If the value of a numeric expression exists then the expression. Numeric expressions. Comparing Numeric Expressions

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, and variable expressions: we will give definitions and give examples of expressions of each type.

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Numeric expressions - what is it?

Acquaintance with numerical expressions begins almost from the very first lessons of mathematics. But their name - numerical expressions - they officially acquire 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 grade 2. There, the representation of numerical expressions is given as follows: 3+5, 12+1−6, 18−(4+6) , 1+1+1+1+1, etc. - it's all numeric expressions, and if we perform the indicated actions in the expression, then we will find expression value.

It can be concluded that at this stage of the study of mathematics, numerical expressions are called records that have mathematical meaning, composed of numbers, brackets and signs of addition and subtraction.

A little later, after getting acquainted with multiplication and division, the entries of numerical expressions begin to contain the signs "·" and ":". Here are some examples: 6 4 , (2+5) 2 , 6:2 , (9 3):3 etc.

And in high school, the variety of entries for numerical expressions grows like a snowball rolling down a mountain. Common and decimal fractions, mixed numbers and negative numbers, degrees, roots, logarithms, sines, cosines and so on.

Let's summarize all the information in the definition of a numeric expression:

Definition.

Numeric expression is a combination of numbers, signs of arithmetic operations, fractional strokes, root signs (radicals), logarithms, notation of 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 constituent parts of the voiced definition.

Absolutely any numbers can participate in numerical expressions: from natural to real, and even complex. That is, in numerical expressions one can meet

Everything is clear with the signs of arithmetic operations - these are the signs of addition, subtraction, multiplication and division, respectively, having the form "+", "−", "·" and ":". In numerical expressions, one of these characters, some of them, or all at once, and more than once, can be present. 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 brackets, there are both numerical expressions in which there are brackets, and expressions without them. If there are brackets 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 the 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 numeric expression, it is also clear that the expression can contain , , log , ln , lg , designations or etc. Here are examples of numerical expressions with them: tgπ , arcsin1+arccos1−π/2 and .

Division in numeric expressions can be denoted with . In this case, there are numerical expressions with fractions. 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 give. For example, let's show a numerical expression with a modulus .

What are literal expressions?

The concept of literal expressions is given almost immediately after getting acquainted with numerical expressions. It is entered like this. In a certain numerical expression, one of the numbers is not written down, but a circle (or a square, or something similar) is put in its place, and it is said that a certain number can be substituted for the circle. Let's take the entry as an example. If you put, for example, the number 2 instead of a square, then you get a numerical expression 3 + 2. So instead of circles, squares, etc. agreed to write letters, and such expressions with letters were called literal expressions. Let's return to our example, if in this entry instead of a square we put the letter a, then we get a literal expression of the form 3+a.

So, if we allow in a numerical expression the presence of letters that denote some numbers, then we get the so-called literal expression. Let us give an appropriate definition.

Definition.

An expression containing letters that denote some numbers is called literal expression.

From this definition it is clear that fundamentally a literal expression differs from a numeric expression in that it can contain letters. Usually, in literal expressions, small letters of the Latin alphabet are used (a, b, c, ...), and when denoting angles, small letters of the Greek alphabet (α, β, γ, ...).

So, literal expressions can be composed of numbers, letters and contain all mathematical symbols that can be found in numerical expressions, such as brackets, root signs, logarithms, trigonometric and other functions, etc. Separately, we emphasize that a literal expression contains at least one letter. But it can also contain several identical or different letters.

Now we 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 we give an example of a literal expression of a complex form: .

Expressions with variables

If in a literal expression a letter denotes a value that does not take on any one specific value, but can take on various meanings, then this letter is called variable and the expression is called variable expression.

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 in the expression x 2 −1 the letter x can 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 as 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, literal expressions have modeled some specific tasks. In algebra, on the other hand, they begin to look at the expression more generally, without being tied to a specific task, with the understanding that this expression fits a huge number of tasks.

In concluding this paragraph, let us pay attention to one more point: according to appearance literal expression, it is impossible to know 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 cells Proc. for general education institutions with adj. to an electron. carrier. At 2 o'clock, Part 1 / [M. I. Moro, M. A. Bantova, G. V. Beltyukova and others] - 3rd ed. - M.: Education, 2012. - 96 p.: ill. - (School of Russia). - ISBN 978-5-09-028297-0.
• Mathematics: studies. for 5 cells. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 p.: ill. ISBN 5-346-00699-0.
• Algebra: textbook for 7 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 17th ed. - M. : Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
• Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.

