Like mathematical formulas. General rules for setting formulas

One of the most complex species set is a set of mathematical formulas. Formulas are texts that include fonts in Russian, Latin and Greek, straight and italic, light, bold, with a large number mathematical and other signs, indices on the top and bottom lines of the font and various large-point characters. The range of fonts for a set of formulas is at least 2 thousand characters. The character table in WORD-98 includes 1148 characters.

The main difference between formula typing and all other types of typing is that formula typing in its classic form is not done in parallel lines, but occupies a certain part of the strip area.

Formula- a mathematical or chemical expression in which the relationship between certain quantities is expressed in a conditional form using numbers, symbols and special signs.

Numbers- signs that denote or express numbers (quantities). Numerals are available in Arabic and Roman numerals.

Arabic numerals: 1, 2. 3, 4, 5, 6, 7, 8, 9, 0. Arabic numerals change their meaning depending on the place they occupy in the series of digital signs. Arabic numerals are divided into two classes - 1st - units, tens, hundreds; 2nd - thousands, tens of thousands, hundreds of thousands, etc.

Roman numerals. There are seven main digital characters: I - one, V - five, X - ten, L - fifty, C - one hundred, D - five hundred, M - one thousand. Roman numerals have constant value, so numbers are obtained by adding or subtracting digital signs. For example: 28 = XXVIII (10 + 10 + 5 + 1 + 1+ 1); 29 = XXIX (10 + 10 -1 + 10); 150 = CL(100 + 50); 200 = SS (100 + 100); 1980 = MDCCCCLXXX (1000 + 500 + 100 + 100 + 100 + 100 + 50 + 10+ 10 + 10); 2002 = MMII (1000 + 1000 + 1 + 1).

Roman numerals usually indicate centuries (XV1st century), volume numbers (Volume IX), chapters (Chapter VII), parts (Part II), etc.

Symbols- letter expressions included in the formula (for example, mathematical symbols: l - length, λ - failure rate (shrinkage), π - ratio of circumference to diameter, etc.; chemical symbols: Al - aluminum, Pb - lead, H - hydrogen, etc.).

Odds- numbers preceding the symbols, for example 2H 2 O; 4sinx. Symbols and numbers often have superscripts (on top line) and subscripts (on the bottom line), which either explain the meaning of the indices (for example, λ c - linear shrinkage, G T - theoretical mass of the casting, C f - actual mass of the casting); or indicate mathematical operations (for example, x 2, y 3, z -2, etc.); or indicate the number of atoms in a molecule and the number of charges of ions in chemical formulas (for example, CH 4). In formulas there are also subscripts to subscripts: superscript to superscript - superscript supraindex, subscript to superscript - superscript subindex, superscript to subscript - subscript subscript and subscript to subscript - subscript subscript.

Signs mathematical operations and ratios - addition “+”, subtraction “-”, equality “=”, multiplication “x”; The action of division is indicated by a horizontal ruler, which will be called a fractional or dividing ruler.

(9.12)

Main line- a line containing the main signs of mathematical operations and relationships.

Classification of formulas.

Mathematical formulas are divided according to the complexity of the set, depending on the composition of the formula (single-line, two-line, multi-line) and its saturation with various mathematical signs and symbols, indices, subindices, supraindices and prefixes. According to the complexity of the set, all mathematical formulas can be conditionally divided into four main groups and one additional:

1 group. One-line formulas (9.13-9.16);

2nd group. Two-line formulas (9.17-9.19). In fact, these files consist of 3 lines;

3rd group. Three-line formulas (9.20-9.23). In fact, these files consist of 5 lines;

4th group. Multiline formulas (9.24-9.26);

Additional group (9.27-9.29).

When assigning formulas to complexity groups, the complexity of typing and the time spent on typing were taken into account.

Group II. Two-line formulas:

(9.29)

Rules for typing mathematical formulas.

When typing mathematical text, you must follow the following basic rules.

Dial numbers in formulas in roman font, for example 2ah; Zu.

Abbreviated trigonometric and mathematical terms, For example sin, cos, tg, ctg, arcsin. Ig, lim etc., type in font Latin alphabet straight light outline.

Abbreviated words in the index type in Russian font on the bottom line.

Abbreviations for physical, metric and technical units of measurement, designated by letters of the Russian alphabet, should be typed in the text in straight font without dots, for example 127 V, 20 kW. The same names, designated by letters of the Latin alphabet, should also be typed in straight font without dots, for example 120 V, 20 kW, unless otherwise indicated in the original.

Symbols (or numbers and symbols), following one after another and not separated by any characters, type without spacers, for example 2xy; 4u.

Punctuation marks In formulas, type in straight light font. Commas inside the formula should be separated from the subsequent element of the formula by 3 p.; the comma is not separated from the previous element of the formula; from the preceding subscript the comma is removed by 1 p.

Ellipsis On the bottom line, type in dots, divided into half-kegel ones. From the previous and subsequent elements of the formula, the points are also half-kegel, for example:

(9.30)

Symbols(or numbers and symbols) following one after another, do not separate, but type without space.

Signs of mathematical operations and ratios, as well as signs of geometric images, such as, = ,< ,> , + , - , beat off the previous and subsequent elements of the formula by 2 p

Abbreviated Math Terms beat off the previous and subsequent elements of the formula by 2 points.

Exponent, immediately following the mathematical term, type close to it, and space after the exponent.

Letters "d" (meaning "differential"), δ (in the meaning of “partial derivative”) and ∆ (in the meaning of “increment”), beat off the previous element of the formula by 2 points, from the subsequent symbol indicated signs don't fight back.

Abbreviated names of physical and technical units of measurement And metric measures in formulas, beat off 3 points from the numbers and symbols to which they relate.

Signs ° , " , " beat off the next symbol (or number) by 2 points; the indicated characters are not separated from the previous symbol.

Punctuation following the formula, do not fight off her.

A line of dots in formulas, type in dots, using half-kegel padding between them.

Formulas typed in a selection with the text are separated from the previous and subsequent texts in half-kegel; When the line is justified, this space does not decrease, but increases. Formulas that follow one after another in the selection with text are also turned off.

Several formulas placed in one line, off in the center, should be separated from each other by a space of no less than a size and no more than 1/2 square.

Small explanatory formulas, typed on the same line with the main formula, should be included in the right edge of the line, or set off by two fonts from the main expression (unless otherwise indicated in the original).

Type serial numbers of formulas in numbers of the same size as one-line formulas, and turn them to the right, for example:

X+Y=2 (9.31)

If the formula does not fit into the line format, and it cannot be hyphenated, it can be typed in a smaller size.

Hyphenations in formulas are undesirable. To avoid hyphenation, it is allowed to reduce the spaces between formula elements. If reducing spaces fails to bring the formula to required format lines, then hyphens are allowed:

1) on the signs of the relationship between the left and right sides of the formula ( = ,>,< );

2) on addition or subtraction signs (+, - );

3) on multiplication signs (x). In this case, the next line begins with the sign where the formula ended in the previous line. When transferring formulas, it is necessary to ensure that the transferred part is not very small, that the expressions enclosed in brackets, expressions related to the signs of the root, integral, and sum are not broken; Separation of indices, exponents, and fractions is not allowed.

In numbered formulas, the formula number, if it is transferred, is placed at the level of the central line of the transferred part of the formula. If the serial numbering does not fit on the line, it is placed in the next one and turned off to the right. Formulas whose numerator or denominator do not fit into the given typesetting format are typed in a font of a smaller size, or in a font of the same size, but in two lines with a hyphen.

If, when transferring a formula, the dividing line or the root ruler breaks, then the place where each line breaks is indicated by arrows.

Arrows cannot be placed near mathematical symbols.

Modern scientific publications are saturated mathematical methods evidence. Scientists enter a large number of formulas and symbols into the text. Distinctive features mathematical formulas – greater semantic concentration, high degree the abstractness of the material contained in them, the specificity of the mathematical language. This is in to a certain extent complicates the reader’s perception of the text and poses many problems for the editor.

A mathematical formula is a symbolic representation of a statement (sentence, judgment). Formulas help replace complex verbal expressions and various operations with quantitative indicators in the text. For this purpose, special designations are used - symbols, which can be divided into three groups:

– conventional letter designations of mathematical and physical-technical quantities;

– symbols of units of measurement of quantities;

– mathematical signs.

