩ b = ∅ - lines a and b do not intersect
(have no common points)Group II SYMBOLS DESIGNATING LOGICAL OPERATIONS
no.
Designation
Content
Symbolic notation example
1
∧
conjunction of sentences; corresponds to the union "and".
Sentence (p∧q) is true if and only if p and q are both trueα∩β = ( K:K∈α∧K∈β) The intersection of surfaces α and β is a set of points (line),
consisting of all those and only those points K that belong to both the surface α and the surface β2
∨
Disjunction of sentences; corresponds to the union "or". Sentence (p∨q)
true when at least one of the sentences p or q is true (i.e. either p or q or both). -
3
⇒
Implication is a logical consequence. The sentence p⇒q means: "if p, then q" (a||c∧b||c)⇒a||b. If two lines are parallel to a third, then they are parallel to each other.
4
⇔
The sentence (p⇔q) is understood in the sense: "if p, then q; if q, then p" А∈α⇔А∈l⊂α.
A point belongs to a plane if it belongs to some line belonging to that plane.
The converse is also true: if a point belongs to some line,
belonging to the plane, then it also belongs to the plane itself.5
∀
The general quantifier reads: for everyone, for everyone, for anyone.
The expression ∀(x)P(x) means: "for any x: property P(x)"∀(ΔABC)( = 180°) For any (for any) triangle, the sum of the values of its angles
at the vertices is 180°6
∃
The existential quantifier reads: exists.
The expression ∃(x)P(x) means: "there is x that has the property P(x)"(∀α)(∃a). For any plane α, there exists a line a not belonging to the plane α
and parallel to the plane α7
∃1
The uniqueness of existence quantifier, reads: there is a unique
(-th, -th)... The expression ∃1(x)(Px) means: "there is a unique (only one) x,
having the property Rx"(∀ A, B)(A≠B)(∃1a)(a∋A, B) For any two different points A and B, there is a unique line a,
passing through these points.8
(px) Negation of the statement P(x) ab(∃α )(α⊃а, b). If lines a and b intersect, then there is no plane a that contains them
9
\
Negative sign ≠ - the segment [AB] is not equal to the segment .a? b - the line a is not parallel to the line b
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