When studying the topic of numerical, literal expressions and expressions with variables, it is necessary to pay attention to the concept expression value. In this article, we will answer the question, what is the value of a numeric expression, and what is called the value of a literal expression and an expression with variables with the selected values ​​of the variables. To clarify these definitions, we give examples.

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What is the value of a numeric expression?

Acquaintance with numerical expressions begins almost from the first lessons of mathematics at school. Almost immediately, the concept of “value of a numerical expression” is introduced. It refers to expressions made up of numbers connected by arithmetic signs (+, −, ·, :). Let us give an appropriate definition.

Definition.

The value of a numeric expression- this is the number that is obtained after performing all the actions in the original numeric expression.

For example, consider the numeric expression 1+2 . After executing , we get the number 3 , it is the value of the numerical expression 1+2 .

Often in the phrase “value of a numerical expression”, the word “numerical” is omitted, and they simply say “value of the expression”, since it is still clear which expression is meant.

The above definition of the meaning of an expression also applies to numerical expressions of a more complex form, which are studied in high school. Here it should be noted that one may encounter numerical expressions, the values ​​of which cannot be specified. This is due to the fact that in some expressions it is impossible to perform the recorded actions. For example, therefore we cannot specify the value of the expression 3:(2−2) . Such numerical expressions are called expressions that don't make sense.

Often in practice, it is not so much the numerical expression that is of interest as its value. That is, the task arises, which consists in determining the value of this expression. In this case, they usually say that you need to find the value of the expression. In this article, the process of finding the value of numerical expressions is analyzed in detail. different kind, and considered a lot of examples with detailed descriptions solutions.

Meaning of literal and variable expressions

In addition to numerical expressions, they study literal expressions, that is, expressions in which one or more letters are present along with numbers. Letters in a literal expression can stand for different numbers, and if the letters are replaced by these numbers, then the literal expression becomes a numeric one.

Definition.

The numbers that replace letters in a literal expression are called the meanings of these letters, and the value of the resulting numerical expression is called the value of the literal expression given the values ​​of the letters.

So, for literal expressions, one speaks not just about the meaning of a literal expression, but about the meaning of a literal expression for given (given, indicated, etc.) values ​​of letters.

Let's take an example. Let's take the literal expression 2·a+b . Let the values ​​of the letters a and b be given, for example, a=1 and b=6 . Replacing the letters in the original expression with their values, we get a numerical expression of the form 2 1+6 , its value is 8 . Thus, the number 8 is the value of the literal expression 2·a+b given the values ​​of the letters a=1 and b=6 . If other letter values ​​were given, then we would get the value of the literal expression for those letter values. For example, with a=5 and b=1 we have the value 2 5+1=11 .

In high school, when studying algebra, letters in literal expressions are allowed to take on different meanings, such letters are called variables, and literal expressions are called expressions with variables. For these expressions, the concept of the value of an expression with variables is introduced for the chosen values ​​of the variables. Let's figure out what it is.

Definition.

The value of an expression with variables for the selected values ​​of the variables the value of a numeric expression is called, which is obtained after substituting the selected values ​​of the variables into the original expression.

Let us explain the sounded definition with an example. Consider an expression with variables x and y of the form 3·x·y+y . Let's take x=2 and y=4 , substitute these variable values ​​into the original expression, we get the numerical expression 3 2 4+4 . Let's calculate the value of this expression: 3 2 4+4=24+4=28 . The found value 28 is the value of the original expression with the variables 3·x·y+y with the selected values ​​of the variables x=2 and y=4 .

If you choose other values ​​of variables, for example, x=5 and y=0 , then these selected values ​​of variables will correspond to the value of the expression with variables equal to 3 5 0+0=0 .

It can be noted that sometimes for various chosen values ​​of variables, one can obtain equal values expressions. For example, for x=9 and y=1, the value of the expression 3 x y+y is 28 (because 3 9 1+1=27+1=28 ), and above we showed that the same value is expression with variables has at x=2 and y=4 .

Variable values ​​can be selected from their respective ranges of acceptable values. Otherwise, substituting the values ​​of these variables into the original expression will result in a numerical expression that does not make sense. For example, if you choose x=0 , and substitute that value into the expression 1/x , you get the numeric expression 1/0 , which doesn't make sense because division by zero is undefined.

