Numbers surround a person everywhere: dates, apartment and house numbers, telephone numbers, cars, time. Identical numbers on the clock are one of the ways the Universe gives a person a sign. In order to correctly interpret the meaning of the signal, it is important to understand at what period of life it appeared.
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Meanings of numbers
Numerology experts say that numbers have magical powers. They predict fate using numbers and make wishes. Those who believe in the magic of numbers have more than once seen in practice how the number of an apartment or car affects a person’s fate. To manage numbers and be able to decipher their meaning, you need to know the meaning of each number separately.
| Numbers | Decoding |
| "Unit" | Number of confidence, driving energy and strength, new beginning |
| "Deuce" | A sign of restraint, patience and gentleness |
| "Troika" | The number of connections between the present and the future, mental activity and meditation. Symbol of creativity |
| "Four" | Denotes organization, hard work, and activity to achieve goals. The fate of a person determines the stability and strength of his position in society |
| "Five" | Symbolizes prudence, caution, attentiveness |
| "Six" | Indicates moral values: kindness, honesty, truthfulness. Symbolizes successful resolution of conflict situations. In Angelic Numerology, six is not a bad number and has nothing to do with the devil. |
| "Seven" | A symbol of good luck and a sign that promises success in business. Indicates the favor of Fate towards a person. |
| "Eight" | Numerologists interpret eight as the number of changes |
| "Nine" | Symbol of wisdom, development of the inner world, gaining and accumulating experience |
| "Zero" | Strengthens the energy of other numbers, symbolizes infinity, eternity, freedom |
To understand what the numbers “say”, you need to find out their general meaning and compare the interpretation with your own situation. For example, a person is about to start a new business and is accompanied by the number “1”: this means that good luck should be expected. Since "zero" enhances "10" it can also be considered a very good numerological sign.
Psychic Alena Kurilova told the “Everything Will Be Good” channel in more detail about how numbers influence a person’s life.
Angelic numerology
Identical numbers on the clock are considered part of angelic numerology. With the help of numerical messages on the dial, guardians help draw attention to the situation. Therefore, time is one of the most effective ways to communicate with the highest forces.
When they see the same numbers on the clock, people make a wish, believing in the magical power of the cherished minute. If we accept angelic numerology as truth, then the interpretation of paired or mirror symbols is much more complicated.
What does the coincidence of numbers on the clock mean:
- a sign from above - you should be more careful and make a balanced decision;
- an Angel's hint to a question or desire;
- part of the rhythm of life, universal existence, a sign of forward movement;
- happy moment;
- a message from the Universe that you should listen to your intuition.
The coincidence of numbers must be an accident. Deliberately waiting for the same numbers is not related to angelic numerology. Only their spontaneous and unexpected appearance can be considered a sign from above.
Interpretation of coincidences
To decipher the combination of repeating numbers on a clock, not only the designation of the numbers is important, but also the time of their appearance. It is especially worth taking a closer look at electronic displays, which, unlike the dial, show exact digital values: 22:22, 11:11, 16:16, etc. The same numbers on the clock are interpreted taking into account the phase of the moon. A rising sign indicates the future, a falling sign indicates the present or past.
From midnight to early morning
In the period from midnight to 5 am, the same numbers on the clock are deciphered as follows.
| Time | Decoding |
| 00:00 | Sign of Fate about a happy time to fulfill desires |
| 01:01 | There is a chance of receiving favorable news or a lucrative offer from the opposite sex |
| 02:02 | The appearance of a friend or ally who will help solve difficult problems and situations; It’s worth taking a closer look at those around you and especially at new acquaintances |
| 03:03 | There is no need to be afraid of change, the highest powers are on your side, implement your plans, implement your plans |
| 04:04 | A sign of fate about the need to “hold your horses”; in the near future you will need to be patient and wait for a more successful opportunity to implement your plans |
| 05:05 | Believe in your strength, but don’t fuss, changes await you |
From morning to lunch
After waking up, the brain works most actively, the connection with the higher mind intensifies, so the same numbers on the clock are most often a response to thoughts, reasoning, and reflection. Also, repeating numbers in the morning promises success in the business you have started.
