Identically equal expressions: definition, examples. Identically equal expressions: definition, examples Identically equal values ​​of the following expressions a4

Both parts of which are identically equal expressions. Identities are divided into alphabetic and numeric.

Identical expressions

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

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

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

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

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

Numerical identity 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

Identical Expression Conversions

All algebraic actions represent the transformation of one algebraic expression into another, identical to the first.

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

Consider the identical transformation of an expression using the example of placing the common factor outside the 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.

Identically Equal Expressions: Definition

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

Definition 1

Identically equal each other 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 to be identically equal to which the same values ​​will correspond.

This is a fairly broad definition that will be correct 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 in addition to integers, there are other types of expressions that will not make sense for certain variables. This gives rise to the concept of admissibility and inadmissibility of certain values ​​of variables, as well as the need to determine the range of admissible values. Let us formulate a more precise definition.

Definition 2

Identically Equal Expressions- these are those expressions, the values ​​of which are equal to each other for any admissible values ​​of the variables included in their composition. Numeric expressions will be identically equal to each other, provided they have the same values.

The phrase "for any valid variable values" refers to all those variable values ​​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

Equally 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, let's look at a few examples of such expressions.

Let's start with numeric expressions.

Example 1

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

Example 2

In the same way, 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 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 displacement property of addition).

Example 4

For example, if a equals 4 and b equals 5, then 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, since they will not be equal for any x.

In the event that the ranges of admissible values ​​of the variables 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 gives 0 in the end. The expressions x 4 and x will be identically equal for any x from the interval [0, + ∞).

But the range of validity in one expression may differ from the range 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 of real numbers, and for the second, the set of all real numbers, with the exception of zero, because then we get 0 in the denominator, and such a division is not defined. These two expressions have a common range of values, formed by the intersection of two separate areas. We can conclude that both expressions x - 1 x x and x - 1 will make sense for any real values ​​of the variables, except 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 on the common range of permissible values ​​these expressions will be identically equal to each other, and for any real x it is impossible to 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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Two expressions are said to be identically equal on a set, if they make sense on this set and all their corresponding values ​​are equal.

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 the identical transformation of the expression.

A task. Find the scope of an expression.

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

NS= (- ¥; -3) È (-3; 3) È (3; + ¥).

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

Solution. a) On the set R these expressions are not identically equal, since for NS= 0 the expression is irrelevant, but the expression NS- 2 has a value of -2.

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

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

but) ; b).

Solution. a) Equality is an identity if;

b) Equality is an identity if.

Having received an idea of ​​identities, it is logical to move on to acquaintance with. In this article, we will answer the question of what are identically equal expressions, and also use examples to 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 7th grade algebra lessons. In a textbook on algebra for 7 classes by the author Yu.N. Makarychev, the following formulation is given:

Definition.

Are expressions whose values ​​are equal for any values ​​of the variables included in them. Numeric expressions that have identical values ​​are also called identically equal.

This definition is used up to grade 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 refined. Let us explain what this is connected with.

In grade 8, the study of other types of expressions begins, which, unlike integer expressions, for some values ​​of the variables may not make sense. This forces us to introduce definitions of permissible and unacceptable values ​​of variables, as well as the range of permissible values ​​of the ODZ of a variable, and, as a consequence, to clarify the definition of identically equal expressions.

Definition.

Two expressions, the values ​​of which are equal for all admissible values ​​of the variables included in them, are called identically equal expressions... Two numeric expressions that have the same value are also called identically equal.

In this definition of 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 make sense at the same time. We will clarify this idea in the next paragraph, considering examples.

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

Definition.

Identically Equal Expressions Are expressions on the left and right sides of the identity.

The meaning of this and the previous definitions coincide.

Examples of identically equal expressions

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

Let's start with identically equal numerical expressions. Numerical expressions 1 + 2 and 2 + 1 are identically equal, since they correspond to equal values ​​3 and 3. Also, expressions 5 and 30: 6 are identically equal, as are expressions (2 2) 3 and 2 6 (the values ​​of the last expressions are equal in force). But the numerical expressions 3 + 2 and 3−2 are not identically equal, since they correspond to the values ​​5 and 1, respectively, and they are not equal.

Now we will 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 on the same values ​​(which follows from the numbers). For example, for 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 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, when x = 1, their values ​​are not equal. Indeed, for x = 1 the expression 2 x is equal to 2 1 = 2, and the expression 3 x is equal to 3 1 = 3.

When the ranges of valid values ​​of variables in expressions coincide, as, for example, in 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.

Algebra: study. for 8 cl. 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.

A. G. Mordkovich Algebra. 7th grade. At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich. - 17th ed., Add. - M .: Mnemozina, 2013 .-- 175 p .: ill. ISBN 978-5-346-02432-3.

Consider two equalities:

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

This equality will hold for any values ​​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 admissible values ​​for this inequality will be the entire set of real numbers, except for zero.

Each of these equalities can be said to be true for any admissible values ​​of the variables a. Such equalities in mathematics are called identities.

Identity concept

Identity is an equality that is true for any valid 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 the 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 that is identically equal to the first. Such a replacement will be an identical transformation.

Examples of identities

Example 1: are the following equalities equal:

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 variables a and b, we will still get correct numerical equalities.

But 4 equality is no longer an identity. Because this equality will not hold for all admissible values. For example, for 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.