It only remains to add that there are expressions with variables whose values ​​do not depend on the values ​​of their constituent variables. For example, the value of an expression with a variable x of the form 2+x−x does not depend on the value of this variable, it is equal to 2 for any chosen value of the variable x from its range of valid values, which in this case is the set of all real numbers.

Bibliography.

• Mathematics: studies. for 5 cells. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 p.: ill. ISBN 5-346-00699-0.
• Algebra: textbook for 7 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 17th ed. - M. : Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
• Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.

In pp. 8.2.1 it was shown that algebraic concepts are means of generalization, a language for describing arithmetic operations. The concept of a mathematical expression is of a different nature than the concepts of addition, subtraction, multiplication and division. The relationship between these concepts can be considered the relationship of form and content: mathematical expressions are one of the forms of sign, written designation of arithmetic operations. A numeric expression can also be considered one of the forms of a number, since each numeric expression has a single numeric value - a number.

Expressions appear in teaching mathematics as soon as records of the form 2 + 3, 4 - 3 appear in the first grade when studying actions.

addition and subtraction. Initially, they are called so: addition record, subtraction record. As you know, these entries also have proper names: "sum", "difference", which can be entered in one lesson along with the corresponding actions or after some time. And the concept of expression as a subject of study should be made only after students already have some practical experience with such records. At the same time, the teacher can use the term "expression" in his speech, without requiring the children to use it, but introducing it into the students' passive vocabulary. This is exactly what happens when Everyday life when children hear a new word related to a visually highlighted object. For example, pointing to addition and subtraction entries a few lessons after the introduction of these actions, the teacher says: “Read these entries, these expressions: ...”, “Find in the textbook under No. ... an expression in which three must be subtracted from seven. ...”, “Consider these expressions (shows on the board). Read the one that allows you to find a number 3 greater than 5, in which there is a number 3 greater than 5; 3 less than 5.

When studying numerical expressions in primary school consider the following concepts and methods of action.

Concepts: mathematical expression, numerical expression (expression), types of numerical expressions(in one action and in several actions; with and without brackets; containing actions of one step and actions of two steps); the numeric value of the expression; rules of procedure; relationship comparison.

Ways of action: reading expressions in one or two steps; recording expressions from dictation in one or two steps; determining the course of action; calculation of the value of expressions according to the rules of the order of actions; comparing two numeric expressions; expression conversion - replacing one expression with another equal to it based on the properties of actions.

Introduction of concepts.Lesson introducing the concept of expression it is helpful to start by discussing the notes. What are the records? Why do people write? Why are you learning to write? What notes do we take when studying mathematics? (Children turn to their notebooks, to a textbook, to pre-prepared cards with examples of records from those that students made during the period of study.) What groups can records be divided into when studying mathematics?

As a result of this discussion, we focus on two main groups of records: the record of numbers and the record of arithmetic operations. Records of arithmetic operations, in turn, are divided into two groups: without calculations and with calculations, i.e. of the form 2 + 3 and 2 + 3 = 5. Based on this classification, we inform students that the record of addition and subtraction of the form 2 + 3 and 7 -5, as well as any record made up of such records, for example, 2 + 3-4, 7 - 5 - 1 and the like, it is customary to call (we agreed to call it) mathematical

expression, or just an expression. Further, as with the introduction of other concepts, it is necessary to perform recognition tasks, teaching a universal educational action - recognizing objects related to the concept being studied. The number of recognizable objects should include those that do not have all the common (essential) properties of the concept and therefore do not represent this concept and falling under the concept, but having different variable (insignificant) properties. For example: 17 - 10, 17 - 10 =, 17 -10 = 7, 17 -; 17 - 5 + 4, 23 - 5 - 4, 23 - (5 + 4), 0 + 0, 18-2-2-2-2-2-2, 18-6= 18-3-3 = 15- 3 = 12.

Since the entries, called expressions, have already been used, read and written by students, it is necessary to generalize the ways in which the expressions in question are read. For example, the expression 17 - 10 can be read as "the difference between the numbers 17 and 10", as a task - "subtract 10 from 17", "reduce the number 17 by 10" or "find a number less than seventeen by ten" and by similar names we teach students to write expressions. In the future, the questions: how to read the written expression and how to write the named expression are discussed with the advent of new types of expressions.

In the same lesson where we introduce the concept of an expression, we also introduce the concept expression value - the number resulting from all its arithmetic operations.

