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, literal expressions and expressions with variables: we will give definitions and give examples of expressions of each type.
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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 decimals, 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 arithmetic operations, fractional lines, root signs (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 numbers: 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 made up of numbers, letters and contain everything mathematical symbols, which can be found in numerical 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 let's give an example of a literal expression complex type: 
.
Expressions with variables
If in a literal expression a letter denotes a quantity that does not take one specific value, but can take different meanings, 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 specific task, with the understanding that this expression fits a huge number of problems.
To conclude this point, let us pay attention to one more point: according to appearance It is impossible to know from a literal expression whether the letters 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.
Entries 2 A + 8, 3A + 5b, A 4 – bс are called expressions with variables. Putting numbers instead of letters, we get numerical expressions. General concept expressions with variables are defined in exactly the same way as the concept of a numerical expression, only, in addition to numbers, expressions with variables can also contain letters.
For expressions with a variable, simplifications are also used: do not put parentheses containing only a number or a letter, do not put a multiplication sign between letters, between numbers and letters, etc.
There are expressions with one, two, three, etc. variables. Designate A(X), IN(x, y) etc.
An expression with a variable cannot be called either a statement or a predicate. For example, about expression 2 A+ 5 it cannot be said whether it is true or false, therefore it is not a statement. If instead of a variable A substitute numbers, we get various numerical expressions, which are also not statements, therefore, this expression is also not a predicate.
Each expression with a variable corresponds to a set of numbers, the substitution of which produces a numerical expression that makes sense. This set is called the domain of definition of the expression.
Example. 8: (4 – X) - domain R\(4), because at X= 4 expression 8: (4 – 4) does not make sense.
If the expression contains multiple variables, for example, X And at, then the domain of definition of this expression is understood as a set of pairs of numbers ( A; b) such that when replacing X on A And at on b the result is a numeric expression that has a value.
Example. , the domain of definition is the set of pairs ( A; b) │A≥ b.
Definition. Two expressions with a variable are said to be identically equal if for any value. Variables from the scope of expressions have their corresponding values equal.
That. two expressions A(X), IN(X) are identically equal on the set X, If
1) sets acceptable values the variables in these expressions are the same;
2) for anyone X 0 of their sets of permissible values, the meanings of expressions at X 0 coincide, i.e. A(X 0) = IN(X 0) is a correct numerical equality.
Example. (2 X+ 5) 2 and 4 X 2 + 20X+ 25 – identically equal expressions.
Designate A(X) º IN(X). Note that if two expressions are identically equal on some set E, then they are identically equal on any subset E 1 М E. It should also be noted that the statement about the identity equality of two expressions with a variable is a statement.
If we connect two identically equal expressions on a certain set with an equal sign, we get a sentence that is called an identity on this set.
True numerical equalities are also considered identities. Identities are the laws of addition and multiplication of real numbers, rules for subtracting a number from a sum and a sum from a number, rules for dividing a sum by a number, etc. Identities are also the rules for operations with zero and one.
Replacing an expression with another that is identically equal to it on some set is called identical transformation of this expression.
Example. 7 X + 2 + 3X = 10 X+ 2 - identical transformation, is not an identical transformation on R.
§ 5. Classification of expressions with a variable
1) An expression made up of variables and numbers using only the operations of addition, subtraction, multiplication, and exponentiation is called an integer expression or polynomial.
Example. (3X 2 + 5) ∙ (2X – 3at)
2) Rational is an expression constructed from variables and numbers using the operations of addition, subtraction, multiplication, division, and exponentiation. A rational expression can be represented as a ratio of two whole expressions, i.e. polynomials. Note that integer expressions are a special case of rational ones.
Example. .
3) Irrational is an expression constructed from variables and numbers using the operations of addition, subtraction, multiplication, division, exponentiation, as well as the operation of extracting the root P-th degree.
Solving problems and some expressions does not always lead to pure numerical answers. Even in the case of trivial calculations, one can arrive at a certain construction called an expression with a variable.
For example, consider two practical problems. In the first case, we have a certain factory that produces 5 tons of milk every day. It is necessary to find how much milk the plant produces in p days.
In the second case, there is a rectangle whose width is 5 cm and length p cm. Find the area of the figure.
