Let's 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 acceptable values for that equality will be the entire set of real numbers.
2. a 12: a 3 = a 2 *a 7 .
This inequality will be true 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 zero.
For each of these equalities it can be argued that it will be true for any admissible values of the variables a. Such equalities in mathematics are called identities.
An identity is an equality that is true for any admissible values of the variables. If you substitute any valid values into this equality instead of variables, you should get a correct numerical equality.
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. Such a replacement will be an identity transformation.
Example 1: are the following equalities identical:
1. a + 5 = 5 + a;
2. a*(-b) = -a*b;
3. 3*a*3*b = 9*a*b;
Not all expressions presented above will be identities. Of these equalities, only 1, 2 and 3 equalities are identities. No matter what 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 valid values. For example, with the values a = 5 and b = 2, the following result will be obtained:
This equality is not true, since the number 3 is not equal to the number -3.
Having gained an idea of identities, it is logical to move on to getting acquainted with. In this article we will answer the question of what identically equal expressions are, and also use examples to understand which expressions are identically equal and which are not.
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The definition of identically equal expressions is given in parallel with the definition of identity. This happens in 7th grade algebra class. In the textbook on algebra for 7th grade by the author Yu. N. Makarychev, the following formulation is given:
Definition.
– these are expressions whose values are equal for any values of the variables included in them. Numeric Expressions to which they answer same values, 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 clarified. Let us explain what this is connected with.
In the 8th grade, the study of other types of expressions begins, which, unlike whole expressions, may not make sense for some values of the variables. This forces us to introduce definitions of permissible and unacceptable values of variables, as well as the range of permissible values of the variable’s variable value, and, as a consequence, to clarify the definition of identically equal expressions.
Definition.
Two expressions whose values are equal for all permissible values of the variables included in them are called identically equal expressions. Two numerical expressions having the same values are also called identically equal.
In this definition of identically equal expressions, it is worth clarifying the meaning of the phrase “for all permissible 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 explain this idea in the next paragraph by looking at examples.
The definition of identically equal expressions in A. G. Mordkovich’s textbook is given a little differently:
Definition.
Identically equal expressions– these are expressions on the left and right sides of the identity.
The meaning of this and the previous definitions coincide.
The definitions introduced in the previous paragraph allow us to give examples of identically equal expressions.
Let's start with identically equal numerical expressions. The numerical expressions 1+2 and 2+1 are identically equal, since 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 latter expressions are equal by virtue of ). 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 let's 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 (as 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 obtain 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 x 1=2, and the expression 3 x is equal to 3 x 1=3.
When the ranges of permissible 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 the variables from these areas, then here everything is clear - these expressions are identically equal for all permissible 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 of ; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
After we have dealt with the concept of identities, we can move on to studying 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.
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The concept of identically equal expressions is usually studied together with the concept of identity itself as part of a school algebra course. Here is the 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, those numerical expressions to which the same values will correspond are considered identically equal.
This is a fairly broad definition that will be true for all integer expressions whose meaning does not change when the values of the variables change. However, later there is a need to clarify this definition, since there are other types of expressions besides integers that will not make sense given certain variables. This gives rise to the concept of admissibility and inadmissibility of certain variable values, as well as the need to determine the range of permissible values. Let us formulate a refined definition.
Definition 2
Identically equal expressions– these are those expressions whose values are equal to each other for any permissible values of the variables included in their composition. Numerical expressions will be identically equal to each other provided the values are the same.
The phrase “for any valid values of the variables” indicates all those values of the variables for which both expressions will make sense. We will explain this point later when we give examples of identically equal expressions.
You can also provide the following definition:
Definition 3
Identically equal expressions are expressions located in the same identity on the left and right sides.
Using the definitions given above, let's look at a few examples of such expressions.
Let's start with numerical expressions.
Example 1
Thus, 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, the expressions 3 and 30 are identically equal: 10, (2 2) 3 and 2 6 (to calculate the value last expressions 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 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 in this case determined by the commutative property of addition).