To summarize the introduction of concepts and plan further work, it is useful to discuss questions in this lesson or in the following lessons: How many expressions are there? How can one expression be similar to another? How can it be different from another? How are all expressions similar to each other? What can expressions tell us? What can you do with expressions? What do you need (can learn) by studying expressions?

Responding to last question formulate with students learning goals future activities: we can learn and we will learn read and write expressions, find expression values, compare expressions.

Reading and writing expressions. Since expressions are records, one must be able to read them. The main ways of reading are set when introducing actions. You can read the expression as a name, as a list of characters, as a task or question. After studying the relations “less (greater) by”, “less (greater) in” between numbers, expressions are also read as statements or questions about the relationship of equality and inequality. Each way of reading reveals a certain facet of the meaning of the corresponding action or actions. Therefore, it is very useful to encourage different ways reading. The reading pattern is set by the teacher when introducing an action or when considering the corresponding concept, property, or relationship.

The basis of reading any expression is reading the expression in one action. Learning to read happens like learning any

mu reading when performing tasks that require such reading. These can be special tasks: "Read the expressions." Reading is necessary when checking the values ​​of the expression (they read the expression as part of equality), when reporting the results of the comparison. The reverse action is also important: writing an expression by its name or the task it sets, the relation. Students perform such actions when conducting mathematical dictations, specially designed to form the ability to write down expressions or as part of tasks for calculating, comparing, etc. Reading mathematical expressions, learning to read expressions is rather not a goal, but a learning tool - a means of developing speech, a means of deepening understanding meaning of actions.

Let's use examples to show how to read the main types of simple expressions:

1) 2 + 3 add three to two; add the numbers two and three; sum
ma numbers two and three; two plus three; find the sum of numbers two and three;

Find the sum of terms two and three; find a number greater than three
than the number two; two increase by three; first term 2, second
term 3, find the sum;

2) 5 - 3 out of five subtract (in no case “subtract 1“!) Three;

The difference between the numbers five and three; five minus three; find the difference
the numbers five and three; minuend five, subtract three, find times
ness; find a number three less than five; five reduce
on three;

3) 2 3 two take the summand three times; take two three times;

Two times three; product of numbers two and three; first
multiplier two, the second - three, find the product; find product
keeping numbers two and three; twice three, three times two; two increase
three times; find a number three times greater than two; first mono
resident two, second three, find the product;

4) 12:4 twelve divided by four; quotient of twelfth
tsat and four private twelve and four); quotient of division
twelve by four; divisible twelve, divisor four, find
quotient (for 13:4 - find the quotient and the remainder); decrease 12 in th
three times; find a number four times less than twelve.

Reading expressions containing more than two actions causes certain difficulties for younger students. In the planned subject results, therefore, the ability to read such expressions can

1 "TAKE OFF, ... 1. whom (what). Take from someone. by force, to deprive someone of something. O. money. O. son. Oh hope. O. someone has their time.(trans.: make someone spend time on something). O. someone's life.(kill). 2. what. Absorb, consume something. The work took a lot of strength from someone. 3. what. Set aside, separate from. O. ladder from the wall.... ". [Ozhegov S.I. Dictionary/ S. I. Ozhegov, N. Yu. Shvedova. - M., 1949 -1994.]

can be placed in an elevated or high level mastery of mathematical speech. Expressions are called with two or more actions on the last action, the components of which are considered expressions. However, some kinds of expressions are included in the texts of the rules. Knowledge of the verbal formulations of the rules also means knowledge of the ways (methods) of reading. For example, the distributive property of multiplication with respect to addition or the rule of multiplying a sum by a number in the very name of the rule gives the name of an expression of the form ( BUT+ □ ) · th. And in the formulation of the property, two types of expressions are called: “The product of a sum by a number is equal to the sum of the products of each term by this number.” Methods for reading expressions in two or more actions can be specified by algorithmic prescriptions. Subsection 4.2 provides an example of such an algorithm. Mastering the ways of reading such expressions occurs when performing the same types of tasks as when learning to read expressions in one action.

