Set symbol designation. Basic mathematical signs and symbols

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 system designations, 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 has wide application 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.

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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! You are aware that the search for a solution to cubic equations in general view 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. AT English 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]

General information

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.

Structure

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.

Standardization

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.

Elements of mathematical notation

Numbers

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.

Superscript and subscript characters

Parentheses, similar symbols, and delimiters

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.

Indices

Depending on the location, superscripts and subscripts are distinguished. The superscript can mean (but does not necessarily mean) exponentiation to , about other uses of .

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 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.

Functions and Operators

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.

Operators and Relations (Unary and Binary)

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.

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 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

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.

Actually numbers

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 tensor analysis

In linear algebra, tensor analysis, differential geometry with indices (in the form of variables) are written

“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:
“... 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:

• time,
• division of quantities,
• the inexhaustibility of the creative nature,
• the very concept of the boundary, pushing beyond it,
• thinking that is unstoppable.

Infinity in most cultures appeared as an abstract quantitative designation for something incomprehensibly large, applied to entities without spatial or temporal boundaries.
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.

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.
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.

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.
The poem is a rhombus.

We -
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!"

(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.
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.

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.
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.

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.
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.

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”.
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”).

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.
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.

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.
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.

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.
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.

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.
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).

Hexagonal Star (Bethlehem)

The letter G is the designation of God (German - Got), the great geometer of the Universe.
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.

Conclusion

Mathematical signs serve primarily to accurately record mathematical concepts and sentences. Their totality constitutes what is called the mathematical language.
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.

Balagin Viktor

With the discovery of mathematical rules and theorems, scientists came up with new mathematical notation, signs. Mathematical signs are symbols designed to record mathematical concepts, sentences and calculations. In mathematics, special symbols are used to shorten the record and express the statement more accurately. In addition to the numbers and letters of various alphabets (Latin, Greek, Hebrew), the mathematical language uses many special symbols invented over the past few centuries.

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MATHEMATICAL SYMBOLS.

I've done the work

7th grade student

GBOU secondary school No. 574

Balagin Viktor

2012-2013 academic year

MATHEMATICAL SYMBOLS.

1. Introduction

The word mathematics came to us from ancient Greek, where μάθημα meant "to learn", "acquire knowledge". And the one who says: "I don't need mathematics, I'm not going to become a mathematician" is wrong. Everyone needs math. Revealing the amazing world of the numbers around us, it teaches us to think more clearly and consistently, develops thought, attention, educates perseverance and will. M.V. Lomonosov said: "Mathematics puts the mind in order." In a word, mathematics teaches us to learn how to acquire knowledge.

Mathematics is the first science that man could master. The oldest activity was counting. Some primitive tribes counted the number of objects using their fingers and toes. The rock drawing, which has survived to our times from the Stone Age, depicts the number 35 in the form of 35 sticks drawn in a row. We can say that 1 stick is the first mathematical symbol.

The mathematical "writing" that we now use - from the notation of unknown letters x, y, z to the integral sign - developed gradually. The development of symbolism simplified the work with mathematical operations and contributed to the development of mathematics itself.

From the ancient Greek "symbol" (Greek. symbolon - a sign, a sign, a password, an emblem) - a sign that is associated with the objectivity it denotes in such a way that the meaning of the sign and its subject matter are represented only by the sign itself and are revealed only through its interpretation.

With the discovery of mathematical rules and theorems, scientists came up with new mathematical notation, signs. Mathematical signs are symbols designed to record mathematical concepts, sentences and calculations. In mathematics, special symbols are used to shorten the record and express the statement more accurately. In addition to the numbers and letters of various alphabets (Latin, Greek, Hebrew), the mathematical language uses many special symbols invented over the past few centuries.

2. Signs of addition, subtraction

The history of mathematical notation begins with the Paleolithic. Stones and bones with notches used for counting date back to this time. The most famous example isishango bone. The famous bone from Ishango (Kongo), dating back to about 20 thousand years BC, proves that already at that time a person performed quite complex mathematical operations. The notches on the bones were used for addition and were applied in groups, symbolizing the addition of numbers.

