Consider two equalities:

1. a 12 * a 3 = a 7 * a 8

This equality will hold for any value of the variable a. The range of valid values ​​for that equality will be the entire set of real numbers.

2. a 12: a 3 = a 2 * a 7 .

This inequality will hold for all values ​​of the variable a, except for a equal to zero. The range of acceptable values ​​for this inequality will be the entire set of real numbers, except for zero.

About each of these equalities, it can be argued that it will be true for any admissible values ​​of the variables a. Such equations in mathematics are called identities.

The concept of identity

An identity is an equality that is true for any admissible values ​​of the variables. If any valid values ​​are substituted into this equality instead of variables, then the correct numerical equality should be obtained.

It is worth noting that true numerical equalities are also identities. Identities, for example, will be properties of actions on numbers.

3. a + b = b + a;

4. a + (b + c) = (a + b) + c;

6. a*(b*c) = (a*b)*c;

7. a*(b + c) = a*b + a*c;

11. a*(-1) = -a.

If two expressions for any admissible variables are respectively equal, then such expressions are called identically equal. Below are some examples of identically equal expressions:

1. (a 2) 4 and a 8 ;

2. a*b*(-a^2*b) and -a 3 *b 2 ;

3. ((x 3 *x 8)/x) and x 10 .

We can always replace one expression with any other expression identically equal to the first one. Such a replacement will be an identical transformation.

Identity Examples

Example 1: Are the following equalities identities:

1. a + 5 = 5 + a;

2. a*(-b) = -a*b;

3. 3*a*3*b = 9*a*b;

Not all of the above expressions will be identities. Of these equalities, only 1,2 and 3 equalities are identities. Whatever numbers we substitute in them, instead of the variables a and b, we still get the correct numerical equalities.

But 4 equality is no longer an identity. Because not for all admissible values ​​this equality will be fulfilled. For example, with the values ​​a = 5 and b = 2, you get the following result:

This equality is not true, since the number 3 does not equal the number -3.

Both parts of which are identically equal expressions. Identities are divided into letter and number.

Identity expressions

The two algebraic expressions are called identical(or identically equal), if for any numerical values ​​of the letters they have the same numerical value. These are, for example, the expressions:

x(5 + x) and 5 x + x 2

Both presented expressions, for any value x will be equal to each other, so they can be called identical or identically equal.

Numeric expressions that are equal to each other can also be called identical. For example:

20 - 8 and 10 + 2

Letter and number identities

Letter identity is an equality that is valid for any values ​​of the letters included in it. In other words, such an equality, in which both parts are identically equal expressions, for example:

(a + b)m = am + bm
(a + b) 2 = a 2 + 2ab + b 2

Numeric Identity- this is an equality containing only numbers expressed in digits, in which both parts have the same numerical value. For example:

4 + 5 + 2 = 3 + 8
5 (4 + 6) = 50

Identity transformations of expressions

All algebraic operations are the transformation of one algebraic expression into another, identical to the first.

When calculating the value of an expression, opening brackets, taking the common factor out of brackets, and in a number of other cases, some expressions are replaced by others that are identically equal to them. The replacement of one expression by another, identically equal to it, is called identical transformation of the expression or simply expression conversion. All expression conversions are performed based on the properties of operations on numbers.

Consider the identical transformation of the expression using the example of taking the common factor out of brackets:

10x - 7x + 3x = (10 - 7 + 3)x = 6x

After we have dealt with the concept of identities, we can proceed to the study of identically equal expressions. The purpose of this article is to explain what it is and to show with examples which expressions will be identically equal to others.

Identical Equal Expressions: Definition

The concept of identically equal expressions is usually studied together with the concept of identity itself in the framework of a school algebra course. Here is a basic definition taken from one textbook:

Definition 1

identically equal each other there will be such expressions, the values ​​of which will be the same for any possible values ​​of the variables included in their composition.