There is an opinion that it is much easier for an editor to work with text that has a lot of formulas than with text without formulas. This is incorrect, because formulas, to an even greater extent than text, can undergo transformations and have various shapes records, and for each specific formula in each specific publication the optimal form must be selected. At the same time, the readership for which the book is intended and the features of each formula are taken into account in order to avoid errors, ambiguities or unreadability. Let's see this using the example of writing one formula.

1. Vehicle operating speed

Tn – time in outfit.

In this form, the formula is convenient, for example, for a university textbook.

2. Vehicle operating speed

where L is the distance traveled by the car during the time it was on duty (at work);

Tn – time in outfit.

Such a record is quite acceptable, for example, for a textbook on course design, the reader of which is already somewhat prepared, and this fragment is part of some calculation methodology.

3. The same formula in production publications for engineering and technical workers may well be included in the selection.

Operating speed of the car v e =L/T n, where L is mileage; Tn – time in outfit.

4. In a textbook for schoolchildren and vocational school students, this formula should have a different form.

The operating speed, which is usually denoted, characterizes the conditional average speed of the rolling stock for the entire time it is on duty (at work) and is determined by the ratio of mileage to the time on duty, i.e.

where L is the distance traveled by the car during the time it was in service;

Tn – time in outfit.

Such a recording allows the student to clearly see how the initial parameters influence the result, i.e. understand which parameters influence the final result in direct proportion, and which vice versa, it is easy to remember the formula and learn the “classical” form of mathematical notation of physical dependence.

5. In popular science literature for the general reader, where there are one or two formulas throughout the entire book, writing in mathematical form seems inappropriate. So it's better to do it this way.

“The operating speed of a vehicle, as one of the most important indicators of its operation, is determined by calculation:

6. B scientific publications, where, for example, the reader needs this formula only as a reminder to explain some phenomena that are not directly related to the calculation of vehicle use indicators, the formula in its traditional form can be omitted altogether, and its meaning is simply conveyed in the words: “Usage vehicle speed, defined as the quotient of mileage divided by time on duty, is one of the most important indicators that have to be taken into account when forming the optimal structure of a transport association’s fleet.”

If we now evaluate the above options, it is not difficult to see that they differ markedly in ease of perception, compactness of construction and labor intensity of publication. Here we will conditionally include the labor intensity of editing, reprinting formulaic originals, and reading into the concept of “labor-intensive publication”. Each option has its own, different from others, indicators of perception, compactness and labor intensity.

Spelling options considered simplest formula, but if it turns out to be more complex, then it is easy to imagine that other options will appear related to the possibility of varying the form of writing indices, highlighting functional groups of parameters in the formula, dividing one complex formula into several simple ones and, conversely, changing the “number of storeys” the formula as a whole and its constituent elements.

Before continuing our discussion about editing mathematical formulas, it is necessary to stipulate what is considered immutable in formulas and what is subject to variation. The special literature states clearly and unambiguously: mathematical formulas must use symbols that are established by the standard or are generally accepted in the industry.

This is certainly true, but we note that only a small part of the symbols are regulated by the standards, and the “commonly accepted” symbols when analyzed specialized literature on one topic most often turn out to be “generally accepted” not in the industry, but within the same organization. This is especially true for indexes.

Many quantities needed only in one branch of science must have their own designations that differ from the designations of similar quantities in other branches of science. To solve this problem, i.e. To individualize a symbol, use indices. An index is added to the main letter designation, indicating a particular meaning. So, Latin letter L or l most often denote length, interval, extent, range, period, etc. If it is necessary to designate a specific concept of length, then a clarifying index is added to the general symbol. For example:

L k – length of the stern part of the boat;

L pr – travel distance;

l e – aileron span;

l ск – length of the shearing section.

The main material for compiling indexes is the lowercase letters of the Russian alphabet. The letters of the Latin alphabet are used much less frequently, and Greek and especially Gothic ones are used very rarely. Quite often, Arabic numerals and mathematical symbols are used in indexes. Based on their location in the letter designation, indices are divided into lower and upper, with the lower ones being preferable. It is better not to use the superscript on the right, since this is the place of the exponent. Most often, strokes are used as superscripts: h?; h??.

Sometimes indices can be located at the top left if it is necessary to distinguish between designations that have exactly the same appearance, and if the designation is already equipped with some indices and degrees. For example, there is a designation for the rotation angles of the rod Q, which, depending on the points of application of force, are provided with subscripts 1, 2, 3, as well as strokes ?, ??, ??? ... - depending on the multiplicity of force application (so, Q1? - the first application of force at point 1; Q 1 ?? - the second application of force at point 1, etc.). If you also need to select the angle of rotation (to the left or right of the rod node), use the upper left indices: ? – to indicate the angle to the left of the node; p – to indicate the angle to the right of the node. So, a letter designation with an index? Q 1 – the first application of force at point 1 when turning the node to the left.

Zero as an index gives the letter designation the meaning of “calculated”, “initial”, “initial”, relating to the center of gravity, etc., and can also be used in the meaning of “standard state of matter”, for example, l 0 – design length, t 0 – initial temperature.

Indexes consisting of several words are abbreviated by initial and characteristic letters. Moreover, if the index consists of two or three abbreviated words, after each of them, except the last one, put a dot, for example S ditch– elevator area.

Now directly about the perception of formulas. It is generally accepted that a well-understood formula is one that is easy to understand and remember. Let's add two additional requirements.

1. Other things being equal, preference should be given to such symbols in formulas that can be easily and unambiguously reproduced in writing (by hand). First of all, this applies to textbooks, the formulas from which the teacher writes on the board, the student writes in notes, etc. Difficulties here usually arise due to the similar style of letters different alphabets and due to the unjustified complexity of the indexes. So, R g.ts is easy to write down and then read. Now let's try to read the entry? e.g. For this it would seem expressive recording there are over 100 (!) reading options, for there are six options for s (“ro” lowercase and uppercase; “pe” lowercase and uppercase; “er” lowercase and uppercase); four options for e ("e" and "el", on line and in index); six options for g (“de” and “zhe”; on the line, in the first and second degree indices). In addition, the entire entry can be read as “? logarithmic."

2. The formula must have good graphic drawing. For example, numbers in the middle of factors (it is better to put them in front), complex exponents and indices, multi-stage indices, and complex formulas reduced to a compact form are poorly perceived.

A special type of graphic distortion, which further worsens the “appearance” of the formula, is violation of typing rules. Wanting to simplify it, sometimes the upper indices are shifted relative to the lower ones (K av tkm). The dots in the indices are often out of place and look like a multiplication sign (D B.P). Inexperienced typesetters type commas after formulas in indexes (A = BC To). The rules for choosing a point size for connections are not followed, as a result of which the formula and explication become inappropriate similar friend on a friend. If letters from different alphabets are found in indexes, they often align poorly (“dance”). The division sign “slash” is often lower in height (smaller than the point size) of the dividend and divisor.

Regarding the main condition for good perceptibility of formulas - facilitating their understanding and memorization - the following recommendations must be taken into account:

– other things being equal, Russian characters, which are the first letter of the encrypted word, are perceived, i.e. are understood and remembered better than Latin or Greek;

– it is undesirable to use abbreviations as symbols, since they are perceived as a work;

– the index should, if possible, reflect as clearly as possible the word or phrase encrypted in it;

The formula is easy to understand and remember, which clearly reflects the dependence of the calculation result on the nature of the change in parameters.

Units physical quantities should be placed only after substituting numerical values of quantities into the formula and carrying out intermediate calculations - when obtaining the final result. For example:

wrong:

s = KTm/s = 1.4 · 290 · 300 m/s = 350 m/s;

Right:

s = CT = 1.4 · 290 · 300 = 350 m/s.

Mathematical symbols are defined as symbols used to record mathematical concepts, sentences, and calculations. Thus, “the ratio of the circumference of a circle to the length of its diameter” is written in the form of a sign.

Mathematical signs are divided into three groups:

1) signs of mathematical objects (points, lines, planes) are usually designated by letters (A, B, C...; a, b, c...; ?, ?, ? ... );

2) signs of addition (+) and subtraction (-); raising to the power a 2 , A 3 etc.; root V; signs of trigonometric functions log, sin, cos, tg, etc.; factorial!; differential and integral dx, ddx,…, ?ydx, module | x |;

3) signs of relations (= – equality, > – more,< – меньше, || – параллельность, ? – перпендикулярность, ? – тождествен–ность, ? – приблизительное равенство).