Example 4
For example, if a is equal to 4 and b is equal to 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 areas of permissible 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 are 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 range of valid values in one expression may be different 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 permissible 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 will 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 ranges. We can conclude that both expressions x - 1 x x and x − 1 will make sense for any real values 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 general range of permissible values these expressions will be identically equal to each other, but for any real x we cannot speak of identical equality.
If we replace one expression with another, which is identically equal to it, then this process is called an identity transformation. This concept is very important, and we will talk about it in detail in a separate material.
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Subject "Proofs of identities» 7th grade (KRO)
Textbook Makarychev Yu.N., Mindyuk N.G.
Lesson Objectives
Educational:
introduce and initially consolidate the concepts of “identically equal expressions”, “identity”, “identical transformations”;
consider ways to prove identities, promote the development of skills to prove identities;
to check students’ assimilation of the material covered, to develop the ability to use what they have learned to perceive new things.
Developmental:
To develop competent mathematical speech of students (to enrich and complicate lexicon when using special mathematical terms),
develop thinking,
Educational: to cultivate hard work, accuracy, and correct recording of exercise solutions.
Lesson type: learning new material
During the classes
1 . Organizing time.
Checking homework.
Homework questions.
Analysis of the solution at the board.
Math is needed
It's impossible without her
We teach, we teach, friends,
What do we remember in the morning?
2 . Let's do a warm-up.
The result of the addition. (Sum)
How many numbers do you know? (Ten)
Hundredth part of a number. (Percent)
Result of division? (Private)
Smallest natural number? (1)
Is it possible to get zero when dividing natural numbers? (No)
Name the largest negative integer. (-1)
What number cannot be divided by? (0)
Result of multiplication? (Work)
Subtraction result. (Difference)
Commutative property of addition. (The sum does not change by rearranging the places of the terms)
Commutative property of multiplication. (The product does not change from rearranging the places of the factors)
Studying new topic(definition with notebook entry)
Let's find the value of the expressions for x=5 and y=4
3(x+y)=3(5+4)=3*9=27
3х+3у=3*5+3*4=27
We got the same result. From the distribution property it follows that, in general, for any values variable values expressions 3(x+y) and 3x+3y are equal.
Let us now consider the expressions 2x+y and 2xy. When x=1 and y=2 they take equal values:
However, you can specify values of x and y such that the values of these expressions are not equal. For example, if x=3, y=4, then
Definition: Two expressions whose values are equal for any values of the variables are called identically equal.
The expressions 3(x+y) and 3x+3y are identically equal, but the expressions 2x+y and 2xy are not identically equal.
The equality 3(x+y) and 3x+3y is true for any values of x and y. Such equalities are called identities.
Definition: An equality that is true for any values of the variables is called an identity.
True numerical equalities are also considered identities. We have already encountered identities. Identities are equalities that express the basic properties of operations on numbers (Students comment on each property, pronouncing it).
a + b = b + a
ab = ba
(a + b) + c = a + (b + c)
(ab)c = a(bc)
a(b + c) = ab + ac
Give other examples of identities
Definition: Replacing one expression with another identically equal expression is called an identical transformation or simply a transformation of an expression.
Identity transformations expressions with variables are performed based on the properties of operations on numbers.
Identical transformations of expressions are widely used in calculating the values of expressions and solving other problems. You have already had to perform some identical transformations, for example, bringing similar terms, opening parentheses.
5 . No. 691, No. 692 (with pronouncing the rules for opening parentheses, multiplying negative and positive numbers)
Identities for choosing a rational solution:(front work)
6 . Summing up the lesson.
The teacher asks questions, and students answer them at will.
Which two expressions are said to be identically equal? Give examples.
What kind of equality is called identity? Give an example.
What identity transformations do you know?
7. Homework. Learn definitions, Give examples of identical expressions (at least 5), write them down in your notebook