Of course, if the plant produces five tons per day, then in p days, according to the simplest mathematical logic, it will produce 5p tons of milk. On the other hand, the area of a rectangle is equal to the product of its sides - that is, in in this case, it's 5 rubles. In other words, in two trivial problems with different conditions, the answer is one whole expression - 5p. Such monomials are called expressions with a variable, since in addition to the numerical part they contain a certain letter called an unknown, or variable. Such an element is denoted by lowercase letters of the Latin alphabet, most often x or y, although this is not important.
The peculiarity of a variable is that it can take any value in practice. Substituting different numbers, we will get the final solution for our problems, for example, for the first one:
p = 2 days, the plant produces 5p = 10 tons of milk;
p = 4 days, the plant produces 5p = 20 tons of milk;
Or for the second:
p = 10 cm, the area of the figure is 5p = 50 cm2
p = 20 cm, the area of the figure is 5p = 100 cm2
It is important to understand that p is not a set of some individual values, but the entire set that will mathematically correspond to the conditions of the problem. The main role of a variable is to replace the missing element in a condition. Any mathematical problem must include some constructions and display the relationship between these constructions in the condition. If the value of an object is missing, then a variable is introduced instead. Moreover, it is an abstract replacement of the very element of the condition (the amount of something represented by a number or expression), and not of functional connections.
If we consider an expression of the form 5p as a neutral and independent object, then the value of p in it can take on any values; in fact, p here is equal to the set of all real numbers.
But in our problems, the answer in the form of 5p is subject to certain mathematical restrictions that follow from the conditions. For example, days and days cannot be negative, so p in both problems is always equal to zero or greater than it. In addition, days cannot be fractional - for the first problem, only those p values that are positive integers are valid.
In the first problem: p is equal to the finite set of all positive integers;
In the second problem: p is equal to the finite set of all positive numbers.
Expressions can include two variables at once, for example:
In this case, a binomial is represented by two monomials, each of which has a variable in its composition, and these variables are different, that is, independent of each other. The value of this expression can only be fully calculated if the values of both variables are available. For example, if x = 2 and y = 4, then:
2x + 3y = 4 + 12 = 16 (with x = 2, y = 4)
It is worth noting that in this expression there are no mathematical or logical restrictions on the values of the variable - both x and y belong to the entire set of real numbers.
IN in general terms, the set of all numbers, when substituting them instead of a variable, the expression retains its meaning and validity, is called the domain of definition (or value) of the variable.
In abstract examples not related to real problems, the domain of definition of a variable is most often either equal to the entire set of real numbers or limited to certain constructions, for example, a fraction. As you know, when the divisor is zero, the entire fraction becomes meaningless. Therefore, a variable in an expression of the form:
cannot be equal to five, because then:
7x/(x - 5) = 7x/0 (for x = 5)
And the fraction will lose its meaning. Therefore, for this expression, the variable x has a domain of definition - the set of all numbers except 5.
Our video tutorial also highlights a special case of using variables when they denote a number of the same order. For example, the numbers 54, 30, 78 can be specified through the variable a, or through the construction ab (with a horizontal bar at the top, to distinguish it from the product), where b specifies units (respectively 4, 0, 8), and - tens (respectively, 5, 3, 7).
In algebra lessons at school we come across expressions various types. As you learn new material, recording expressions becomes more diverse and complex. For example, we got acquainted with powers - powers appeared in expressions, we studied fractions - fractional expressions appeared, etc.
For the convenience of describing the material, expressions consisting of similar elements were given specific names in order to distinguish them from the whole variety of expressions. In this article we will get acquainted with them, that is, we will give an overview of the basic expressions studied in algebra lessons at school.
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Monomials and polynomials
Let's start with expressions called monomials and polynomials. At the time of this writing, the conversation about monomials and polynomials begins in 7th grade algebra lessons. The following definitions are given there.
Definition.
Monomials numbers, variables, their powers are called natural indicator, as well as any works compiled from them.
Definition.
Polynomials is the sum of the monomials.
For example, the number 5, the variable x, the power z 7, the products 5 x and 7 x x 2 7 z 7 are all monomials. If we take the sum of monomials, for example, 5+x or z 7 +7+7·x·2·7·z 7, then we get a polynomial.