All these signs, except for object signs, are used only in formulas; it is prohibited to use them in the text instead of words of the corresponding meaning. Object signs in the text can be used with the words: at point A, on plane a, from angle x.

Often after the formula there is an explication - a decoding of the symbols included in the formula. Its elements are arranged in the sequence in which the symbols are read in the formula. It is recommended to group the same letters with different indices together. When deciphering fractional formula expressions, first explain the letter designations of the numerator and then the denominator.

If it is necessary to decipher the meaning of a symbol on the left side of the equation, it is recommended to do this in the previous formula of the part of the sentence. Unfortunately, this recommendation is not always followed.

Let us give examples from the magazine “Military Economic Bulletin” (2002. No. 12).

The cost of transporting weapons and equipment is calculated using the formula

W p.e.t. = In p.v.t? With p.v.t? D p (29)

Where W p.e.t.– costs for transportation of the same type of weapons and equipment, rub.; In p.v.t.– quantity of transported weapons (equipment) of this type, units; From p.v.t.– cost of transportation of 1 unit of weapons (equipment) per 1 km in rubles; D P– range of transportation of weapons (equipment), km.

The calculation is made for each type of weapon (equipment) separately.

In addition, to secure the transported weapons and equipment on the platform, fastening materials are used - wire, nails, staples, wooden beams or special fastening devices. To purchase them you also need cash. The cost of purchasing fastening materials is calculated using the formula

W km = V p.v.t? Ts k.k.m, (30)

where Z km – costs for purchasing fastening materials, rub.; In p.v.t – quantity of transported weapons and equipment, units; Ts k.k.m – price of 1 set of fastening material (per unit of equipment), rub.

The costs of purchasing fastening materials (fastening devices) are calculated separately only if they are not included in the prices for transportation of weapons and equipment.

Transportation costs personnel during exercises, various types of transport are determined by the formula

Z p.l.s = V hp? With p.h? D p, (31)

where Z p.l.s – costs of transporting personnel on a specific type of transport, rub.; In hp - the number of personnel transported on a specific type of transport, units; C p.h - the cost of transporting one person per 1 km by a specific type of transport, rub.; D p – range of transportation of personnel, km.

And in the first, and in the second, and in the third formulas, the symbol on the left side of the equations should be deciphered in the text preceding the formula. The symbol B everywhere denotes the quantity of transported weapons or personnel, units. Symbol C – the cost of transporting 1 person, 1 weapon per 1 km; D – distance of transportation of weapons and personnel, km. It would be necessary to give the decoding of the symbols once, without repeating it after each formula.

After the formula, a comma is placed before the explication, and the explication begins with the word where, followed by the designation of the first quantity and its decoding, etc. It is recommended to put a semicolon at the end of each transcript, and a period at the end of the last one. Designations of units of physical quantities in decodings are separated from the text by a comma. For example:

The inductance of a multilayer coil is determined by the formula

Where? – number of turns; D – average winding diameter, mm; l – winding length, mm; h – winding height, mm.

The explanation for the formulas is not standard. In the scientific literature you can find various versions of it - from the simplest to the complex, relating to one formula and several. If the formulas in a sentence are separated by text, it is better to separate the general explanation for them into an independent sentence. For example:

In vector form, these equations can be represented as follows: equation of motion of the center of mass

and the equation of motion of the aircraft relative to the center of mass

The following notations are adopted in these equations: V – vector of the speed of movement of the aircraft relative to inertial space;

R – vector external forces, acting on aircraft; G – vector of gravity forces;

M is the vector of the moment of external forces relative to the center of mass of the aircraft.

In scientific, reference, and encyclopedic publications, in order to use paper more economically, the explication can be placed in a selection.

Careful checking and correct processing of formulas and symbols found in the text requires a lot of attention from the editor. It is necessary not only to ensure the correctness and accuracy of all designations and numerical indicators, but also to achieve the greatest clarity and clarity in design, to avoid ambiguities or the possibility of different interpretations.

It is generally accepted that the author is entirely responsible for the correctness of the data provided, but the editor of the publishing house is obliged to make a complete or selective control check formulas Problems in textbooks and teaching aids are thoroughly tested. Equalities can be checked by substituting the corresponding values.

To competently edit a formulaic text, it is not enough just knowledge about the mathematical construction of the formula, about the use symbols and so on. It is also necessary to know the printing requirements for formulas, since compliance with them helps to make the formulas understandable, expressive, and compact.

The editor must know how best to arrange the formula, how to move it if it does not fit on one line, what formulas should be numbered, etc.

There are two types of formulas: inside text lines and as separate lines in the middle of the typesetting format. Placing formulas in the selection helps to save a lot of space. Therefore, if short, simple formulas do not have independent meaning and are not numbered, but are included in separate lines, they can be arranged in a selection with the text. For example:

From the continuity condition we find

This text can be arranged like this:

This technique is especially effective with a large typesetting format (it allows you to save up to 70-80% of the area), however, this technique is not recommended for use when the formulas are multi-line or multi-story.

Several formulas placed in a row, in which the same or similar quantities are calculated, are aligned or using the equal sign:

p xx= ?R+ ?div? + 2?? 1 ;

r yy= ?R+ ?div? + 2?? 2 ;

p zz= ?R+ ?div? + 2?? 3;

or by magnitude, which is the basis of comparison:

150°? ? ?210°;

330°? ? ?360°.

If a formula is being converted, and the formula itself is multi-line, intermediate groups should be placed one below the other so that the progress of the transformations is better visible. For example:

Numbering of formulas. Very often it is necessary to operate with formulas not only where they are located, but also in the previous or subsequent presentation. In order to avoid citing it in full each time you refer to a formula, the formulas are numbered. Typically, continuous numbering is used for a limited number of the most important formulas. Numbering all formulas in a row clutters the book.

IN big works(textbooks, monographs) sometimes sequential numbering of formulas by chapter is used, the so-called double numbering. In this case, the first digit of the numbered formula must correspond to the chapter number, the second - the serial number of the formula within the chapter, for example: the 12th formula in Chapter 2 is numbered (2.12), the 5th formula in Chapter 3 is (3.5) and etc. In exceptional cases, when the next formula is a variation of the previously given main one, lettered numbering of the formulas with an Arabic numeral and a lowercase straight letter of the Russian alphabet is allowed. The number and letter are written together and are not separated by a comma, for example: 17a, 17b, etc.

The serial numbers of all formulas must be written in Arabic numerals in parentheses (Roman numerals are not used for numbering formulas) at the right edge of the page without departing from the formula to its number.

formula (4.15) shows...

In the case of numbering a group of formulas or a system of equations with one serial number this number, enclosed in parentheses, is placed at the level of the middle of the combined group of formulas or system of equations at the right edge of the page. In this case, parentesis (curly brace) is used.

The serial number of the formula when transferring is placed at last line. For example:

Integrating equation (2.17) once, we obtain

Multiplication sign in formulas. Coefficients and symbols in formulas, as a rule, are not separated by any signs, but are written together. The dot as a sign of multiplication by the middle line is not placed before and between alphabetic symbols, before brackets and between factors in brackets, before fractional expressions, written through and after a horizontal line. For example:

A dot on the middle line as a multiplication sign is placed only in exceptional cases:

– between numerical factors: 18 · 242.5 · 8;

– when the argument of a trigonometric function is followed by a letter designation: Jtg c · a sin b;

– to separate factors from expressions related to

to the signs of the radical, integral, logarithm, etc.:

In general, the expression cos? t? that or

usually presented in the form that cos? t or

Unless there is a special purpose for writing factors in a certain sequence, so as not to disrupt the harmony of the previous conclusion or mathematical analysis.

The oblique cross (?) as a multiplication sign is used in formulas:

– when specifying dimensions: room area 4 ? 3m;

– when recording vector product vectors: huh? b;

– when transferring a formula from one line to another at the multiplication sign.

Transferring formulas. If the formula given in the manuscript is so long that it does not fit on one line on the publication page (without hyphenation), they usually require that the author outline possible places for hyphenation. It is preferable to do the transfer first of all on the signs of mathematical relations: = ? , ?, ?,?, ?, >, <, >> etc.

If it is not possible to divide the formula into lines using these signs, it should be divided using the + or - operation signs. Less desirable, although acceptable, is dividing formulas into lines using ± and multiplication signs. It is not customary to divide a line at a division sign (two dots). If a formula is divided at the multiplication sign, it is shown not with a dot, but with an oblique cross (?).