Ancient Egypt already had a much more advanced system of notation. For example, inpapyrus of ahmesas a symbol for addition, the image of two legs walking forward in the text is used, and for subtraction - two legs walking backward.The ancient Greeks denoted addition by writing side by side, but from time to time they used the slash symbol “/” for this and a semi-elliptic curve for subtraction.

The symbols for the arithmetic operations of addition (plus "+'') and subtraction (minus "-'') are so common that we almost never think that they did not always exist. The origin of these symbols is unclear. One of the versions is that they were previously used in trading as signs of profit and loss.

It is also believed that our signcomes from one of the forms of the word “et”, which in Latin means “and”. Expression a+b written in Latin like this: a et b . Gradually, due to frequent use, from the sign " et " remains only " t ", which, over time, turned into"+ ". The first person who may have used the signas an abbreviation for et, was the astronomer Nicole d'Orem (author of The Book of the Sky and the World) in the middle of the fourteenth century.

At the end of the fifteenth century, the French mathematician Chiquet (1484) and the Italian Pacioli (1494) used “'' or " '' (denoting "plus") for addition and "'' or " '' (denoting "minus") for subtraction.

The subtraction notation was more confusing, since instead of a simple “” in German, Swiss and Dutch books sometimes used the symbol “÷” with which we now denote division. Several books of the seventeenth century (for example, those of Descartes and Mersenne) used two dots “∙ ∙” or three dots “∙ ∙ ∙” to indicate subtraction.

The first use of the modern algebraic sign “” refers to a German manuscript on algebra from 1481, which was found in the library of Dresden. In a Latin manuscript from the same time (also from the Dresden library), there are both characters: "" and " - " . The systematic use of the signs "” and “-” for addition and subtraction occurs inJohann Widmann. The German mathematician Johann Widmann (1462-1498) was the first to use both signs to mark the presence and absence of students in his lectures. True, there is evidence that he "borrowed" these signs from a little-known professor at the University of Leipzig. In 1489, in Leipzig, he published the first printed book (Mercantile Arithmetic - “Commercial Arithmetic”), in which both signs were present. and , in the work "A quick and pleasant account for all merchants" (c. 1490)

As a historical curiosity, it is worth noting that even after the adoption of the signnot everyone used this symbol. Widman himself introduced it as a Greek cross(the sign we use today) whose horizontal stroke is sometimes slightly longer than the vertical one. Some mathematicians such as Record, Harriot and Descartes used the same sign. Others (eg Hume, Huygens, and Fermat) used the Latin cross "†", sometimes placed horizontally, with a crossbar at one end or the other. Finally, some (such as Halley) used a more decorative look " ».

3. Equal sign

The equal sign in mathematics and other exact sciences is written between two expressions that are identical in size. Diophantus was the first to use the equal sign. He denoted equality with the letter i (from the Greek isos - equal). ATancient and medieval mathematicsequality was indicated verbally, for example, est egale, or they used the abbreviation “ae” from the Latin aequalis - “equal”. Other languages ​​also used the first letters of the word “equal”, but this was not generally accepted. The equal sign "=" was introduced in 1557 by a Welsh physician and mathematician.Robert Record(Recorde R., 1510-1558). The symbol II served in some cases as a mathematical symbol for equality. The record introduced the symbol "='' with two identical horizontal parallel lines, much longer than those used today. The English mathematician Robert Record was the first to use the symbol "equality", arguing with the words: "no two objects can be equal to each other more than two parallel segments." But even inXVII centuryRene Descartesused the abbreviation "ae".François Vietthe equals sign denotes subtraction. For some time, the spread of the Record symbol was hindered by the fact that the same symbol was used to indicate parallel lines; in the end, it was decided to make the symbol of parallelism vertical. The sign received distribution only after the works of Leibniz at the turn of the 17th-18th centuries, that is, more than 100 years after the death of the person who first used it for thisRoberta Record. There are no words on his tombstone - just a carved "equal" sign.

Related symbols for approximate equality "≈" and identity "≡" are very young - the first was introduced in 1885 by Günther, the second - in 1857Riemann

4. Signs of multiplication and division

The multiplication sign in the form of a cross ("x") was introduced by an Anglican priest-mathematicianWilliam Otred in 1631. Before him, the letter M was used for the multiplication sign, although other designations were proposed: the rectangle symbol (Erigon, ), asterisk ( Johann Rahn, ).