Working with monomials and polynomials often involves doing things with them. Thus, on the set of monomials, the multiplication of monomials and the raising of a monomial to a power are defined, in the sense that as a result of their execution a monomial is obtained.
Addition, subtraction, multiplication, and exponentiation are defined on the set of polynomials. How these actions are determined and by what rules they are performed, we will talk in the article Actions with Polynomials.
If we talk about polynomials with a single variable, then when working with them, dividing a polynomial by a polynomial has significant practical significance, and often such polynomials have to be represented as a product; this action is called factoring the polynomial.
Rational (algebraic) fractions
In the 8th grade, the study of expressions containing division by an expression with variables begins. And the first such expressions are rational fractions, which some authors call algebraic fractions.
Definition.
Rational (algebraic) fraction is a fraction whose numerator and denominator are polynomials, in particular monomials and numbers.
Here are some examples of rational fractions: and 
. By the way, any ordinary fraction is a rational (algebraic) fraction.
Addition, subtraction, multiplication, division and exponentiation are introduced on a variety of algebraic fractions. How this is done is explained in the article Actions with algebraic fractions.
It is often necessary to perform transformations of algebraic fractions, the most common of which are reduction and reduction to a new denominator.
Rational expressions
Definition.
Expressions with powers (power expressions) are expressions containing degrees in their notation.
Here are some examples of expressions with powers. They may not contain variables, for example, 2 3 , 
. Power expressions with variables also take place: 
and so on.
It wouldn't hurt to familiarize yourself with how it's done. converting expressions with powers.
Irrational expressions, expressions with roots
Definition.
Expressions containing logarithms are called logarithmic expressions.
Examples of logarithmic expressions are log 3 9+lne , log 2 (4 a b) , 
.
Very often, expressions contain both powers and logarithms, which is understandable, since by definition a logarithm is an exponent. As a result, expressions like this look natural: .
To continue the topic, refer to the material converting logarithmic expressions.
Fractions
In this section we will look at the expressions special type- fractions.
The fraction expands the concept. Fractions also have a numerator and denominator located above and below the horizontal fraction line (to the left and right of the slanted fraction line), respectively. Only unlike ordinary fractions, the numerator and denominator can contain not only integers, but also any other numbers, as well as any expressions.
So, let's define a fraction.
Definition.
Fraction is an expression consisting of a numerator and a denominator separated by a fractional line, which represent some numerical or alphabetic expressions or numbers.
This definition allows you to give examples of fractions.
Let's start with examples of fractions whose numerators and denominators are numbers: 1/4, 
, (−15)/(−2) . The numerator and denominator of a fraction can contain expressions, both numerical and alphabetic. Here are examples of such fractions: (a+1)/3, (a+b+c)/(a 2 +b 2), 
.
But the expressions 2/5−3/7 are not fractions, although they contain fractions in their notations.
General expressions
In high school, especially in problems of increased difficulty and problems of group C in the Unified State Exam in mathematics, you will come across expressions of a complex form, containing in their notation simultaneously roots, powers, logarithms, and trigonometric functions, and so on. For example, 
or 
. They appear to fit several types of expressions listed above. But they are usually not classified as one of them. They are considered expressions general view
, and when describing they simply say an expression, without adding additional clarifications.
Concluding the article, I would like to say that if a given expression is cumbersome, and if you are not entirely sure what type it belongs to, then it is better to call it simply an expression than to call it an expression that it is not.
Bibliography.
- 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.
- Mathematics. 6th grade: educational. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
- 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.
- Algebra: 9th grade: educational. for general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2009. - 271 p. : ill. - ISBN 978-5-09-021134-5.
- Algebra and the beginning of analysis: Proc. for 10-11 grades. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M.: Education, 2004. - 384 pp.: ill. - ISBN 5-09-013651-3.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
ALGEBRA
Lessons for 7th grade
Lesson #14
Subject. Expressions with variables
Goal: to improve students’ ability to work with expressions containing variables (calculating the values of expressions, finding the ODZ of expressions with variables).
Lesson type: application of skills.
During the classes
I. Verification homework
@ You should especially carefully check the completion of task No. 2 (to compose an expression with variables) and No. 3 (to find the ODZ of a variable in the expression).