Particular attention is paid to the issue of transferring equations, the right or left parts of which are presented in the form of fractions with long numerators and denominators or with cumbersome radical expressions. Such equations must be transformed, bringing them to a form convenient for transfer.

It is advisable to represent fractions with a long numerator and a short denominator so that the numerator is written as a polynomial in parentheses, and the unit divided by the denominator is placed outside the brackets. For example, the equation

easily brought to mind

With a short numerator and a long denominator, it is recommended to replace individual complex elements with simplified notations. For example: instead of

If the formula includes a fraction with a long numerator and a long denominator, then for transfer either use both recommended conversion methods, or replace the horizontal fraction bar with a division sign (two dots). IN the latter case the formula will look like

(a 1 x+ a 2 y+ ... + a i h) : (b 1 x+ b 2 y+ ... + b i h).

can be written like this:

(a 1 x+ b 1 x 2 + ... + nxn) 1/2 .

The signs on which the transfer is made are placed twice: at the end of the first line and at the beginning of the transferred part. For example:

If the formula is interrupted at an accent, it is also repeated at the beginning of the next line. If the equal sign comes before the minus sign, the translation is done at the equal sign. If a formula contains several expressions in parentheses, it is recommended to carry over at the + or – sign in front of the brackets.

Despite all the efforts of editors and proofreaders, errors in the text with formulas still remain. A typical mistake when transferring formulas is separating the argument from the function. For example:

Of course, one cannot demand from a typesetter that he differentially evaluate a record of type f(x - y): without context it is impossible to say what it means: the product of two functions f and (x - y) or the dependence of a function f on the argument (x - y). However, it is known that trigonometric functions without an argument they have no meaning, so they are not used without them. And placing a multiplication sign between a function and its argument is a gross mistake.

In the example given, the editor could not foresee the mistakes made. In the first case, the transfer of the formula was caused by an oversight by the typesetter when breaking it into two lines; in the second, the formula was in the text itself, and it was almost impossible to foresee its transfer in this place during editing. But in the layout the editor was obliged to correct this error.

Capacity printed sheet with formulas is 2-3 times less than the capacity of a printed sheet of text, which increases the cost of publication. Publishing practice has rational methods providing formulas that give a tangible economic effect. Formulas, as a rule, are typed in a red line with padding at the top and bottom. This leads to an increase in paper consumption and an increase in the cost of typing and installing formulas.

Including formulas in the middle of the format is advisable in two cases: a) the formula needs emphasis; b) due to its complexity and cumbersomeness, the formula cannot be typed along with the text. Formulas that need to be paid attention to are usually numbered. However, formulas are often turned off unnecessarily.

For example, text

can be placed on one line.

Significant compaction of the set can be achieved even when this would seem to be prevented by the numbering of formulas. For example:

With this arrangement of formulas, finding its number is not difficult.

In such a case, all formulas can be placed in one line under one number:

Changing links to them is easy. If, for example, you need to refer to a formula for expressing a coordinate, you can write: “according to the second of formulas (3).”

The transformation methods inherent in the nature of the formula itself allow you to present almost any formula of any complexity in a form convenient for typing. Simplest fraction

turns out to be inconvenient to type. But it can be written either through a slash 1/2, or as a decimal fraction 0.5, or as a power of 2 -1 . All options are equal, but the first one is most widespread.

It is believed that in editions of works scientific literature you can convert any fractions into one-line expressions like: (a + b)/c; (A + B)/(c + d), etc. There is a clear benefit in paper consumption. Converting multistory fractions is especially useful. For example, fraction

can be converted to the form (a/b + c/d)/(e/f + g/h) -1 .

In order to save paper, this compactness is given great attention. However, there was some overkill here: huge imperceptible formulas and formulas of ambiguous interpretation began to appear in the press.

Incomprehensible formulas are the result of sometimes thoughtless translation of complex two- and three-story formulas into one-line ones using the “slash” sign and negative indicators degrees.

Formulas of ambiguous interpretation are obtained in those cases when the denominator after the slash contains a product.

A striking example of careless handling of the “slash” sign is in Appendix 1 to OST 29.115-88 “Author's and text publishing originals. Are common technical requirements" The authors of the standard consider the formula possible

convert like this:

This is incorrect, because it becomes unclear which symbols are in the numerator and which are in the denominator. If this ambiguity is eliminated (with the help of additional brackets), the formula will turn out to be even less perceivable. This option will, perhaps, become suitable only for some special compact publication, in which the formula is given only so that, without thinking about its meaning, one can substitute numbers and get the result.

Let's look at another “textbook” example:

If we simply replace the horizontal slash with a slash, we get

A = B/CX and A = B/CX,

those. different formulas became the same.

To prevent this from happening, in the first formula you need to put the product in the denominator in parentheses, and in the second, move X forward or write B/C in parentheses:

A = B/(CX) and A = XB/C = (B/C) X.

Many people believe that the second formula in option A = B/ CX can be left unchanged, because according to the rules of arithmetic, the actions here will be performed in the order of the signs. We cannot agree with this, since in technical literature there has long been a stereotype of perceiving the expression behind a slash as a single whole. For example, specific fuel consumption has always been designated as follows: g/kWh, where “h (as)” is actually in the denominator, although according to the rules of arithmetic it is in the numerator.

If in the expression A = B/ CX the slash is replaced with a division sign (two dots), this is also not good, because C and X will be typed without a space and will be mistaken by many for the product (A = B: CX).

As was agreed, the labor intensity of formulas (cost-effectiveness) will include the labor intensity of not only typing, but also editing, reprinting the original formula, and reading. To be fair, this should also include the laboriousness of checking formulas by the author in the layout, when he sometimes has to spend hours checking formulas that have become unrecognizable after editing. It is obvious, for example, how much more difficult it is to check the second formula than the first:

before conversion

after conversion? = 4( A/C):[(1+A/C) 2 +B 2 /C(?/? r ?? r /?) 2 ].

Of course, the fact that the complexity of formulas usually comes down only to the cost of the set is to some extent understandable: the cost of the set is a quantitative and external indicator of the preparation of the publishing original. The remaining indicators of labor intensity are not calculated and are internal to the publishing house.

To minimize the labor intensity of editing, it is necessary to ensure that authors present material that meets the following requirements:

– formulas are written by hand in block letters, neat and clear (if the author was unable to implement computer set);

– division signs in complex formulas have the appearance of a horizontal line. Such formulas are easy to check, analyze and make a decision, having, of course, agreed with the author on the expediency of giving the formula a more compact form;

– formulas are marked;

– the necessary clarifications have been made in the margins (“e” is not “el”, etc.);

– the number of letters and signs that require additional explanation in the margins is reduced to a minimum in the formulas.

A lot of extra paper is spent on detailed presentations of mathematical operations and calculations. In such cases, the number of formulas can be reduced - it is not always necessary to give all intermediate transformations if they are elementary in nature. For example, instead of a whole series of transformations of the formula

it's enough to write

You can also save paper by grouping formulas. So, formulas

?x= ?? + 2Ge x;

?y= ?? + 2Ge y;

?z= ?? + 2Ge z;

?y z= ??y z;

?x z= ??x z;

?x y= ??x y;

can be grouped more compactly:

?x= ?? + 2Ge x; ?yz= ??yz;

?y= ?? + 2Ge y; ?xz= ??xz;

?z= ?? + 2Ge z; ?xy= ??xy.

Punctuation in a text with formulas is not yet sufficiently systematized, since formulas are often considered as an independent part, artificially interspersed into a sentence. Unsystematicity and inconsistency can be easily eliminated if formulas and individual symbols are considered as members of a sentence. From this position, each formula must be regarded as a syntactic unit included in a sentence, and punctuation marks must be placed accordingly.

Formulas, as already mentioned, are either located inside text lines or are switched off in the middle of the typing format. If there are formulaic expressions inside the text, then when arranging punctuation marks, the signs of mathematical operations should be considered as nominal part a compound nominal predicate in which the copula is omitted. For example:

If? Z,C< ?X,C, That M(y, z, s) = Mu?x, s.

Punctuation marks are placed taking into account the fact that mathematical symbols< (меньше), = (равно) являются именной частью ска–зуемого. Связка «есть» опущена, так как сказуемое имеет значение настоящего времени.