Also, such numerical expressions are considered identically equal, which will correspond to the same values.

This is a fairly broad definition, which will be true for all integer expressions, the meaning of which does not change when the values ​​of the variables change. However, later it becomes necessary to clarify this definition, since there are other types of expressions besides integers that will not make sense with certain variables. This gives rise to the concept of the admissibility and inadmissibility of certain values ​​of variables, as well as the need to determine the range of admissible values. Let us formulate a refined definition.

Definition 2

Identical equal expressions are those expressions whose values ​​are equal to each other for any valid values ​​of the variables included in their composition. Numeric expressions will be identically equal to each other, provided that the values ​​are the same.

The phrase "for any admissible values ​​of the variables" indicates all those values ​​of the variables for which both expressions will make sense. We will explain this position later, when we give examples of identically equal expressions.

You can also specify the following definition:

Definition 3

Identical equal expressions are expressions located in the same identity on the left and right sides.

Examples of expressions that are identically equal to each other

Using the definitions given above, consider a few examples of such expressions.

Let's start with numeric expressions.

Example 1

Thus, 2 + 4 and 4 + 2 will be identically equal to each other, since their results will be equal to (6 and 6).

Example 2

In the same way, the expressions 3 and 30 are identically equal: 10 , (2 2) 3 and 2 6 (to calculate the value of the last expression, you need to know the properties of the degree).

Example 3

But the expressions 4 - 2 and 9 - 1 will not be equal, since their values ​​are different.

Let's move on to examples of literal expressions. A + b and b + a will be identically equal, and this does not depend on the values ​​of the variables (the equality of expressions in this case is determined by the commutative property of addition).

Example 4

For example, if a is 4 and b is 5, the results will still be the same.

Another example of identically equal expressions with letters is 0 · x · y · z and 0 . Whatever the values ​​of the variables in this case, when multiplied by 0 , they will give 0 . The unequal expressions are 6 x and 8 x because they won't be equal for any x .

In the event that the ranges of allowable values ​​of the variables will coincide, for example, in the expressions a + 6 and 6 + a or a b 0 and 0, or x 4 and x, and the values ​​of the expressions themselves will be equal for any variables, then such expressions are considered identically equal. So, a + 8 = 8 + a for any value of a, and a · b · 0 = 0 too, since multiplying any number by 0 results in 0. The expressions x 4 and x will be identically equal for any x from the interval [ 0 , + ∞) .

But the scope of a valid value in one expression may differ from the scope of another.

Example 5

For example, let's take two expressions: x − 1 and x - 1 · x x . For the first of them, the range of admissible values ​​of x will be the entire set real numbers, and for the second - the set of all real numbers, except for zero, because then we get 0 in the denominator, and such a division is not defined. These two expressions have a common range, formed by the intersection of two separate ranges. It can be concluded that both expressions x - 1 · x x and x − 1 will make sense for any real values ​​of the variables, except for 0 .

The basic property of the fraction also allows us to conclude that x - 1 x x and x - 1 will be equal for any x that is not 0 . This means that these expressions will be identically equal to each other on the general range of admissible values, and for any real x one cannot speak of identical equality.

If we replace one expression with another that is identically equal to it, then this process is called identity transformation. This concept is very important, and we will talk about it in detail in a separate article.

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Having got an idea about identities , it is logical to move on to acquaintance with . In this article, we will answer the question of what identically equal expressions are, and also, using examples, we will figure out which expressions are identically equal and which are not.

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What are identically equal expressions?

The definition of identically equal expressions is given in parallel with the definition of identity. This happens in algebra class in 7th grade. In the textbook on algebra for 7 classes, the author Yu. N. Makarychev gives the following wording:

Definition.

are expressions whose values ​​are equal for any values ​​of the variables included in them. Numeric expressions that correspond to the same values ​​are also called identically equal.