Later Leibnizreplaced the cross with a dot (end17th century) so as not to be confused with the letter x ; before him, such symbolism was found inRegiomontana (15th century) and an English scientistThomas Harriot (1560-1621).

To indicate the action of divisionBranchpreferred the slash. Colon division began to denoteLeibniz. Before them, the letter D was also often used.fibonacci, the feature of the fraction, which was also used in Arabic writings, is also used. Division in the form obelus ("÷") was introduced by a Swiss mathematicianJohann Rahn(c. 1660)

5. Percent sign.

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

6. Sign of infinity

The current infinity symbol "∞" has come into useJohn Wallis in 1655. John Wallispublished a large treatise "The Arithmetic of the Infinite" (lat.Arithmetica Infinitorum sive Nova Methodus Inquirendi in Curvilineorum Quadraturam, aliaque Difficiliora Matheseos Problemata), where he introduced the symbol he inventedinfinity. It is still not known why he chose this particular sign. One of the most authoritative hypotheses relates the origin of this symbol to the Latin letter "M", which the Romans used to represent the number 1000.The symbol for infinity is called "lemniscus" (lat. ribbon) by the mathematician Bernoulli about forty years later.

Another version says that the drawing of the "eight" conveys the main property of the concept of "infinity": movement without end . Along the lines of the number 8, you can make endless movement, like on a cycle track. In order not to confuse the introduced sign with the number 8, mathematicians decided to place it horizontally. Happened. This notation has become standard for all mathematics, not just algebra. Why is infinity not denoted by zero? The answer is obvious: no matter how you turn the number 0, it will not change. Therefore, the choice fell on 8.

Another option is a serpent devouring its tail, which, one and a half thousand years BC in Egypt, symbolized various processes that have no beginning and no end.

Many believe that the Möbius strip is the progenitor of the symbolinfinity, since the infinity symbol was patented after the invention of the "Möbius strip" device (named after the nineteenth century mathematician Möbius). Möbius strip - a strip of paper that is curved and connected at the ends, forming two spatial surfaces. However, according to available historical information, the infinity symbol began to be used to represent infinity two centuries before the discovery of the Möbius strip.

7. Signs coal a and perpendicular sti

Symbols " corner" and " perpendicular» came up with 1634French mathematicianPierre Erigon. His perpendicular symbol was upside down, resembling the letter T. The angle symbol was reminiscent of the icon, gave it a modern formWilliam Otred ().

8. Sign parallelism and

Symbol " parallelism» known since ancient times, it was usedHeron and Pappus of Alexandria. At first, the symbol was similar to the current equal sign, but with the advent of the latter, to avoid confusion, the symbol was rotated vertically (Branch(1677), Kersey (John Kersey ) and other mathematicians of the 17th century)

9. Pi

The generally accepted notation for a number equal to the ratio of the circumference of a circle to its diameter (3.1415926535...) was first formedWilliam Jones in 1706, taking the first letter of the Greek words περιφέρεια -circle and περίμετρος - perimeter, which is the circumference of a circle. Liked this abbreviationEuler, whose works fixed the designation definitively.

10. Sine and cosine

The appearance of sine and cosine is interesting.

Sinus from Latin - sinus, cavity. But this name has a long history. Indian mathematicians advanced far in trigonometry in the region of the 5th century. The word "trigonometry" itself did not exist, it was introduced by Georg Klugel in 1770.) What we now call the sine roughly corresponds to what the Indians called ardha-jiya, translated as a semi-bowstring (i.e., half-chord). For brevity, they simply called it - jiya (bowstring). When the Arabs translated the works of the Hindus from Sanskrit, they did not translate the "string" into Arabic, but simply transcribed the word in Arabic letters. It turned out to be a jib. But since short vowels are not indicated in Arabic syllabic writing, j-b really remains, which is similar to another Arabic word - jaib (cavity, sinus). When Gerard of Cremona translated the Arabs into Latin in the 12th century, he translated this word as sinus, which in Latin also means sinus, deepening.