It is more difficult to place punctuation marks in a sentence with a formula highlighted on a separate line. Particularly controversial is the placement of the sign before the formula.

Let's take the most general case, i.e. formulaic text of the following type (Fig. 2), and consider punctuation marks before the formula, between several formulas, after the formula and in the post-formula text.

Rice. 2. General case formulaic text

There may not be any sign in front of the formula; there may be a comma or a colon. After the text preceding the formula, usually no punctuation marks are placed if the formula is a member of a sentence, which, according to the rules of punctuation, should not be separated from the preceding words by punctuation marks. For example:

We characterize the efficiency of the channel by the value

A comma is usually placed before a formula if the preformula text ends with introductory words. For example: But for VNA lattices always?1 = 0, therefore,

d 2 = ?? ?i p+ G p = f(?, t?) And G p = f(?, t ?) ? f(d 2).

A comma is also used when the formula ends with subordinate clause, participle or participial phrase.

Now if R ex and e e both are equal to zero,

From formula (36) we obtain, by introducing flow coefficients,

The most controversial issue of punctuation in text with formulas is the placement of a colon before the formula. In Russian, a colon is placed before homogeneous members sentences after a generalizing word, in non-union complex sentences, with direct speech and the use of quotations.

A colon may be placed before a formula in the following cases.

1. If there is a generalizing word before several formulas; in its absence, a colon should be placed before several formulas only in cases where it is necessary to warn the reader that what follows is a list of several formulas:

Applying the superposition theorem to equation (8.32), we obtain two types of convolution integral, or Duhamel integral:

From equation (3) we get:

2. If the formulaic text can be considered as non-union difficult sentence, in which the formula, being the second part, either explains the meaning of the first part (mental formulation of words is possible), or contains the reason or justification for what is said in the first part (mental formulation of words is possible because, since, since).

Let's substitute expression (3.57) into the formula for B 0 :

We assume that With he, there is a linear function:

Between formulas it is customary to put a semicolon or a comma, depending on which sign is used throughout the work.

In systems of equations united by parentheses, punctuation marks can be omitted, considering the system as a single member of the sentence. For example: From a system of equations

it is possible to determine the values of constant coefficients.

If a system of equations ends a sentence or an explication is given after the system, such a system is considered as a listing of formulas and they are separated from each other by the corresponding sign.

Sometimes two formulas are connected by the conjunction or. The conjunction or is used in Russian in two meanings: as a separative and as a clarifying one. A dividing conjunction or (single or repeating) indicates the need to choose one of the concepts that are expressed by homogeneous members and exclude or replace each other. There is no comma or comma before a single separating conjunction.

If the conjunction or has a clarifying meaning, then a comma is required before the single conjunction.

The editor needs to determine in what sense the author used the conjunction or between formulas. Sometimes it is not difficult to understand that the second formula, joined by the conjunction or, is simply a transformed first formula, and a comma is needed. This happens in cases when, instead of letter designations, they are substituted into the same formula numeric values. For example:

…we apply equation (2) and after rearranging the terms we get

Such designs are rare. Therefore, to check the identity of the formulas, the editor has to make some mathematical transformations. They are elementary (do not go beyond the course high school) and can be done by any editor. Let's look at a few examples.

From the trigonometry course we know that 2 sin ? 2 cos ? 2 is the formula double angle sine, i.e. 2 sin?2 cos?2 = sin 2?2. Consequently, in the second formula 2 sin ?2 cos ?2 is replaced by sin 2?2, which means that the formulas are identical and a comma must be inserted.

Here the right side of the first equation is reduced by cos ?2. The formulas are also identical, and a comma is needed.

Placing a comma before a conjunction or in in this case requires no explanation.

In this regard, we will consider recommendations for “processing mathematical text, in particular formulas, which allows, without harm to the content and assimilation of the material, to achieve either a reduction in the number of formulas, or simplifying their writing, reducing the space they occupy in the book.”

Sometimes it is necessary to highlight a whole series of formulas that are consistently obtained as a result of mathematical transformations, the nature of which is clear to the reader without additional explanation. As a rule, all such formulas are turned off in the middle of the strip format, and the formulas themselves are connected with words or, i.e., from, etc., each of which occupies a separate line. However, the same text will take up a much smaller area if you remove the connecting words (replace them with semicolons) and arrange the formulas more compactly.

For example:

By arranging formulas in a selection, we naturally save paper. But the author proposes, at the same time, to remove clarifying conjunctions and words, and to separate formulas from each other with a semicolon, thereby violating mathematical meaning. In the first example we are dealing with the transformation of one formula into another form, i.e. the last formula was obtained by successive transformations first. In the second example, the semicolon indicates that we have several independent formulas that are not related in meaning to other formulas. As you can see, the author's recommendation led to an error.

After the formula there should be the punctuation mark that is necessary for the meaning.

There are restrictions on the use of some punctuation marks. Directly to formulas, symbols, symbols, mathematical terms, units of measurement, etc. Punctuation marks used as or similar to mathematical symbols cannot be adjacent.

Thus, the dash (-) coincides in spelling with the mathematical sign of the subtraction operation (-), the colon (:) - with the division sign (:), Exclamation point(!) – with a factorial sign (!).

A comma cannot be placed between two formulas typed in a selection, the first of which ends with a number, and the second begins with a number; a comma also cannot be placed between the listed quantities expressed in Arabic numerals, since it can be mistaken for separator mark decimal fraction. In these cases, the comma must be replaced with a semicolon.

Formulas or individual letter symbols in the text that have large, long subscripts must be separated by a semicolon, even when the meaning requires a comma, otherwise the comma will be mistaken for a sign included in the index, especially with fuzzy printing.

For example:

l?e1; l?22; l?y+1.

To eliminate possible errors when typing mathematical symbols and letter symbols, you need an accurate editor-marking of all symbols, marks and inscriptions that help the typesetter quickly and accurately determine which alphabet a particular letter belongs to, whether it is lowercase or uppercase. naya, straight or italic, bold or light, etc.

Marking is necessary due to the fact that in the Russian and Latin alphabets there are letters and signs that are exactly the same or very similar to each other, both in handwriting and in typewriting, but differ in printing reproduction. Thus, in handwriting, especially when writing quickly by hand, there is almost no difference between the uppercase and lowercase letters C and s, K and k, O and o, P and r, S and s, V and v, W and w, Z and z, y and y, x and x. The letter O and 0 (zero) and the degree sign ° are similar in spelling; Russian letter Z and number 3; Roman I and Arabic 1 (unit); Russian letter x (ha), Latin x (ix) and multiplication sign (x), etc.

In addition to a clear outline, all letters and signs that are similar to each other must be appropriately marked in the manuscript with special proofreading marks. Capital letters, for example, are underlined with two strokes below (X), lowercase letters - with two strokes above ( x). In all cases where the outline of letters may raise doubts among the editor or typesetter, explanatory inscriptions should be made in the margins of the manuscript or directly next to the letters between the lines: letter, number, zero, sign. deg., sign. multiply, el, not el, etc.

Letters of the Latin alphabet in mathematical formulas are typed in italics and underlined in the manuscript with a wavy line. Greek letters circled in red, signs of a German gothic font– a green rectangle.

Some physical and mathematical quantities and notations are usually typed in the Roman alphabet, for example, the Mach numbers M, Reynolds Re, Prindtl Rg, etc., trigonometric, hyperbolic, inverse circular and inverse hyperbolic functions, names of temperature scales °C, °Ra, °K, °F, generally accepted conditional mathematical abbreviations of maximum and minimum (max, min), optimal value of a quantity (opt), constancy of quantity (const), limit signs (lim), decimal, natural and other logarithms (lg, log, Log, In, Zn), determinant (det), etc.

The arrangement of formulas and their parts according to the technical rules of the set is subject to the following:

– in formulas consisting of single-line and fractional parts, symbols and signs of the main line and dividing lines are located according to midline formulas; Moreover, if there is no clearly defined center line in the formula, it is considered to be a horizontal line passing through the middle of the height of the formula;

– groups of similar formulas and formulas united by paranthesis are equated by an equal sign or another sign of relations;

– the numerator and denominator are turned off in the center of the dividing line;

– in columns of formula determinants with different widths, they are turned off in the center of the column format.

