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.
Encyclopedic YouTube
1 / 5
✪ Sign / in math
✪ Mathematics Grade 3. Table of digits of multi-digit numbers
✪ Sets in math
✪ Mathematics 19. Math fun - Shishkin school
Subtitles
Hey! 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, thereby 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 can 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]
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 implied 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
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 the reading of 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.
| 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 D |
| 2 | ≅ | 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 α |
|
| 8 | → | 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 s |
| no. | Designation | Content | Symbolic notation example | An example of symbolic notation in geometry |
|---|---|---|---|---|
| 1 | M,N | Sets | - | - |
| 2 | A,B,C,... | Set elements | - | - |
| 3 | { ... } | Consists of... | 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 N | A ∈ 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 numbers | a⊂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 | ∪ | Union | 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) |
| 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 true | α∩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 β |
| 2 | ∨ | 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 α |
| 7 | ∃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 |
The development of mathematical symbolism was closely connected with the general development of the concepts and methods of mathematics. First Mathematical signs there were signs for depicting numbers - numbers, the emergence of which, apparently, preceded writing. The most ancient numbering systems - Babylonian and Egyptian - appeared as early as 3 1/2 millennia BC. e.
First Mathematical signs for arbitrary values appeared much later (starting from the 5th-4th centuries BC) in Greece. Quantities (area, volumes, angles) were shown as segments, and the product of two arbitrary homogeneous quantities - as a rectangle built on the corresponding segments. In "Beginnings" Euclid (3rd century BC) quantities are indicated by two letters - the initial and final letters of the corresponding segment, and sometimes even one. At Archimedes (3rd century BC) the latter method becomes common. Such a designation contained the possibilities for the development of literal calculus. However, in classical ancient mathematics, literal calculus was not created.
The beginnings of letter representation and calculus arise in the late Hellenistic era as a result of the liberation of algebra from geometric form. Diophantus (probably 3rd century) wrote down an unknown ( X) and its degrees with the following signs:
[ - from the Greek term dunamiV (dynamis - strength), denoting the square of the unknown, - from the Greek cuboV (k_ybos) - cube]. To the right of the unknown or its degrees, Diophantus wrote the coefficients, for example, 3x5 was depicted
(where = 3). When adding, Diophantus attributed terms to each other, for subtraction he used a special sign; Diophantus denoted equality with the letter i [from the Greek isoV (isos) - equal]. For example, the equation
(x 3 + 8x) - (5x 2 + 1) =X
Diophantus would write it like this:
(here
means that the unit does not have a multiplier in the form of a power of the unknown).
A few centuries later, the Indians introduced various Mathematical signs for several unknowns (abbreviations for the names of colors denoting unknowns), square, square root, subtracted number. So the equation
3X 2 + 10x - 8 = x 2 + 1
In recording Brahmagupta (7th century) would look like:
Ya va 3 ya 10 ru 8
Ya va 1 ya 0 ru 1
(ya - from yavat - tawat - unknown, va - from varga - square number, ru - from rupa - rupee coin - a free member, a dot above the number means the number to be subtracted).
The creation of modern algebraic symbolism dates back to the 14th-17th centuries; it was determined by the successes of practical arithmetic and the study of equations. In various countries spontaneously appear Mathematical signs for some actions and for powers of an unknown quantity. Many decades and even centuries pass before one or another convenient symbol is developed. So, at the end of 15 and. N. Shuke and L. Pacioli used addition and subtraction signs
(from lat. plus and minus), German mathematicians introduced modern + (probably an abbreviation of lat. et) and -. Back in the 17th century can count about ten Mathematical signs for the multiplication operation.
were different and Mathematical signs unknown and its degrees. In the 16th - early 17th centuries. more than ten notations competed for the square of the unknown alone, for example se(from census - a Latin term that served as a translation of the Greek dunamiV, Q(from quadratum), , A (2), , Aii, aa, a 2 etc. Thus, the equation
x 3 + 5 x = 12
the Italian mathematician G. Cardano (1545) would have the form:
from the German mathematician M. Stiefel (1544):
from the Italian mathematician R. Bombelli (1572):
French mathematician F. Vieta (1591):
from the English mathematician T. Harriot (1631):
In the 16th and early 17th centuries equal signs and brackets come into use: square (R. Bombelli , 1550), round (N. Tartaglia, 1556), curly (F. viet, 1593). In the 16th century the modern form takes the notation of fractions.