Seeing 11:11 on the clock before starting an important task promises success. Do not doubt the decision - fate gives the go-ahead.
During the day
You can find out what the same numbers on the clock mean in the daytime from the table.
Evening time
Signs of fate at this time of day relate to unfinished business, relationships with loved ones, or answers to questions posed during the day.
Mirror numbers
Mirror numbers are endowed with a lesser degree of magical meaning, but if a person sees them often, it is worth paying attention to this. Such coincidences indicate a certain delay in time and space. Perhaps, having started a business, you will have to return to the starting point or change your plan of action.
| Time | Decoding |
| 01:10 | Don’t place high hopes on the near future; results will not come immediately |
| 02:20 | Restrain your emotions, watch your words, there is a chance of saying too much |
| 03:30 | Improving relationships with the opposite sex |
| 04:40 | Not a good day |
| 05:50 | Don't take risks, beware of natural elements |
| 10:01 | A reliable friend will appear in your life |
| 12:21 | The day promises new acquaintances |
| 13:31 | Feel free to make a wish |
| 15:51 | Possible love relationships |
| 20:02 | Time to rest |
| 21:12 | Plan life changes |
| 23:32 | Pay attention to your own health |
Video “What numbers bring good luck: secrets of a numerologist”
Numbers carry positive or negative energy. What numbers can be considered successful, said the author of unique techniques in numerology, author of the book “The Digitized World” Sergei Kuznetsov. Video from the Pravda channel.
§ 6. Numerical and letter expressions. Formula
Addition, subtraction, multiplication, division - arithmetic operations (or arithmetic operations). These arithmetic operations correspond to the signs of arithmetic operations:
+ (read " plus") - sign of the addition operation,
- (read " minus") is the sign of the subtraction operation,
∙ (read " multiply") is the sign of the multiplication operation,
: (read " divide") is the sign of the division operation.
A record consisting of numbers interconnected by arithmetic signs is called numerical expression. Numerical expressions may also contain parentheses. For example, entry 1290 : 2 - (3 + 20 ∙ 15) is a numeric expression.
The result of performing actions on numbers in numerical expression is called the value of a numeric expression. Performing these actions is called calculating the value of a numeric expression. Before writing the value of a numerical expression, put equal sign"=". Table 1 shows examples of numerical expressions and their meanings.
Table 1
A record consisting of numbers and small letters of the Latin alphabet interconnected by signs of arithmetic operations is called literal expression. This entry may contain parentheses. For example, record a+b - 3 ∙c is a literal expression. Instead of letters, you can substitute various numbers into a letter expression. In this case, the meaning of the letters may change, so the letters in the letter expression are also called variables.
By substituting numbers instead of letters into the literal expression and calculating the value of the resulting numerical expression, they find the meaning of a literal expression for given letter values(for given values of variables). Table 2 shows examples of letter expressions.
A literal expression may have no meaning if substituting the values of the letters results in a numeric expression whose value cannot be found for natural numbers. This numerical expression is called incorrect for natural numbers. It is also said that the meaning of such an expression is “ undefined" for natural numbers, and the expression itself "doesn't make sense". For example, the literal expression a-b does not matter when a = 10 and b = 17. Indeed, for natural numbers, the minuend cannot be less than the subtrahend. For example, if you have only 10 apples (a = 10), you cannot give away 17 of them (b = 17)! Table 2 (column 2) shows an example of a literal expression. By analogy, fill out the table completely.
table 2
For natural numbers the expression is 10 -17 incorrect (does not make sense), i.e. the difference 10 -17 cannot be expressed as a natural number. Another example: you cannot divide by zero, so for any natural number b, the quotient b: 0 undefined.