A set of mathematical formulas is subject to rules that require the following:

– type one-line formulas in a font of the same typeface and size as the font of the main text, and their fractional parts in a font whose size is 2 points smaller;

– do not separate symbols that are not separated by mathematical signs and numbers from each other (12ab);

– do not separate from the preceding element: a) expressions in parentheses from the opening bracket; b) indices and exponents of a symbol or digit (if a symbol or digit has both an upper and lower index, the upper index may be placed after the lower index, i.e., with space for the width of the lower index);

c) a radical expression from the radical sign; d) punctuation marks, if the preceding element is single-line; e) parentheses closing from those enclosed in expression brackets; f) factorial;

– do not separate from the subsequent element: a) the differential sign from the following function designation or arguments: dX; b) the integral sign from the next integral sign: JJ; c) increment sign from the following designation of functions or arguments, including in brackets: D/(x); d) radical sign from the radical expression following it; e) parentheses opening from the expression enclosed in brackets; f) function sign from the following function designation or arguments, including those in brackets: / (x);

– beat off by 2 points from the preceding and subsequent elements: a) single and double vertical rulers | a + b | ? | a | + | b |; x || A ||; b) a differential sign along with the following and not separated from it designation of the function or arguments; c) the integral sign along with the following and not separated from it designation of the function or arguments;

G) mathematical notation(sin, lg, etc.) together with the exponent that is not separated from them (sin 2?); e) the increment sign along with the designation of the function or arguments that follows and does not separate from it; e) attached signs (the space can be increased to 12 points if the connections to the sign are larger than its width); g) a radical sign together with a radical expression;

h) brackets together with the expression enclosed in them and not separated from the closing bracket by an exponent or index;

i) relationship signs (=,<, ~ и т.д.);

– beat off from the previous element by 2 points: punctuation mark from the dividing line;

– move 3 points away from the previous element in the designation of units of physical quantities in book publications (15 km/h);

– set the comma inside the formula 3 points from the subsequent element;

- do not beat off horizontally: a) the denominator from the dividing line, except in cases where the exponent of the denominator is closely adjacent to the dividing line and when it is allowed to set off both the denominator and the numerator by 1-2 points from it; b) superscript or subscript marks from symbols; c) connections to additional signs from these signs; d) the numerator from the dividing line, with the exception of cases when the lower index is closely adjacent to the dividing line and when it is allowed to offset both the numerator and the denominator from it by 1-2 points.

4.2. Chemical formulas

Chemical formulas are images of the composition of chemically individual substances using chemical symbols and numbers. They are empirical (denote the molecule of a substance, its atomic weight, the nature of the bond between atoms) and structural (show the structure of the substance).

All symbols of chemical elements are typed in letters of the Latin alphabet in straight font, for example, C1 - chlorine, Cu - copper, etc. Letter designations of coefficients included in chemical formulas and indices are typed in italics. The numbers preceding the formula of a chemical compound, and the numbers included in the index, are in straight font without spacers, on– example: C m+ n ;C n H 2n ;8H 2 0.

If under the formula of a chemical compound the verbal name of the compound or element is given, it should be turned off in the middle and typed in a straight font with a lowercase letter, size 6, for example:

(SN 3 SOO) 2 Ca

calcium acetate salt

The writing of chemical symbols in the text must be unified. They should be typed either only in words (nitrogen, chlorine), or in symbols, but accompanied by words (nitrogen N, chlorine C1). If the chemical composition of a substance is indicated, first the percentage content of the chemical element is given, then its designation (for example, 0.8% Si, 3% Cu).

If there are a large number of components, the percentage designation (%) is given first, and then the symbol of each component and its percentage content (without the %) sign. For example: chemical composition of steel,%: Cr 5.2; Ni 4.42; Cu 4.13; Si 0.66, etc.

In combination with chemical formulas and terms, Russian, Latin and Greek prefixes are found. Prefixes attached to chemical terms with a hyphen are typed in italics, prefixes written together are typed in roman font. For example: anti-diazotate; trinitro-tert-butyltoluene; β-ethyl-pyridine; 1,4-dihydronaphthalene; cyclohexane. In combination with formulas, prefixes are typed in italics and attached to the formula with a hyphen. For example: iso-C 4 H 9; cis-C 7 H 14 .

There are two types of structural formulas: open (Fig. 3) and ring (Fig. 4).

The task of a proofreader reading a text with structural formulas is to achieve an exact correspondence of the set to the original, to monitor the correctness of the geometric figure, the accuracy of the placement of connection signs (rulers) and the uniformity of the arrangement and design of formulas in the text.

It is not customary to put punctuation marks before and after chemical formulas typed in the red line.

Shifts of empirical formulas are allowed on the signs =, > ,-,+, -, and they should be repeated at the beginning of the next line. It is not allowed to transfer the formula on the connection sign (=).

Structural formulas cannot be split by transfer.

Reading texts with various formulas- a difficult task, since it is necessary to know not only the symbolism accepted in this field of science, the conditions of construction, but also the rules for the set of formulas. It is recommended that the proofreader read formulaic texts alone in order to visually see how this or that symbol should have been typed, how the formula should be constructed and positioned. Before you start reading the stripes, you need to familiarize yourself with the following:

– common system symbols and designations in this publication;

– the peculiarities of writing symbols and notations in the original, so that during the reading process one does not confuse one sign with another;

– principles of layout, placement of formulas in the text, methods of their design in this publication in order to achieve uniformity.

The set of chemical formulas is subject to the following technical rules:

– chemical formulas are typed in 8-point font when typing the main text in 10-point (or 8-point) font;

– horizontal, vertical and oblique connection signs must be equal in length to the font size of the formula itself, except for cases when the structural features of the formula itself require increasing the connection sign so that it reaches the middle of the connected chemical symbols without interruption from them or with tapping 2 points when you need to visually equalize the distances;

– signatures under the formulas of chemical compounds are typed in font size 6 and centered on the designation of the chemical compound or the entire formula with a 4-point offset from the formula;

– if the height of the formulas of the compounds in the formula is different, the signatures are aligned along the top line of the signature for the connection with the greatest height;

– the inscriptions above the reaction direction arrow and the signatures under it are typed in font size 6 without space from the arrow and are turned off in its center.

Without further ado, here it is:

It is usually called Euler's identity in honor of the great Swiss mathematician Leonhard Euler (1707 - 1783). It can be seen on T-shirts and coffee mugs, and several surveys of mathematicians and physicists have given it the name “the greatest equation” (Crease, Robert P., “The greatest equations ever”).

The sense of beauty and elegance of the identity comes from the fact that it combines in a simple form the five most important numbers of mathematical constants: - the base natural logarithm, — Square root from and . Looking at it carefully, most people think about the exponent: what does it mean to raise a number to an imaginary power? Patience, patience, we'll get there.

To explain where this formula comes from, we must first obtain the more general formula found by Euler, and then show that our equality is just a special case of this formula. The general formula is amazing in itself and has many wonderful applications in mathematics, physics and engineering.

The first step in our journey is to understand that most functions in mathematics can be represented as infinite sum by degrees of argument. That's an example:

Here it is measured in radians, not degrees. We can get a good approximation for specific meaning, using only the first few terms of the series. This is an example of a Taylor series, and it is quite easy to derive this formula using mathematical analysis. Here I do not assume knowledge of mathematical analysis, so I ask the reader to take it on faith.

The corresponding formula for cosine is:

The number is a constant equal to , and Euler was the first to recognize its fundamental importance in mathematics and derive the last formula (the previous two were found by Isaac Newton). Books have been written about number (for example, Maor, E. (1994). e, the story of a number. Princeton University Press), you can also read about it.

Around 1740, Euler looked at these three formulas, arranged approximately as we see them here. It is immediately clear that each term in the third formula also appears in any previous one. However, half of the terms in the first equalities are negative, while every term in the last is positive. Most people would have left it that way, but Euler saw a pattern in all this. He was the first to put together the first two formulas:

Pay attention to the sequence of signs in this series: , it is repeated in groups of 4. Euler noticed that the same sequence of signs is obtained when we raise the imaginary unit to integer powers:

This meant that we could replace in the last formula with and get:

Now the signs correspond to the signs in the previous formula, and the new series coincides with the previous one, except that the expansion terms are multiplied by . That is, we get exactly

This is a surprising and mysterious result, and suggests that there is a close connection between number and sines and cosines in trigonometry, although it was known only from problems not involving geometry or triangles. Apart from its elegance and strangeness, however, it would be difficult to overestimate the importance of this formula in mathematics, which has increased since its discovery. It appears everywhere, and a book of about 400 pages (Nahin P. Dr. Euler's Fabulous Formula, 2006) was recently published describing some of the applications of this formula.