Finding the value of expressions. Procedure rules. Since the beginning of the study of arithmetic operations and the appearance of expressions, the rule has been implicitly accepted: actions must be performed from left to right in the order they are written. The problem of the order of actions is revealed when there are difficulties in denoting certain objective situations by expression. For example, you need to take 7 blue dice, 2 fewer white dice and find out how many dice are taken in total. We perform almost all actions, denoting the number of cubes with numbers, and actions with signs of arithmetic operations. Let's count 7 blue cubes. To take 2 fewer white ones, let's move two blue dice away for a while and, by pairing, take as many white dice as there are blue ones without two. Combine white and blue cubes. Our actions with cubes in arithmetic notation: 7 + 7-2. But in such a record, the actions must be performed in the order of the record, and these are not the actions for which we made the record! There is a contradiction. We need that first 2 is subtracted from 7 (we find out the required number of white cubes), and then the result of subtracting 7 and 2 is added to 7 - the number of blue cubes. How to be?

The way out of this and similar situations can be as follows: you need to somehow select the action or actions that need to be performed not in the order of writing from left to right in the expression record. And there is such a way. This is parentheses, which are just invented for situations where actions in an expression need to be performed out of order from left to right. With brackets, the mathematical notation of our practical action with dice will look like this: 7 + (7 - 2). Actions written in brackets are usually performed first. To master and assign this property of brackets, we compose different expressions with students, put brackets in them in different ways, calculate, compare the results. Replacement

tea: sometimes changing the order of actions does not change the value of the expression, and sometimes it does. For example, 12 - 6 + 2 = 8, (12 - 6) + 2 = 8, 12 - (6 + 2) = 4.

When introducing brackets, the generally accepted rules for the order of actions are clearly not yet studied, although two rules are already practically applied: a) if there is only addition and subtraction in an expression without brackets, then the actions are performed in the order they are written from left to right; b) actions in parentheses are performed first.

Again, the problem of the order of operations becomes acute after the appearance of expressions containing the operations of multiplication and (or) division and the operations of addition and (or) subtraction. During this period, the need for order rules can be recognized by students and it is during this period that students can already discuss this problem, formulate and understand the generally accepted formulations of order rules.

You can create an understanding of the need for such rules by experimenting with a multi-step expression. For example, let's calculate the value of the expression 7 - 3 2 + 15: 5, performing actions in three different sequences: 1) - + (in the order of writing); 2) - + ·: (first addition and subtraction, then multiplication and division); 3) ·: - + (first multiplication and division, then addition and subtraction). As a result, we get three different values: 1) 4 (remaining 3); 2) 13 (rest. 3); 3) 6. Discussing the situation with the students, we conclude: it is necessary to agree and accept only one sequence as a generally accepted rule of action. And since the values ​​of expressions were calculated even before us, and even more than one hundred years, then, probably, such agreements already exist. We find them in the textbook.

Next, we discuss with students the need for knowledge of these rules and the ability to apply them. Having justified such a need for themselves, students may well try to determine for themselves the types academic work, performing which, they will be able to remember the rules and learn to follow them accurately. Such a definition of the types of educational work can be outlined in group work, and in the same lesson some types of such work can be performed. In the process of group work, students get acquainted with the content of the corresponding pages of the textbook and notebook for independent work to the textbook, they can supplement the learning tasks themselves, complete some of them, test themselves and then make a group work report on what they have already mastered as a result of group work. For example: “In our group, everyone learned to determine the order of actions in expressions without brackets in three or four actions, referring to the text of the rule in the textbook, and to designate this order with action numbers above the action signs in the expression.” Then the goal is to learn how to find the meanings of such "big" expressions - in three or four or more actions in many lessons for students.

students perform learning activities to achieve it. The method of finding the values ​​of a compound expression can be represented in an algorithmic form.

Algorithm for finding the value of a numeric expression(set by verbal prescription in the form of a list of steps).

1. If a the expression contains parentheses, then perform actions in brackets as in an expression without brackets. 2. If a there are no parentheses in the expression, then: a) if in the expression only addition and (or) subtraction or only multiplication and (or) division, then perform these steps in order from left to right; b) if the expression contains actions from the addition - subtraction group and from the multiplication - division group, then first perform multiplication and division in order from left to right, then Perform addition and subtraction in order from left to right. 3. The result of the last action is called the value of the expression.

A special role in learning is played by methods for finding the values ​​of expressions based on the properties of actions. Such methods consist in the fact that first the expressions are transformed based on the properties of the actions, and only then the rules of the order of actions are applied. For example, you need to find the value of the expression: 23 + 78 + 77. According to the rules of the order of actions, you must first add 78 to 23, and add 17 to the result. However, the commutative and associative properties or the rule “You can add numbers in any order” allows us to replace this expression equal to it with another order of operations 23 + 77 + 78. Having performed the actions in accordance with the rules of the order of operations, we easily get the result 100 + 78 = 178.