Mathematical laws, properties, some rules and relationships are often written in literal form (i.e., in the form of a literal expression). In these cases, the literal expression is called formula. For example, if the sides of a heptagon are equal a,b,c,d,e,f,g, then the formula (literal expression) to calculate its perimeter p has the form: 
p =a+b+c +d+e+f+g
With a = 1, b = 2, c = 4, d = 5, e = 5, f = 7, g = 9, the perimeter of the heptagon p = a + b + c + d + e + f + g = 1 + 2 + 4 + 5 +5 + 7 + 9 = 33.
With a = 12, b = 5, c = 20, d = 35, e = 4, f = 40, g = 18, the perimeter of the other heptagon is p = a + b + c + d + e + f + g =12 + 5 + 20 + 35 + 4 + 40 + 18= 134.
Block 6.1. Dictionary
Compile a dictionary of new terms and definitions from § 6. To do this, write words from the list of terms below in the empty cells. In the table (at the end of the block), indicate the numbers of the terms in accordance with the numbers of the frames. It is recommended to carefully review § 6 before filling in the cells of the dictionary.
4. The result of performing actions on numbers in numerical expression.
- The value of a numeric expression that is obtained by substituting variables into a literal expression.
- A numeric expression whose value cannot be found for natural numbers.
10.A numerical expression whose value for natural numbers can be found.
- An alphabet whose small letters are used to write alphabetic expressions.
List of terms and definitions
Answer table
Block6 .2. Match
Match the task in the left column with the solution in the right. Write your answer in the form: 1a, 2d, 3b...
IN option 1
IN option 2
Block 3. Facet test. Numeric and alphabetic expressions
Facet tests replace collections of problems in mathematics, but differ favorably from them in that they can be solved on a computer, the solutions can be checked, and the result of the work can be immediately found out. This test contains 70 problems. But you can solve problems by choice; for this there is an evaluation table, which indicates simple tasks and more difficult ones. Below is the test.
- Given a triangle with sides c,d,m, expressed in cm
- Given a quadrilateral with sides b,c,d,m, expressed in m
- The speed of the car in km/h is b, travel time in hours is d
- The distance traveled by the tourist in m hours is With km
- The distance covered by the tourist, moving at speed m km/h is b km
- The sum of two numbers is greater than the second number by 15
- The difference is less than the one being reduced by 7
- A passenger liner has two decks with the same number of passenger seats. In each of the rows of the deck m seats, rows on deck on n more than seats in a row
- Petya is m years old, Masha is n years old, and Katya is k years younger than Petya and Masha together
- m = 8, n = 10, k = 5
- m = 6, n = 8, k = 15
- t = 121, x = 1458
- The meaning of this expression
- The literal expression for the perimeter is
- Perimeter expressed in centimeters
- Formula for the distance s traveled by a car
- Formula for speed v, tourist movement
- Formula for time t, tourist movement
- Distance traveled by the car in kilometers
- Tourist speed in kilometers per hour
- Tourist travel time in hours
- The first number is...
- The subtrahend is equal to...
- Expression for the largest number of passengers that a liner can carry in k flights
- The largest number of passengers that an aircraft can carry in k flights
- Letter expression for Katya's age
- Katya's age
- The coordinate of point B, if the coordinate of point C is t
- The coordinate of point D, if the coordinate of point C is t
- The coordinate of point A, if the coordinate of point C is t
- Length of segment BD on the number line
- Length of segment CA on the number line
- Length of segment DA on the number line
Answers (equal, has the form, undefined):
a)1; b)s=b∙d; at 9; d) 40; d)b+c +d+m; e) 7; g) the expression does not make sense (incorrect) for natural numbers; h) 2 ∙m(m+n) ∙k; And) (m+n) -k; j) 6; l) 15; m) 3760; m)t - 3; o) the figure cannot be a triangle; n) 22; R) t - 3 ∙ 7; c) 0; t) 32; y) 59600; t) 6019; x) 2880; v) 10378; h)1440; w) you cannot divide by zero; y) 13; s) 1800; e) 496; u) 2; i) 12; aa) 14; bb) 5; cc) 35; dd) 79200; her) 1900; LJ) 118; zz) 18; ii) 12800; kk) 98; ll) 1458; mm) v =c:m; nn) 100; oo) 19900; pp)t =b:m; pp) 2520; ss)c +d+m; tt)x; yy) 1579; ff)t+2; xx) 10206; cc) 135; hh)t + 2 ∙ 7; shsh) 7 ∙x; schshch)x - 2; ыы) 7 ∙x - 2 ∙ 7; uh)t+x ∙ 7; yuyu) 10192; yaya)t+x; aaa) 123; bbb) 1456; www) 10327.