Note that the old question about imaginary exponents has now been resolved: to raise to an imaginary power, simply put the imaginary number in Euler's formula. If the base is a number other than , only a minor modification is required.

One of the most complex types of typing is a set of mathematical formulas. Formulas are texts that include fonts on Russian, Latin and Greek bases, roman and italic, light, bold, with a large number of mathematical and other characters, indices on the upper and lower lines of the font and various large-point characters. The range of fonts for a set of formulas is at least 2 thousand characters. The character table in WORD-98 includes 1148 characters.

The main difference between formula typing and all other types of typing is that formula typing in its classic form is not done in parallel lines, but occupies a certain part of the strip area.

Formula- a mathematical or chemical expression in which the relationship between certain quantities is expressed in a conditional form using numbers, symbols and special signs.

Numbers- signs that denote or express numbers (quantities). Numerals are available in Arabic and Roman numerals.

Roman numerals. There are seven main digital characters: I - one, V - five, X - ten, L - fifty, C - one hundred, D - five hundred, M - one thousand. Roman numerals have a constant value, so numbers are obtained by adding or subtracting digits. For example: 28 = XXVIII (10 + 10 + 5 + 1 + 1+ 1); 29 = XXIX (10 + 10 -1 + 10); 150 = CL(100 + 50); 200 = SS (100 + 100); 1980 = MDCCCCLXXX (1000 + 500 + 100 + 100 + 100 + 100 + 50 + 10+ 10 + 10); 2002 = MMII (1000 + 1000 + 1 + 1).

Roman numerals usually indicate centuries (XV1st century), volume numbers (Volume IX), chapters (Chapter VII), parts (Part II), etc.

Odds- numbers preceding the symbols, for example 2H 2 O; 4sinx. Symbols and numbers often have superscripts (on the top line) and subscripts (on the bottom line), which either explain the meaning of the indices (for example, λ c - linear shrinkage, G T - theoretical mass of the casting, C f - actual mass of the casting); or indicate mathematical operations (for example, x 2, y 3, z -2, etc.); or indicate the number of atoms in a molecule and the number of charges of ions in chemical formulas (for example, CH 4). In formulas there are also subscripts to subscripts: superscript to superscript - superscript supraindex, subscript to superscript - superscript subindex, superscript to subscript - subscript subscript and subscript to subscript - subscript subscript.

Signs of mathematical operations and ratios - addition “+”, subtraction “-”, equality “=”, multiplication “x”; The action of division is indicated by a horizontal ruler, which will be called a fractional or dividing ruler.

(9.12)

Main line- a line containing the main signs of mathematical operations and relationships.

Classification of formulas.

Mathematical formulas are divided according to the complexity of the set, depending on the composition of the formula (single-line, two-line, multi-line) and its saturation with various mathematical signs and symbols, indices, subindices, supraindices and prefixes. According to the complexity of the set, all mathematical formulas can be divided into four main groups and one additional one:

1 group. One-line formulas (9.13-9.16);

2nd group. Two-line formulas (9.17-9.19). In fact, these files consist of 3 lines;

3rd group. Three-line formulas (9.20-9.23). In fact, these files consist of 5 lines;

4th group. Multiline formulas (9.24-9.26);

Additional group (9.27-9.29).

When assigning formulas to complexity groups, the complexity of typing and the time spent on typing were taken into account.

IIgroup. Two-line formulas:

(9.29)

Rules for typing mathematical formulas.

When typing mathematical text, you must follow the following basic rules.

Dial numbers in formulas in roman font, for example 2ah; Zu.

Abbreviated trigonometric and mathematical terms, For example sin, cos, tg, ctg, arcsin. Ig, lim etc., type in the Latin alphabet with a straight light font.

Abbreviated words in the index type in Russian font on the bottom line.

Abbreviations for physical, metric and technical units of measurement, designated by letters of the Russian alphabet, should be typed in the text in straight font without dots, for example 127 V, 20 kW. The same names, designated by letters of the Latin alphabet, should also be typed in straight font without dots, for example 120 V, 20 kW, unless otherwise indicated in the original.

Symbols (or numbers and symbols), following one after another and not separated by any characters, type without spacers, for example 2xy; 4u.

Ellipsis On the bottom line, type in dots, divided into half-kegel ones. From the previous and subsequent elements of the formula, the points are also half-kegel, for example:

(9.30)

Symbols(or numbers and symbols) following one after another, do not separate, but type without space.

Signs of mathematical operations and ratios, as well as signs of geometric images, such as, = ,< ,> , + , - , beat off the previous and subsequent elements of the formula by 2 p

Abbreviated Math Terms beat off the previous and subsequent elements of the formula by 2 points.

Exponent, immediately following the mathematical term, type close to it, and space after the exponent.

Letters « d"(meaning "differential"), δ (in the meaning of “partial derivative”) and ∆ (in the meaning of “increment”) are separated from the previous element of the formula by 2 points; the indicated signs are not separated from the subsequent symbol.

Abbreviated names of physical and technical units of measurement And metric measures in formulas, beat off 3 points from the numbers and symbols to which they relate.

Signs ° , " , " beat off the next symbol (or number) by 2 points; the indicated characters are not separated from the previous symbol.

Punctuation following the formula, do not fight off her.

A line of dots in formulas, type in dots, using half-kegel padding between them.

Several formulas placed in one line, off in the center, should be separated from each other by a space of no less than a size and no more than 1/2 square.

Type serial numbers of formulas in numbers of the same size as one-line formulas, and turn them to the right, for example:

X+Y=2 (9.31)

If the formula does not fit into the line format, and it cannot be hyphenated, it can be typed in a smaller size.

Hyphenations in formulas are undesirable. To avoid hyphenation, it is allowed to reduce the spaces between formula elements. If reducing spaces fails to bring the formula to the desired line format, then hyphens are allowed:

on the signs of the relationship between the left and right sides of the formula ( = ,>,< );

on addition or subtraction signs (+, - );

on multiplication signs (x). In this case, the next line begins with the sign where the formula ended in the previous line. When transferring formulas, it is necessary to ensure that the transferred part is not very small, that the expressions enclosed in brackets, expressions related to the signs of the root, integral, and sum are not broken; Separation of indices, exponents, and fractions is not allowed.

If, when transferring a formula, the dividing line or the root ruler breaks, then the place where each line breaks is indicated by arrows.

Arrows cannot be placed near mathematical symbols.

Single-line and multi-line formulas.

In one-line formulas, the main line (without indexes and prefixes) should be typed in the same font size as the main text of the publication (unless otherwise indicated in the original).

The center point of all letters, numbers and signs of the main line of a one-line formula must be on the same line, which is called the middle line. When determining the center line, connections to the characters of the main line are not taken into account.

Subscripts and exponents in a multiline formula are aligned along the main line of the font.

Single-line formulas are switched off in the middle of the format, i.e. in the red line (if there are no special instructions in the original) and beat each other by 4 - 6 points.

A group of formulas with the same type of left or right part is aligned by the ratio sign, while the longest formula is typed first and included in the red line, the rest are equalized by it, for example:

(9.32)

When typing multi-line formulas, if the main text is typed kg. 10 p., then the central line is typed with a body, the numerator and denominator - with a petit.

The ruler separating the numerator from the denominator in a two-line formula must be equal in length to the longer of these expressions or longer than it by no more than 2 - 4 points. The minimum length of the ruler is equal to the font size with which the fraction is typed. Ruler size - 2 points, thin.

In a multi-line fraction, the main line should be 4 points longer than the dividing lines in the numerator and denominator, for example:

(9.33)

The numerator and denominator are turned off in the middle of the main dividing line.

The numerator and denominator do not deviate from the line, with the exception of the denominator, which is dominated by capital letters and exponents.

Explanations for formulas that begin with the word “where” are typed either on one line with the first character and a half-point space from it, then all subsequent explanations are aligned along the dash line, for example:

A is the amount of solution;

B - number of additives;

or with the word “where” justified to the left edge of a separate line, for example:

A is the amount of solution;

B is the number of additives.

Indices and exponents.

The formulas contain first-order indices (indices) and second-order indices (subindices and supraindices - index to index).

Most formulas, single-line and multi-line, contain 1st order indices: superscript and subscript one below the other.