Actually mathematical activity, the mathematical development of students occurs precisely when they are looking for rational or original ways transformations of expressions with subsequent convenient calculations. Therefore, it is necessary to develop a habit among students in any non-calculative calculations, to look for ways to simplify calculations, transform expressions so that instead of cumbersome, ugly calculations, the desired value of the expression is found using simple and beautiful cases of calculation. Tasks are formulated for this as follows: "Compute in a convenient (or rational) way ...".

Finding the values ​​of literal expressions - an important skill that forms ideas about the variable and is the basis for understanding the functional dependence in the future. A very convenient form of tasks for finding the values ​​of literal expressions and for observing the dependence of the value of an expression on the values ​​of the letters included in it is tabular. For example, according to Table. 8.1 students can establish a number of dependencies: if the values a are consecutive numbers, then the values 2a there are consistent even numbers, and the values 3a - every third number starting from value 3a at the smallest value a and etc.

Table 8.1

Expression comparison. Relations that connect the values ​​of expressions are transferred to expressions. The main comparison is finding the values ​​of compared expressions and comparison of expression values. Comparison algorithm:

1. Find the values ​​of the compared expressions. 2. Compare the received numbers. 3. Transfer the result of comparing numbers to expressions. If necessary, put the appropriate sign between the expressions. End.

As well as when finding the values ​​of expressions, comparison methods based on the properties of arithmetic operations, the properties of numerical equalities and inequalities are valued, since such a comparison requires deductive reasoning and therefore ensures the development of logical thinking.

For example, you need to compare 73 + 48 and 73 + 50. The property is known: “If one term is increased or decreased by several units, then the sum will increase or decrease by the same number of units.” Therefore, the value of the first expression is less than the value of the second, which means that the first expression is less than the second, and the second is greater than the first. We compared expressions without finding the values ​​of the expressions, without performing any arithmetic operations, by applying the well-known property of addition. For such cases, it is useful to compare expressions written using generic symbology. Compare expressions. © + F and © + (F+ 4), © + F and © + (F- 4).

Interesting methods of comparison are based on the transformation of the compared expressions - replacing them with equal ones. For example: 18 4 and 18 + 18 + 18 + 18; 25 (117 - 19) and 25 117 - 19; 25 (117 -119) and 25 117 - - 19 117, etc. By transforming the expression in one part based on the properties of actions, we get expressions that can already be compared by comparing numbers - components of the same action.

Example. 126 + 487 and 428 + 150. For comparison, we use the commutative property. We get: 487 + 126 and 428 and 150. Let's transform the first expression: 487 + 132 = (483 + 4) + (130 - 4) = 483 + 4 + 130 -4 = 483 + 130 = (483 - 20) + (130 + 20) = 463 + 150. Now you need to compare the expressions 463 + 150 and 428 + 150.

Formula

Addition, subtraction, multiplication, division - arithmetic operations (or arithmetic operations). These arithmetic operations correspond to the signs of arithmetic operations:

+ (read " plus") - the sign of the addition operation,

- (read " minus") - the sign of the subtraction operation,

∙ (read " multiply") - the sign of the multiplication operation,

: (read " divide") is the sign of the division operation.

A record consisting of numbers interconnected by signs of arithmetic operations is called numerical expression. Parentheses can also be present in a numeric expression. For example, entry 1290 : 2 - (3 + 20 ∙ 15) is a numeric expression.

The result of performing operations on numbers in a 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 numeric expression, put equal sign"=". Table 1 shows examples of numeric expressions and their meanings.

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, the entry a +b - 3 ∙c is a literal expression. Instead of letters in a literal expression, you can substitute various numbers. In this case, the meaning of the letters can change, so the letters in the literal expression are also called variables.

Substituting numbers instead of letters into the literal expression and calculating the value of the resulting numerical expression, they find the value of a literal expression given the values ​​of the letters(for the given values ​​of the variables). Table 2 shows examples of literal expressions.

A literal expression may not have a value if, by substituting the values ​​of the letters, a numeric expression is obtained whose value for natural numbers cannot be found. Such a numerical expression is called incorrect for natural numbers. They also say that the meaning of such an expression " undefined" for natural numbers, and the expression itself "doesn't make sense". For example, the literal expression a-b does not matter for a = 10 and b = 17. Indeed, for natural numbers, the minuend cannot be less than the subtrahend. For example, having 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 in the table completely.