TEST INDICATORS. Number of tasks 70, completion time 2 - 3 hours, total points: 1 ∙ 22 + 2 ∙ 24 + 3 ∙ 24 = 142. For the facet test, you can use the following rating scale.
Educational game "Dungeon Treasures"
On the playing field is an illustration for R. Kipling’s book “Mowgli”. Five of the chests have padlocks, and on their reverse sides the number of points the team gets if they manage to “open the chest” is indicated. This number is different for each chest: for wooden - 1 point, for tin - 2, for copper - 3, for silver - 4, for gold - 5. To open the chest, you must complete the “White Cobra task”.
The task is common to all chests
Read how the money in each chest was spent and write a letter expression for that money. Then substitute the values of the variables and calculate the amount of money that was in the chest at first. This number must be entered in the response of the computer version of the game. The answers are under lock and key!
Wooden chest. Was purchased A books for 50 rubles, b paintings at a price of 250 rubles, d chairs for 300 rubles. There are 250 rubles left in the chest. Variable values: a = 40, b = 8, d = 20.
Tin chest.
It was purchased to renovate the school d kg of paint for 120 rubles, k bags of cement at a price of 200 rubles, m lamps at a price of 280 rubles. There was still a sum of money left in the chest, like in a wooden chest, but rounded up to thousands. Values 
variables: d= 12, k = 16, m = 25.
Copper chest. From this chest they took the amount of money in the tin chest, rounded to hundreds. If you add 5,200 rubles to it, then with this money you can buy m tables by price n rubles and 5 computers for the price R rubles Variable values: m = 10,n= 400 (rubles), p = 6000 (rubles).
Silver chest.
From the silver chest they took an amount of money equal to the amount of money in the copper chest rounded to the nearest thousand. Then they reported 12,000 rubles and bought x microscopes by price y rubles and r
chemical kits by price z rubles .
Variable values: x = 15, y = 8600 (rub), r = 16, z = 1500 (rub).
Golden chest. With the money from this chest, the mathematics classroom was repaired, which took an amount of money equal to the money in the silver chest. With the remaining money it was planned to buy for the gym: mats at a price r( rubles) , the balls are not p( rubles), sports uniforms at a price z(rubles). Each of the items k things . However, the price of the ball and uniform increased by m rubles Therefore, I had to take out 5,200 rubles on credit. Variable values: k = 20, r = 3200, m = 200, p = 400, z = 1200.
iʞwɐε ɐн imıqw doɔdʎʞ ǝɯɓǝʚɐн wɐҺɐɓɐε ʞ ıqɯǝʚɯо qɯɐнεʎ ıqƍоɯҺ
Educational game "Leopold the cat's lessons"
Fatty and Genius set up ambushes in various places on the playing field; they are numbered on the field. There are five ambushes in total. Hover your cursor over the ambush number and receive tasks. Enter your answers into the windows on the screen. If the answers are correct, then the ambush has been found, and the mice ask Leopold for forgiveness. In case of an error, the game must be repeated.
Trap No. 1
Identify each of the unshaded shares and enter the answer. Use slashes to write fractions. For example: 1/2, 1/3, 1/4, etc.