This definition is used up to class 8, it is valid for integer expressions, since they make sense for any values ​​of the variables included in them. And in grade 8, the definition of identically equal expressions is specified. Let's explain what it is connected with.

In grade 8, the study of other types of expressions begins, which, unlike integer expressions, may not make sense for some values ​​of variables. This makes it necessary to introduce definitions of admissible and invalid values ​​of variables, as well as the range of admissible values ​​of the ODV of a variable, and as a result, to clarify the definition of identically equal expressions.

Definition.

Two expressions whose values ​​are equal for all admissible values ​​of their variables are called identically equal expressions. Two numeric expressions that have the same value are also said to be identically equal.

IN this definition identically equal expressions, it is worth clarifying the meaning of the phrase "for all admissible values ​​of the variables included in them." It implies all such values ​​of variables for which both identically equal expressions simultaneously make sense. This idea will be clarified in the next section by considering examples.

The definition of identically equal expressions in A. G. Mordkovich's textbook is given a little differently:

Definition.

Identical equal expressions are expressions on the left and right sides of the identity.

In meaning, this and the previous definitions coincide.

Examples of identically equal expressions

The definitions introduced in the previous subsection allow us to bring examples of identically equal expressions.

Let's start with identically equal numerical expressions. The numeric expressions 1+2 and 2+1 are identically equal because they correspond to equal values ​​3 and 3 . The expressions 5 and 30:6 are also identically equal, as are the expressions (2 2) 3 and 2 6 (the values ​​of the last expressions are equal due to ). But the numerical expressions 3+2 and 3−2 are not identically equal, since they correspond to the values ​​5 and 1, respectively, but they are not equal.

Now we give examples of identically equal expressions with variables. These are the expressions a+b and b+a . Indeed, for any values ​​of the variables a and b, the written expressions take the same values ​​(which follows from the numbers). For example, with a=1 and b=2 we have a+b=1+2=3 and b+a=2+1=3 . For any other values ​​of the variables a and b, we will also get equal values ​​of these expressions. The expressions 0·x·y·z and 0 are also identically equal for any values ​​of the variables x , y and z . But the expressions 2 x and 3 x are not identically equal, since, for example, at x=1 their values ​​are not equal. Indeed, for x=1, the expression 2 x is 2 1=2 , and the expression 3 x is 3 1=3 .

When the areas of allowable values ​​of variables in expressions coincide, as, for example, in the expressions a+1 and 1+a , or a b 0 and 0 , or and , and the values ​​of these expressions are equal for all values ​​of variables from these areas, then here everything is clear - these expressions are identically equal for all admissible values ​​of the variables included in them. So a+1≡1+a for any a , the expressions a b 0 and 0 are identically equal for any values ​​of the variables a and b , and the expressions and are identically equal for all x from ; ed. S. A. Telyakovsky. - 17th ed. - M. : Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.

Two expressions are called identically equal on a set if they make sense on that set and all their corresponding values ​​are equal.

An equality in which the left and right sides are identically equal expressions is called identity.

Replacing one expression with another, identically equal to it on a given set, is called identical transformation of the expression.

A task. Find the scope of expression .

Solution. Since the expression is a fraction, to find its scope, you need to find those values ​​of the variable X, for which the denominator vanishes, and eliminate them. By solving the equation X 2 - 9 = 0, we find that X= -3 and X= 3. Therefore, the domain of this expression consists of all numbers other than -3 and 3. If we denote it by X, then we can write:

X= (-¥; -3) È (-3; 3) È (3; +¥).

A task. Are expressions and X- 2 identically equal: a) on the set R; b) on the set of non-zero integers?

Solution. a) on the set R these expressions are not identically equal, since when X= 0 the expression has no value, and the expression X- 2 has a value of -2.

b) On the set of integers other than zero, these expressions are identically equal, since = .

A task. At what values X the following equalities are identities:

but) ; b) .

Solution. a) Equality is an identity if ;

b) Equality is an identity if .