A significant step forward in the development of mathematical symbolism was the introduction by Vieta (1591) Mathematical signs for arbitrary constants in the form of capital consonants of the Latin alphabet B, D, which made it possible for him for the first time to write down algebraic equations with arbitrary coefficients and operate with them. Unknown Viet depicted vowels in capital letters A, E, ... For example, the record Vieta
In our symbols it looks like this:
x 3 + 3bx = d.
Viet was the creator of algebraic formulas. R. Descartes (1637) gave the signs of algebra a modern look, denoting unknowns with the last letters of lat. alphabet x, y, z, and arbitrary given quantities - in initial letters a, b, c. He also owns the current record of the degree. Descartes' notation had a great advantage over all the previous ones. Therefore, they soon received universal recognition.
Further development Mathematical signs was closely connected with the creation of infinitesimal analysis, for the development of the symbolism of which the basis was already prepared to a large extent in algebra.
Dates of occurrence of some mathematical signs
| sign | meaning | Who introduced | When introduced |
| Signs of individual objects | |||
| ¥ | infinity | J. Wallis | 1655 |
| e | base of natural logarithms | L. Euler | 1736 |
| p | ratio of circumference to diameter | W. Jones L. Euler | 1706 |
| i | square root of -1 | L. Euler | 1777 (in press 1794) |
| i j k | unit vectors, orts | W. Hamilton | 1853 |
| P (a) | angle of parallelism | N.I. Lobachevsky | 1835 |
| Signs of Variable Objects | |||
| x,y,z | unknowns or variables | R. Descartes | 1637 |
| r | vector | O. Koshy | 1853 |
| Signs of individual operations | |||
| + | addition | German mathematicians | Late 15th century |
| – | subtraction |
||
| ´ | multiplication | W. Outred | 1631 |
| × | multiplication | G. Leibniz | 1698 |
| : | division | G. Leibniz | 1684 |
| a 2 , a 3 ,…, a n | degrees | R. Descartes | 1637 |
| I. Newton | 1676 |
||
| | roots | K. Rudolph | 1525 |
| A. Girard | 1629 |
||
| Log | logarithm | I. Kepler | 1624 |
| log | B. Cavalieri | 1632 |
|
| sin | sinus | L. Euler | 1748 |
| cos | cosine |
||
| tg | tangent | L. Euler | 1753 |
| arc sin | arcsine | J. Lagrange | 1772 |
Sh | hyperbolic sine | V. Riccati | 1757 |
Ch | hyperbolic cosine |
||
| dx, ddx, … | differential | G. Leibniz | 1675 (in press 1684) |
d2x, d3x,… |
|||
| | integral | G. Leibniz | 1675 (in press 1686) |
| | derivative | G. Leibniz | 1675 |
| ¦¢x | derivative | J. Lagrange | 1770, 1779 |
| y' |
|||
| ¦¢(x) |
|||
| Dx | difference | L. Euler | 1755 |
| | partial derivative | A. Legendre | 1786 |
| | definite integral | J. Fourier | 1819-22 |
| | sum | L. Euler | 1755 |
| P | work | K. Gauss | 1812 |
| ! | factorial | K. Crump | 1808 |
| |x| | module | K. Weierstrass | 1841 |
| lim | limit | W. Hamilton, many mathematicians | 1853, early 20th century |
| lim |
|||
| n = ¥ |
|||
| lim |
|||
| n ® ¥ |
|||
| x | zeta function | B. Riemann | 1857 |
| G | gamma function | A. Legendre | 1808 |
| AT | beta function | J. Binet | 1839 |
| D | delta (Laplace operator) | R. Murphy | 1833 |
| Ñ | nabla (Hamilton operator) | W. Hamilton | 1853 |
| Signs of variable operations | |||
| jx | function | I. Bernoulli | 1718 |
| f(x) | L. Euler | 1734 |
|
| Signs of individual relationships | |||
| = | equality | R. Record | 1557 |
| > | more | T. Harriot | 1631 |
| < | smaller |
||
| º | comparability | K. Gauss | 1801 |
| | parallelism | W. Outred | 1677 |
| ^ | perpendicularity | P. Erigon | 1634 |
AND. newton in his method of fluxes and fluent (1666 and following years) introduced signs for successive fluxions (derivatives) of magnitude (in the form
and for an infinitesimal increment o. Somewhat earlier, J. Wallis (1655) proposed the infinity sign ¥.