Trap No. 2
Convert to Arabic numerals and solve: 
- IX+III = ?
- VI - IV = ?
- II + X1 = ?
- X - V = ?
|
Trap No. 3 Solve the chain
Substitute the values of the variables in your answer. At what value of the variable a is the literal expression 4 ? |
Trap No. 4 Solve the chain
4 becomes incorrect if all variables are natural numbers ? |
Trap No. 5 Solve the chain
Substitute the values of the variables in your answer. At what value of the variable with literal expression 4 becomes incorrect if all variables are natural numbers ? |
Answers to the game "Leopold's Lessons"
Trap 1: 1/2, 1/3, 2/3, 7/8.
Trap 2. 12, 2, 13 5.
Trap 3. 6
Trap 4. 15.
Numerical values of quantities in the text must be indicated with the required degree of accuracy, while in a series of quantities it is necessary to align the number of decimal places. It is unacceptable to give the following series of values: 10; 20; 16.7; 13.14. This series should look like this: 10.00; 20.00; 16.70; 13.14. The text of the work should not contain values in which the number of significant figures is more than three. 86.7897 should not be specified. For use in the text of the work, it is better to round the value to 86.8. It’s even better if the values are expressed in whole numbers. Therefore, in economic calculations, percentages expressed in whole numbers are more often used, which provide sufficient accuracy, and when describing socio-economic processes, per milles are used.
In the text of the work, numerical values of quantities with the designation of units of physical quantities and units of counting should be written in numbers, and a number without designation of physical quantities and units of counting from one to nine should be written in a word. For example: “The selection of documents is carried out five times, and the total amount for monetary documents must be at least 9 rubles.”, “The selection is carried out 15 times.” It is unacceptable to separate a unit of physical quantity from a numerical value (transfer them to different lines or pages), except for units of physical quantities placed in tables.
If the text to characterize an indicator provides a range of numerical values expressed in the same units of measurement, then the measurement units are indicated after the last numerical value of the range, for example: “the number of overpayments in the amount of 100 to 500 rubles.”
If the text of the work contains a number of numerical values expressed in the same units of measurement, then the units of measurement are indicated only after the last numerical value, for example: “200, 300, 4000 rubles.”
Conventional letters, images or signs must comply with those adopted in current legislation or state standards.
Rules for applying formulas
The text of the work usually uses mathematical formulas using the designation of parameters. Before designating the parameter, give its explanation, for example: “pair correlation coefficient r”. Formulas must have continuous numbering in Arabic numerals, which are written at the formula level on the right in parentheses. One formula is designated “(1)”. Numbering of formulas within a chapter of a thesis or coursework question is allowed. In this case, the formula number consists of the chapter or question number and the formula number, separated by a dot, for example: “(3.1)”. References in the text to serial numbers of formulas are given in parentheses, for example, “...in formula (1).”
Explanations of the symbols included in the formula should be given directly below the formula. The values of each character are given on a new line in the order in which they are given in the formula. The first line of the transcript should begin with the word “where” without a colon after it, for example:
where r is the pair correlation coefficient;
X Y- the average value of the product of the factor and the indicator;
* - average value of the indicator;
U - average factor value;
<т, - среднеквадратическое отклонение показателя; - среднеквадратическое отклонение фактора.
It is allowed to move the formula to the next line only on the signs of the operations being performed. In this case, the used character is repeated at the beginning of the next line. When transferring a formula to the multiplication sign, use the “x” sign. The order of presentation of mathematical equations in the text of the work is the same as the formulas.
In medicine and healthcare, signs expressed by numbers are often used, which can take on different numerical values in different units of the population, often repeated in several units. In each given population and in these specific conditions, this feature is characterized by a certain value (level), which differs from the value of this feature in another population, in the presence of other conditions. Pulse, blood pressure, body temperature, duration of temporary disability, length of hospital stay differ (vary) in patients even with the same diagnosis.