The cosine appeared automatically, because the Hindus called him koti-jiya, or ko-jiya for short. Koti is the curved end of a bow in Sanskrit.Modern abbreviations and introduced William Oughtredand fixed in the works Euler.

The tangent/cotangent designations are of much later origin (the English word tangent comes from the Latin tangere, to touch). And even until now there is no unified designation - in some countries the designation tan is more often used, in others - tg

11. Abbreviation "What was required to prove" (ch.t.d.)

Quod erat demonstrandum » (kwol erat lamonstranlum).
The Greek phrase means "what had to be proved", and the Latin - "what had to be shown." This formula ends every mathematical reasoning of the great Greek mathematician of Ancient Greece, Euclid (III century BC). Translated from Latin - which was required to prove. In medieval scientific treatises, this formula was often written in an abbreviated form: QED.

12. Mathematical notation.

13. Conclusion

Mathematical science is necessary for a civilized society. Mathematics is found in all sciences. Mathematical language is mixed with the language of chemistry and physics. But we still understand it. We can say that we begin to study the language of mathematics together with our native speech. Mathematics has become an integral part of our life. Thanks to the mathematical discoveries of the past, scientists create new technologies. The surviving discoveries make it possible to solve complex mathematical problems. And the ancient mathematical language is clear to us, and discoveries are interesting to us. Thanks to mathematics, Archimedes, Plato, Newton discovered physical laws. We study them at school. In physics, too, there are symbols, terms inherent in physical science. But mathematical language is not lost among physical formulas. On the contrary, these formulas cannot be written without knowledge of mathematics. Through history, knowledge and facts are preserved for future generations. Further study of mathematics is necessary for new discoveries. To use the preview of presentations, create a Google account (account) and sign in: https://accounts.google.com

Slides captions:

Mathematical symbols The work was done by a student of the 7th grade of school No. 574 Balagin Viktor

A symbol (Greek symbolon - a sign, a sign, a password, an emblem) is a sign that is associated with the objectivity it designates so that the meaning of the sign and its subject matter are represented only by the sign itself and are revealed only through its interpretation. Signs are mathematical conventions designed to record mathematical concepts, sentences and calculations.

Bone of Ishango Part of the papyrus of Ahmes

+ − Plus and minus signs. Addition was denoted by the letter p (plus) or the Latin word et (conjunction "and"), and subtraction by the letter m (minus). The expression a + b was written in Latin like this: a et b.

subtraction notation. ÷ ∙ ∙ or ∙ ∙ ∙ Rene Descartes Marin Mersenne

A page from Johann Widmann's book. In 1489, Johann Widmann published the first printed book in Leipzig (Mercantile Arithmetic - “Commercial Arithmetic”), in which both + and - signs were present.

Addition notation. Christian Huygens David Hume Pierre de Fermat Edmund (Edmond) Halley

Equal sign Diophantus was the first to use the equal sign. He denoted equality with the letter i (from the Greek isos - equal).

Equal sign Proposed in 1557 by the English mathematician Robert Record "No two objects can be equal to each other more than two parallel segments." In continental Europe, the equal sign was introduced by Leibniz

× ∙ Multiplication sign Introduced in 1631 by William Oughtred (England) in the form of an oblique cross. Leibniz replaced the cross with a dot (end of the 17th century) so as not to confuse it with the letter x. William Otred Gottfried Wilhelm Leibniz

Percent. Matthieu de la Porte (1685). One hundredth of a whole, taken as a unit. "percentage" - "pro centum", which means - "one hundred". "cto" (short for cento). The typesetter mistook "cto" for a fraction and typed "%".

Infinity. John Wallis John Wallis introduced the symbol he invented in 1655. The serpent devouring its tail symbolized various processes that have no beginning and no end.

The symbol for infinity began to be used to represent infinity two centuries before the discovery of the Möbius strip A Möbius strip is a strip of paper that is curved and connected at its ends to form two spatial surfaces. August Ferdinand Möbius

Angle and Perpendicular. Symbols were invented in 1634 by the French mathematician Pierre Erigon. Erigon's angle symbol resembled an icon. The perpendicular symbol has been reversed, resembling the letter T . These signs were given their modern form by William Oughtred (1657).