For natural numbers, the expression 10 -17 wrong (doesn't 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) for calculating its perimeter p looks like:

p=a +b +c +d+e +f +g

For a = 1, b = 2, c = 4, d = 5, e = 5, f = 7, g = 9, the perimeter of the heptagon is p = a + b + c + d + e + f + g = 1 + 2 + 4 + 5 +5 + 7 + 9 = 33.

For a = 12, b = 5, c = 20, d = 35, e = 4, f = 40, g = 18, the perimeter of another heptagon is p = a + b + c + d + e + f + g = 12 + 5 + 20 + 35 + 4 + 40 + 18 = 134.

Block 1. Dictionary

Make a dictionary of new terms and definitions from the paragraph. To do this, in the empty cells, enter the words from the list of terms below. In the table (at the end of the block), indicate the numbers of terms in accordance with the numbers of the frames. It is recommended to carefully review the paragraph before filling in the cells of the dictionary.

1. Operations: addition, subtraction, multiplication, division.

2. Signs "+" (plus), "-" (minus), "∙" (multiply, " : " (divide).

3. A record consisting of numbers that are interconnected by signs of arithmetic operations and in which brackets may also be present.

4. The result of performing operations on numbers in numerical terms.

5. The sign before the value of a numeric expression.

6. An entry consisting of numbers and small letters of the Latin alphabet, interconnected by signs of arithmetic operations (brackets may also be present).

7. Common name letters in a literal expression.

8. The value of a numeric expression, which is obtained by substituting variables into a literal expression.

9. Numeric expression whose value for natural numbers cannot be found.

10. Numeric expression whose value for natural numbers can be found.

11. Mathematical laws, properties, some rules and ratios written in literal form.

12. An alphabet whose small letters are used to write literal expressions.

Block 2. Match

Match the task in the left column with the solution in the right. Write down the answer in the form: 1a, 2d, 3b ...

Block 3. Facet test. Numeric and alphabetic expressions

Faceted tests replace collections of problems in mathematics, but compare favorably with them in that they can be solved on a computer, check solutions and immediately find out the result of the work. This test contains 70 tasks. But you can solve problems by choice, for this there is an evaluation table, which indicates simple tasks and more difficult. Below is a test.

1. Given a triangle with sides c,d,m, expressed in cm
2. Given a quadrilateral with sides b,c,d,m expressed in m
3. The speed of the car in km/h is b, travel time in hours is d
4. Distance traveled by a tourist m hours, is with km
5. The distance traveled by a tourist moving at a speed m km/h is b km
6. The sum of two numbers is greater than the second number by 15
7. The difference is less than the reduced by 7
8. A passenger liner has two decks with the same number of passenger seats. In each of the deck rows m seats, rows on deck on n more than seats in a row
9. Petya is m years old Masha is n years old, and Katya is k years younger than Petya and Masha together
10. m=8, n=10, k=5
11. m=6, n=8, k=15
12. t=121, x=1458

1. The value of this expression
2. The literal expression for the perimeter is
3. Perimeter expressed in centimeters
4. Formula for the distance s traveled by the car
5. Velocity formula v, tourist movements
6. Time formula t, tourist movements
7. Distance traveled by car in kilometers
8. Tourist speed in kilometers per hour
9. Travel time in hours
10. The first number is...
11. Subtracted equals….
12. Expression for most passengers that the liner can carry for k flights
13. The largest number of passengers that an airliner can carry in k flights
14. Letter expression for Katya's age
15. Katya's age
16. The coordinate of point B, if the coordinate of point C is t
17. The coordinate of point D, if the coordinate of point C is t
18. The coordinate of point A, if the coordinate of point C is t
19. The length of the segment BD on the number line
20. The length of the segment CA on the number line
21. The length of the segment DA on the number line

Numeric expression is any record of numbers, arithmetic signs and brackets. A numeric expression can also consist of just one number. Recall that the basic arithmetic operations are "addition", "subtraction", "multiplication" and "division". These actions correspond to the signs "+", "-", "∙", ":".

Of course, in order for us to get a numerical expression, the notation from numbers and arithmetic signs must be meaningful. So, for example, such an entry 5: + ∙ cannot be called a numeric expression, since this is a random set of characters that does not make sense. On the contrary, 5 + 8 ∙ 9 is already a real numerical expression.