The value of the studied characteristic can take either discrete (discontinuous) or continuous numerical values. Examples of discrete quantities in which the values are expressed as integers: the number of children in the family, the number of patients in the ward, the number of bed days, the number of any medical devices in the institution, pulse. Examples of continuously changing quantities, when the values are expressed in fractional quantities, can gradually transform into one another: height, body weight, temperature, blood pressure.
The values obtained during the study are first recorded chaotically, that is, in the order in which the researcher receives them. A series in which the ordering and the corresponding frequencies are compared (by degree of increasing or decreasing) is called variational. Individual quantitative expressions of a characteristic are called options(V), and the numbers showing how often these options are repeated are frequencies(R).
For a generalized numerical characteristic of the characteristic being studied in a population of subjects, average values are calculated, the advantage of which is that one value characterizes a large set of homogeneous phenomena.
There are several types of averages: arithmetic average, geometric average, harmonic average, progressive average, chronological average. In addition to the indicated averages, sometimes special averages of a relative nature - mode and median - are used as generalizing values of a variation series.
Fashion (Mo) is the most frequently repeated option. Median (Me) - the value of the variant dividing the variation series in half; on either side of it there is an equal number of options.
The most commonly used is the arithmetic average. The arithmetic mean, which is calculated in a variation series, where each option occurs only once (or all options occur with the same frequency) is called simple arithmetic mean. It is determined by the formula:
M - arithmetic mean;
V- the value of the variational characteristic;
n is the total number of observations.
If one or more options are repeated in the series under study, then the weighted arithmetic mean is calculated. In this case, the weight of each option is taken into account and the higher the frequency of a given option, the greater its influence on the arithmetic average. This average is calculated using the formula.
Writing the conditions of problems using the notation accepted in mathematics leads to the appearance of so-called mathematical expressions, which are simply called expressions. In this article we will talk in detail about numeric, alphabetic and variable expressions: we will give definitions and give examples of expressions of each type.
Page navigation.
Numerical expressions - what are they?
Acquaintance with numerical expressions begins almost from the very first mathematics lessons. But they officially acquire their name - numerical expressions - a little later. For example, if you follow the course of M.I. Moro, then this happens on the pages of a mathematics textbook for 2 grades. There, the idea of numerical expressions is given as follows: 3+5, 12+1−6, 18−(4+6), 1+1+1+1+1, etc. - this is all numeric expressions, and if we perform the indicated actions in the expression, we will find expression value.
We can conclude that at this stage of studying mathematics, numerical expressions are records with a mathematical meaning made up of numbers, parentheses and addition and subtraction signs.
A little later, after becoming familiar with multiplication and division, records of numerical expressions begin to contain the signs “·” and “:”. Let's give a few examples: 6·4, (2+5)·2, 6:2, (9·3):3, etc.
And in high school, the variety of recordings of numerical expressions grows like a snowball rolling down a mountain. They contain ordinary and decimal fractions, mixed numbers and negative numbers, powers, roots, logarithms, sines, cosines, and so on.
Let's summarize all the information into the definition of a numerical expression:
Definition.
Numeric expression is a combination of numbers, signs of arithmetic operations, fractional lines, signs of roots (radicals), logarithms, notations for trigonometric, inverse trigonometric and other functions, as well as brackets and other special mathematical symbols, compiled in accordance with the rules accepted in mathematics.
Let us explain all the components of the stated definition.
Numerical expressions can involve absolutely any number: from natural to real, and even complex. That is, in numerical expressions one can find
Everything is clear with the signs of arithmetic operations - these are the signs of addition, subtraction, multiplication and division, respectively having the form “+”, “−”, “·” and “:”. Numerical expressions may contain one of these signs, some of them, or all of them at once, and moreover, several times. Here are examples of numerical expressions with them: 3+6, 2.2+3.3+4.4+5.5, 41−2·4:2−5+12·3·2:2:3:12−1/12.