The creator of the modern symbolism of differential and integral calculus is G. Leibniz. He, in particular, belongs to the currently used Mathematical signs differentials
dx, d 2 x, d 3 x
and integral
A huge merit in creating the symbolism of modern mathematics belongs to L. Euler. He introduced (1734) into general use the first sign of the variable operation, namely the sign of the function f(x) (from lat. functio). After Euler's work, the signs for many individual functions, such as trigonometric functions, acquired a standard character. Euler owns the notation for constants e(base of natural logarithms, 1736), p [probably from Greek perijereia (periphereia) - circumference, periphery, 1736], imaginary unit
(from the French imaginaire - imaginary, 1777, published in 1794).
In the 19th century the role of symbolism is growing. At this time, signs of the absolute value |x| (TO. Weierstrass, 1841), vector (O. Cauchy, 1853), determiner
(BUT. Cayley, 1841) and others. Many theories that arose in the 19th century, such as Tensor Calculus, could not be developed without suitable symbolism.
Along with the specified standardization process Mathematical signs in modern literature one can often find Mathematical signs used by individual authors only within the scope of this study.
From the point of view of mathematical logic, among Mathematical signs the following main groups can be outlined: A) signs of objects, B) signs of operations, C) signs of relations. For example, the signs 1, 2, 3, 4 depict numbers, that is, objects studied by arithmetic. The addition sign + by itself does not represent any object; it receives subject content when it is indicated which numbers are added: the notation 1 + 3 depicts the number 4. The sign > (greater than) is the sign of the relationship between numbers. The sign of the relation receives a quite definite content when it is indicated between which objects the relation is considered. To the above three main groups Mathematical signs adjoins the fourth: D) auxiliary signs that establish the order of combination of the main signs. A sufficient idea of such signs is given by brackets indicating the order in which actions are performed.
The signs of each of the three groups A), B) and C) are of two kinds: 1) individual signs of well-defined objects, operations and relations, 2) general signs of “non-repetitive” or “unknown” objects, operations and relations.
Examples of signs of the first kind can serve (see also the table):
A 1) Notation of natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9; transcendental numbers e and p; imaginary unit i.
B 1) Signs of arithmetic operations +, -, ·, ´,:; root extraction, differentiation
signs of sum (union) È and product (intersection) Ç of sets; this also includes the signs of the individual functions sin, tg, log, etc.
1) Equals and inequality signs =, >,<, ¹, знаки параллельности || и перпендикулярности ^, знаки принадлежности Î элемента некоторому множеству и включения Ì одного множества в другое и т.п.
Signs of the second kind depict arbitrary objects, operations and relations of a certain class or objects, operations and relations subject to some predetermined conditions. For example, when writing the identity ( a + b)(a - b) = a 2 -b 2 letters a and b denote arbitrary numbers; when studying functional dependence at = X 2 letters X and y - arbitrary numbers related by a given ratio; when solving the equation
X denotes any number that satisfies the given equation (as a result of solving this equation, we learn that only two possible values \u200b\u200b+1 and -1 correspond to this condition).
From a logical point of view, it is legitimate to call such general signs signs of variables, as is customary in mathematical logic, without being afraid of the fact that the “region of change” of a variable may turn out to consist of a single object or even “empty” (for example, in the case of equations with no solution). Further examples of such signs are:
A 2) Designation of points, lines, planes and more complex geometric shapes with letters in geometry.
B 2) Notation f, , j for functions and notation of operator calculus, when one letter L depict, for example, an arbitrary operator of the form:
The notation for "variable ratios" is less common, and is used only in mathematical logic (cf. Algebra of logic ) and in relatively abstract, mostly axiomatic, mathematical studies.
Lit.: Cajori, A history of mathematical notations, v. 1-2, Chi., 1928-29.
Article about the word Mathematical signs" in the Great Soviet Encyclopedia has been read 39765 times