The value of a numeric expression.

Let's say right away that if we perform the actions indicated in a numerical expression, then as a result we will get a number. This number is called the value of a numeric expression.

Let's try to calculate what we get as a result of performing the actions of our example. According to the order of performing arithmetic operations, we first perform the multiplication operation. Multiply 8 by 9. We get 72. Now we add 72 and 5. We get 77.
So, 77 - meaning numerical expression 5 + 8 ∙ 9.

Numerical equality.

You can write it this way: 5 + 8 ∙ 9 = 77. Here we first used the sign "=" ("Equal"). Such a notation, in which two numerical expressions are separated by the sign "=", is called numerical equality. Moreover, if the values ​​of the left and right parts of the equality are the same, then the equality is called faithful. 5 + 8 ∙ 9 = 77 is the correct equality.
If we write 5 + 8 ∙ 9 = 100, then this will already be false equality, since the values ​​of the left and right sides of this equality no longer coincide.

It should be noted that in a numeric expression, we can also use parentheses. Parentheses affect the order in which actions are performed. So, for example, we modify our example by adding brackets: (5 + 8) ∙ 9. Now we first need to add 5 and 8. We get 13. And then multiply 13 by 9. We get 117. Thus, (5 + 8) ∙ 9 = 117.
117 – meaning numerical expression (5 + 8) ∙ 9.

To correctly read an expression, you need to determine which action is performed last to calculate the value of a given numeric expression. So, if the last action is a subtraction, then the expression is called "difference". Accordingly, if the last action is the sum - "sum", division - "private", multiplication - "product", exponentiation - "degree".

For example, the numerical expression (1 + 5) (10-3) reads like this: “the product of the sum of the numbers 1 and 5 and the difference between the numbers 10 and 3.”

Examples of numeric expressions.

Here is an example of a more complex numeric expression:

\[\left(\frac(1)(4)+3.75 \right):\frac(1.25+3.47+4.75-1.47)(4\centerdot 0.5)\]

In this numerical expression, prime numbers, ordinary and decimal fractions are used. The symbols for addition, subtraction, multiplication and division are also used. The fraction bar also replaces the division sign. With apparent complexity, finding the value of this numerical expression is quite simple. The main thing is to be able to perform operations with fractions, as well as carefully and accurately do calculations, observing the order of actions.

In brackets we have the expression $\frac(1)(4)+3.75$ . Let's transform decimal 3.75 in ordinary.

$3.75=3\frac(75)(100)=3\frac(3)(4)$

So, $\frac(1)(4)+3.75=\frac(1)(4)+3\frac(3)(4)=4$

Further, in the numerator of the fraction \[\frac(1.25+3.47+4.75-1.47)(4\centerdot 0.5)\] we have the expression 1.25 + 3.47 + 4.75-1.47. To simplify this expression, we apply the commutative law of addition, which says: "The sum does not change from a change in the places of the terms." That is, 1.25+3.47+4.75-1.47=1.25+4.75+3.47-1.47=6+2=8.

In the denominator of the fraction, the expression $4\centerdot 0,5=4\centerdot \frac(1)(2)=4:2=2$

We get $\left(\frac(1)(4)+3.75 \right):\frac(1.25+3.47+4.75-1.47)(4\centerdot 0.5)=4: \frac(8)(2)=4:4=1$

When do numeric expressions not make sense?

Let's consider one more example. In the denominator of a fraction $\frac(5+5)(3\centerdot 3-9)$ the value of the expression $3\centerdot 3-9$ is 0. And, as we know, division by zero is impossible. Therefore, the fraction $\frac(5+5)(3\centerdot 3-9)$ has no value. Numeric expressions that don't have a meaning are said to "have no meaning".

If we use letters in addition to numbers in a numerical expression, then we will have an algebraic expression.

Publication date: 08/30/2014 10:58 UTC

• Geometry, solution book for the book by Balayan E.N. "Geometry. Tasks on ready-made drawings for preparing for the OGE and the Unified State Examination: Grades 7-9, Grade 7, Balayan E.N., 2019
• Geometry trainer, Grade 7, to the textbook by Atanasyan L.S. etc. “Geometry. Grades 7-9”, Federal State Educational Standard, Glazkov Yu.A., Yegupova M.V., 2019