As for parentheses, there are both numeric expressions that contain parentheses and expressions without them. If there are parentheses in a numeric expression, then they are basically
And sometimes brackets in numerical expressions have some specific, separately indicated special purpose. For example, you can find square brackets denoting the integer part of a number, so the numerical expression +2 means that the number 2 is added to the integer part of the number 1.75.
From the definition of a numerical expression it is also clear that the expression may contain , , log , ln , lg , notations or etc. Here are examples of numerical expressions with them: tgπ , arcsin1+arccos1−π/2 and 
.
Division in numerical expressions can be indicated by . In this case, numerical expressions with fractions take place. Here are examples of such expressions: 1/(1+2) , 5+(2 3+1)/(7−2,2)+3 and 
.
As special mathematical symbols and notations that can be found in numerical expressions, we present . For example, let's show a numerical expression with the modulus 
.
What are literal expressions?
The concept of letter expressions is given almost immediately after becoming familiar with numerical expressions. It is entered approximately like this. In a certain numerical expression, one of the numbers is not written down, but instead a circle (or square, or something similar) is placed, and it is said that a certain number can be substituted for the circle. For example, let's look at the entry. If you put, for example, the number 2 instead of a square, you get the numerical expression 3+2. So instead of circles, squares, etc. agreed to write down letters, and such expressions with letters were called literal expressions. Let's return to our example, if in this entry we put the letter a instead of a square, we get a literal expression of the form 3+a.
So, if we allow in a numerical expression the presence of letters that denote certain numbers, then we get a so-called literal expression. Let us give the corresponding definition.
Definition.
An expression containing letters that represent certain numbers is called literal expression.
From this definition it is clear that a literal expression fundamentally differs from a numeric expression in that it can contain letters. Typically, small letters of the Latin alphabet (a, b, c, ...) are used in letter expressions, and small letters of the Greek alphabet (α, β, γ, ...) are used when denoting angles.
So, literal expressions can be composed of numbers, letters and contain all the mathematical symbols that can appear in numeric expressions, such as parentheses, root signs, logarithms, trigonometric and other functions, etc. We emphasize separately that a literal expression contains at least one letter. But it can also contain several identical or different letters.
Now let's give some examples of literal expressions. For example, a+b is a literal expression with the letters a and b. Here is another example of the literal expression 5 x 3 −3 x 2 +x−2.5. And here is an example of a complex literal expression: 
.
Expressions with variables
If in a literal expression a letter denotes a quantity that does not take on one specific value, but can take on different values, then this letter is called variable and the expression is called expression with variable.
Definition.
Expression with variables is a literal expression in which the letters (all or some) denote quantities that take on different values.
For example, let the letter x in the expression x 2 −1 take any natural values from the interval from 0 to 10, then x is a variable, and the expression x 2 −1 is an expression with the variable x.
It is worth noting that there can be several variables in an expression. For example, if we consider x and y to be variables, then the expression 
is an expression with two variables x and y.
In general, the transition from the concept of a literal expression to an expression with variables occurs in the 7th grade, when they begin to study algebra. Up to this point, letter expressions modeled some specific tasks. In algebra, they begin to look at the expression more generally, without reference to a specific problem, with the understanding that this expression fits a huge number of problems.
In conclusion of this point, let us pay attention to one more point: by the appearance of a literal expression it is impossible to know whether the letters included in it are variables or not. Therefore, nothing prevents us from considering these letters as variables. In this case, the difference between the terms “literal expression” and “expression with variables” disappears.
Bibliography.
- Mathematics. 2 classes Textbook for general education institutions with adj. per electron carrier. At 2 p.m. Part 1 / [M. I. Moro, M. A. Bantova, G. V. Beltyukova, etc.] - 3rd ed. - M.: Education, 2012. - 96 p.: ill. - (School of Russia). - ISBN 978-5-09-028297-0.
- Mathematics: textbook for 5th grade. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 pp.: ill. ISBN 5-346-00699-0.
- Algebra: textbook for 7th grade general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
- Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
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