No. 2. The expression looks like: 6n - 50m. If m = 2, n = 30, then
6 30 - 2 50 = 180 - 100 = 80 (k).
Answer. For 80 kopecks.
@ No. 3. For students, the moment of transition from the condition under which the expression does not make sense (the divisor or denominator is equal to zero) to the conditions when the expression makes sense is quite difficult (that is, from the set of any numbers we exclude those values of the variable for which the expression doesn't make sense):
1) 2x - 5 makes sense for any value of x, because it is an integer expression;
2) makes sense for all x except 0;
3) makes sense for all x except x = -3, for x = -3 x + 3 = 0;
4) makes sense for any value of x, because it is an entire expression.
II. Updating of reference knowledge
@ Instead of routine (and not very effective) frontal questioning, you can organize work in pairs (or groups) with such a task.
The given expressions are: ; 25: (3.5 + a); (3.5 + a) : 25.
Compare them and find as many differences as possible. During the presentation of the results of the work, students reproduce the content of the main concepts of the topic:
1. Numeric expressions and expressions with variables.
2. The meaning of numeric expressions and expressions with variables.
3. Expressions that don't make sense
III. Improving skills
@ In this lesson we continue to work on improving students’ skills:
a) calculate the values of expressions with variables;
b) find the values of the variables at which the expression makes sense;
c) compose expressions with certain conditions.
We select a higher level of tasks.
Doing writing exercises
1. Find the value of the expression if:
1) x = 4; in = 1.5;
2) x = -1; y = ;
3) x = 1.4; y = 0;
4) x = 1.3; y = -2.6.
2. It is known that a - b = 6; c = 5. Find the value of the expression:
1) a - b + 3 c ;
3. 2) c (b - a);
4.
3)
;
5. 4) .
6. At what values of the variable does the expression make sense:
1) ;
2) ;
3) ;
4) ;
5) ;
6)
;
7)
?
@ Since students do not yet have the ability to solve equations by factoring polynomials, solve fractional equations, systems of equations, we solve problems using reasoning with approximately the following content: since the variable is in the denominator of the expression (the expression is fractional), then for the expression to make sense, it is necessary that the denominator was not equal to 0. But since x2 cannot be a negative number, the sum of x 2 + 1 cannot equal 0 for any value of x, so x2 + 1 does not equal 0 for any value of x.
Therefore, the expression makes sense for any x (etc.).
7. Write an expression to solve the problem.
a) The perimeter of the rectangle is 16 cm, one of its sides is m cm. What is the area of the rectangle?
b) From two cities, the distance between which is S km, two cars drove towards each other. The speed of one of them is v 1 km/h, and the speed of the second is v 2 km/h. In how many hours will they meet?
8. Write as an expression:
1) the sum of the product of numbers a and b and number c;
2) the difference between the number c and the proportion of numbers a and b;
3) the product of the difference between the numbers x and y and their sum;
4) the share of the sum of a and b and their difference.
IV. Diagnostics of assimilation
Independent work (multi-level)
1. Find the meaning of the expression:
A. 3 x - 5 if x = -1. (2 points)
B., if a = 3.5. (3 6.)
B. 
, if m + n = 8, r = 3. (4 6.)
2. Make up an expression that corresponds to the condition:
A. Difference of numbers 5 and 7b. (2 points)
B. Analysis of the product of the numbers -0.2 and a and the number 0.8. (According to b.)
B. The speed of a boat in still water is v km/h. River flow speed in km/h. How long will it take the boat to travel S km over the course of the river? (4 points)
3. Find at what values of the variable mass the expression makes sense:
A. 2a + 5. (2 b.)
B. . (3 points)
IN. . (4 points)
@ While doing the work, students must choose only one task (A, B, C) from the three proposed. We evaluate accordingly: A - 2 points, B - 3 points; B - 4 points. (The student has the right to choose tasks of different levels, for example No. 1 - A, No. 2 - B, No. 3 - B.)
V. Reflection
We check that tasks are completed correctly. (Students receive a table with solutions and answers and check their work.)
Task No. |
Condition (expression) |
Variable value |
Numeric expression |
Expression value |
Number of points |
|
|
||||||
|
m + n = 8 |
|||||
5a - 7b |
||||||
(-0.2 and -0.8) |
||||||
= -16