Parallelism. The symbol was used by Heron of Alexandria and Pappus of Alexandria. At first, the symbol was similar to the current equal sign, but with the advent of the latter, to avoid confusion, the symbol was rotated vertically. Heron of Alexandria

Pi. π ≈ 3.1415926535... William Jones in 1706 π εριφέρεια - circumference and π ερίμετρος - perimeter, that is, the circumference of the circle. This reduction pleased Euler, whose works fixed the designation completely. William Jones

sin Sinus and cosine cos Sinus (from Latin) - sinus, cavity. koti-jiya, or ko-jiya for short. Koti - the curved end of the bow Modern short designations were introduced by William Otred and fixed in the works of Euler. "arha-jiva" - among the Indians - "half-string" Leonard Euler William Otred

What was required to prove (ch.t.d.) "Quod erat demonstrandum" QED. This formula ends every mathematical reasoning of the great mathematician of Ancient Greece, Euclid (III century BC).

We understand the ancient mathematical language. In physics, too, there are symbols, terms inherent in physical science. But mathematical language is not lost among physical formulas. On the contrary, these formulas cannot be written without knowledge of mathematics.

Each of us from the school bench (more precisely, from the 1st grade of elementary school) should be familiar with such simple mathematical symbols as greater sign and less sign, as well as the equals sign.

However, if it is rather difficult to confuse something with the latter, then about how and in what direction are the signs more and less written (less sign and sign over, as they are sometimes called) many immediately after the same school bench and forget, because. they are rarely used by us in everyday life.

But almost everyone sooner or later still has to face them, and to "remember" in which direction the character they need is written is obtained only by turning to their favorite search engine for help. So why not answer this question in detail, at the same time telling the visitors of our site how to remember the correct spelling of these signs for the future?

It is about how the greater-than sign and the less-than sign are spelled that we want to remind you in this short note. It will also not be superfluous to say that how to type greater than or equal signs on 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 next time to “google” again, and 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 - signs more and less are written like this, as shown in the image below.

And now we will tell a little more about how to understand this and remember it for the future.

In general, the logic of understanding is very simple - which side (larger or smaller) the sign in the direction of writing looks to the left - such is the sign. Accordingly, the sign more to the left looks with a wide side - a larger one.

An example of using the greater than sign:

• 50>10 - number 50 more number 10;
• student attendance in this semester was >90% of classes.

How to write a less than sign, perhaps, is not worth explaining again. It is exactly the same as the greater than sign. If the sign looks to the left with a narrow side - a smaller one, then the sign is smaller in front of you.
An example of using the less than sign:

• 100<500 - число 100 меньше числа пятьсот;
• came to the meeting<50% депутатов.

As you can see, everything is quite logical and simple, so now you should not have questions about which way to write the greater than sign and the less than sign in the future.

Greater than or equal/less than or equal sign

If you have already remembered how the sign you need is written, then it will not be difficult for you to add one dash to it from below, so you will get a sign "less or equal" or sign "more or equal".

However, regarding these signs, some 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 to" denoting as ">=" , which, in principle, is often quite acceptable, but can be made more beautiful and more correct.

In fact, in order to type these characters, there are special characters that can be entered on any keyboard. Agree, the signs "≤" and "≥" look much better.

Greater than or equal sign on keyboard

In order to write "greater than or equal to" on the keyboard with one character, you don't even need to go into the table of special characters - just put a greater than sign while holding down the key "alt". Thus, the keyboard shortcut (entered in the English layout) will be as follows.

Or you can just copy the icon from this article if you need to use it once. Here he is, please.

≥

Less than or equal sign on keyboard

As you probably already guessed, you can write "less than or equal" on the keyboard by analogy with the greater than sign - just put the less than sign while holding down the key "alt". The keyboard shortcut to be entered in the English layout will be as follows.

Or just copy it from this page, if it's easier for you, here it is.

≤

As you can see, the rule for writing greater than and less than signs is quite easy to remember, and in order to type the greater than or equal and less than or equal icons on the keyboard, you just need to press an additional key - everything is simple.

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