In terms of their size, the indices are noticeably less than a letter and numbers of the main line, in addition, they must protrude beyond the font line of the main line. When typing the main line in kg font. 10 p. and 8 p. indices are typed in kg font. 6 p., when typing the main line in kg font. 6 points. The point of indices and exponents should be 4 points, while the index is lowered below the main line by 2 points, and the exponents are raised above the line by 2 points.

Double (upper and lower) indices must be located strictly one below the other.

Supraindices and subindices typed in kg font. 4 p.

Subscripts and exponents are typed close to the expression to which they relate. If the integrand to a power is one-line, the integral sign is typed in kg font. 10 points, if two-line - in kg font. 12 p., for example:

(9.34)

Sum sign Σ in the connection to the top line with a one-line exponent, it is typed in kg font. 6 p. or 8 p., with two lines - in kg font. 10 p., for example:

(9.35)

Brackets (round, square and curly) must be straight, the size of the brackets is chosen so that they can close the entire expression contained in them. The parentheses are separated from the preceding symbols in the formula by 2 p, the symbols enclosed in brackets are not separated from the brackets, and the exponent placed behind the bracket is not separated from the bracket. Consecutive parentheses do not separate from each other.

Large font signs.

Root sign √ The font size should be 2 points larger than the font size used to type the radical expression.

The root ruler is drawn with a two-point ruler, equal in length to the radical expression or 1-2 points longer,

(9.36)

Signs Σ , S(sum signs) and P(product sign) are typed in a straight font with a larger size, so when typing formulas kg. 8 or 10 points - the indicated characters are typed in kg font. 12 points, when typed in kg font. 6 points - prefixes in one-line formulas are typed in kg font. 10 points, in two-line ones - 16 - 20 points depending on the height of the formula, and in multi-line formulas - with a font size that allows you to cover the smaller part of the formula if the numerator and denominator of the formula are not the same in height, for example (formula 9.37) :

Indices above and below signs Σ , S, P are typed in kg font. 6 points and placed in the middle of the sign, for example:

(9.39)

Signs Σ , S(sum signs) and P(product sign) are separated from the previous and subsequent elements of the formula by 2 points.

Integral sign ∫ typed in a larger font size as follows: when typing a one-line formula in kg font. 6 p. - ∫ typed in kg font. 12 p.; when typing a one-line formula in kg font. 8 p. or 10 p. - ∫ typed in kg font. 14 or 16 p.; in two-line forms - ∫ typed in a font whose size is selected depending on the height of the integrand, and the middle of the character should always be on the center line of the formula, for example:

(9.40)

The size of an integral without subkeys for a formula height of 36 points should be 28 points, and for a formula height of 48 points - 36. The indices above and below the integral signs are also typed in kg font. 6 p, placed close to ∫ and turn off in the middle.

Integral same as signs Σ , S(sum signs) and P(product sign), is separated from the previous and subsequent elements of the formula by 2 points, and this space in the case of long indices can be increased to 12 points. The signs of the integral are not separated from each other.

Vertical rulers, single or double, must be exactly equal to the height of the expression contained in them, for example:

(9.41)

The space between lines in a group of formulaic expressions must be equal to half font size, and between columns of numbers - at least font size.

Rulers are chosen with a 2 point font.

When typing matrices, vertical rulers take two-point double ones, for example:

(9.42)

Formula expressions in matrix columns are turned into a red line or aligned to the left edge of the columns.

Vertical rulers are separated from the expressions contained in them by half-pointers, curly brackets by 6 points.

All horizontal rulers in formulas are always typed with two-point thin lines.

The length of the fraction ruler should be such that the largest part of the fraction (numerator and denominator) is covered by the ruler.

Mathematician Ian Stewart, in his new book In Search of the Unknown: 17 Equations That Changed the World, examines some of the most important equations of all time and provides examples of their practical applications.

According to the Pythagorean Theorem in right triangle square of the length of the hypotenuse equal to the sum squares of leg lengths.

Importance: The Pythagorean theorem is the most important equation in geometry, which connects it with algebra and is the basis of trigonometry. Without it, it would be impossible to create accurate cartography and navigation.

Modern use: Triangulation is still used today to accurately determine relative locations for GPS navigation.

A logarithm is the power to which the base must be raised to obtain an argument.

Importance: Logarithms were a real revolution, allowing astronomers and engineers to make calculations more quickly and accurately. With the advent of computers, they have not lost their importance as they are still essential for scientists.

Modern use: Logarithms are an important component for understanding radioactive decay.

Fundamental theorem of analysis or Newton - Leibniz formula gives the relationship between two operations: taking a definite integral and calculating the antiderivative.

Importance: The analysis theorem actually created modern world. Calculus is important in our understanding of how to measure solids, curves and areas. It is the basis of many natural laws and a source of differential equations.

Modern use: Any math problem where an optimal solution is required. Essential for medicine, economics and computer science.

Newton's classical theory of gravity describes gravitational interaction.

Importance: The theory allows one to calculate the gravitational force between two objects. Although it was later supplanted by Einstein's theory of relativity, the theory is still needed to practically describe how objects interact with each other. We still use it to this day to design the orbits of satellites and spacecraft.

Modern use: Allows you to find the most energy-efficient ways to launch satellites and space probes. Also makes satellite TV possible.

Complex numbers

Complex numbers are an extension of the field of real numbers.

Importance: Many modern technologies, including digital cameras, could not have been invented without complex numbers. They also provide the analysis that engineers need to solve practical problems in aviation.

Modern use: Widely used in electrical engineering and complex mathematical theories.

Importance: Contributed to the understanding of topological space, in which only the properties of continuity are considered. An essential tool for engineers and biologists.

Modern use: Topology is used to understand the behavior and function of DNA.

Importance: The equation is the basis of modern statistics. The natural and social sciences could not exist in their present form without him.

Modern use: Used in clinical trials to determine the effectiveness of drugs versus negative side effects.

Differential equation describing the behavior of waves.

Importance: Waves are studied to determine the time and location of earthquakes and to predict ocean behavior.

Modern use: Oil companies use explosives and then read data from subsequent sound waves to identify geological formations.

Importance: Equation allows you to break down, refine and analyze complex patterns.

Modern use: Used for compressing JPEG image information, as well as for detecting the structure of molecules.

Navier-Stokes equations

On the left side of the equation is the acceleration of a small amount of fluid, on the right side are the forces that act on it.

Importance: Once computers became powerful enough to solve this equation, they opened up a complex and very useful area of physics. It is especially useful for creating better aerodynamics in vehicles.

Modern use: Among other things, the equation has helped in the improvement of modern passenger aircraft.

Describe the electromagnetic field and its relationship with electric charges and currents in vacuum and continuous media.

Importance: Helped in understanding electromagnetic waves, which contributed to the creation of many of the technologies we use today.

Modern use: Radar, television and modern communications.

All energy and heat will disappear over time.

Importance: Essential to our understanding of energy and the universe through the concept of entropy. The discovery of the law helped improve the steam engine.

Modern use: Helped prove that matter consists of atoms, physicists still use this knowledge.

Energy equals mass times the speed of light squared.

Importance: Probably the most famous equation in history. It completely changed our perspective on matter and reality.

Modern use: Helped create nuclear weapon. Used in GPS navigation.

Schrödinger equation

Describes matter as a wave rather than a particle.

Importance: It turned the ideas of physicists upside down - particles can exist in a range of possible states.

Modern use: Significant contributions to the use of semiconductors and transistors, and thus to most modern computer technology.

Estimates the amount of data in a piece of code by calculating the probability of its symbols.

Importance: This is the equation that opened the door to the Information Age.

Modern use: Pretty much anything to do with finding errors in coding (programming).

Assessing intergenerational changes in populations of living things with limited resources.

Importance: Helped in the development of , which completely changed our understanding of how natural systems work.

Modern use: Used for earthquake modeling and weather forecasting.

Black-Scholes model

One of the options pricing models.

Importance: Helped create several trillion dollars. According to some experts, misuse of the formula (and its derivatives) contributed to the financial crisis. In particular, the equation makes several assumptions that do not hold true in real financial markets.

Modern use: Even after the crisis are used to determine prices.

Instead of a conclusion

There are many other important equations and formulas in the world that have changed the fate of humanity in general and our personal lives in particular. Among them, the Hodgkin-Huxley model, the Kalman filter and, of course, the Google search engine equation. We hope that we have been able to show how important mathematics is and how invaluable its contribution is to all people.