While studying algebra, we came across the concepts of a polynomial (for example ($y-x$,$\ 2x^2-2x$, etc.) and algebraic fraction (for example $\frac(x+5)(x)$, $\frac(2x ^2)(2x^2-2x)$,$\ \frac(x-y)(y-x)$, etc.) The similarity of these concepts is that both polynomials and algebraic fractions contain variables and numerical values, and arithmetic is performed. actions: addition, subtraction, multiplication, exponentiation. The difference between these concepts is that in polynomials division by a variable is not performed, but in algebraic fractions division by a variable can be performed.
Both polynomials and algebraic fractions are called rational algebraic expressions in mathematics. But polynomials are whole rational expressions, and algebraic fractions are fractional rational expressions.
It is possible to obtain an entire algebraic expression from a fractional-rational expression using an identity transformation, which in this case will be the main property of a fraction - the reduction of fractions. Let's check this in practice:
Example 1
Convert:$\ \frac(x^2-4x+4)(x-2)$
Solution: This fractional rational equation can be transformed by using the basic property of fractional reduction, i.e. dividing the numerator and denominator by the same number or expression other than $0$.
This fraction cannot be reduced immediately; the numerator must be converted.
Let's transform the expression in the numerator of the fraction, for this we use the formula for the square of the difference: $a^2-2ab+b^2=((a-b))^2$
The fraction looks like
\[\frac(x^2-4x+4)(x-2)=\frac(x^2-4x+4)(x-2)=\frac(((x-2))^2)( x-2)=\frac(\left(x-2\right)(x-2))(x-2)\]
Now we see that the numerator and denominator have a common factor - this is the expression $x-2$, by which we will reduce the fraction
\[\frac(x^2-4x+4)(x-2)=\frac(x^2-4x+4)(x-2)=\frac(((x-2))^2)( x-2)=\frac(\left(x-2\right)(x-2))(x-2)=x-2\]
After reduction, we found that the original fractional rational expression $\frac(x^2-4x+4)(x-2)$ became a polynomial $x-2$, i.e. whole rational.
Now let us pay attention to the fact that the expressions $\frac(x^2-4x+4)(x-2)$ and $x-2\ $ can be considered identical not for all values of the variable, because in order for a fractional rational expression to exist and to be able to reduce by the polynomial $x-2$, the denominator of the fraction must not be equal to $0$ (as well as the factor by which we are reducing. In this example, the denominator and the factor are the same, but This doesn't always happen).
The values of the variable at which the algebraic fraction will exist are called the permissible values of the variable.
Let's put a condition on the denominator of the fraction: $x-2≠0$, then $x≠2$.
This means that the expressions $\frac(x^2-4x+4)(x-2)$ and $x-2$ are identical for all values of the variable except $2$.
Definition 1
Identically equal expressions are those that are equal for all valid values of the variable.
An identical transformation is any replacement of the original expression with an identically equal one. Such transformations include performing actions: addition, subtraction, multiplication, putting a common factor out of brackets, bringing algebraic fractions to a common denominator, reducing algebraic fractions, bringing similar terms, etc. It is necessary to take into account that a number of transformations, such as reduction, reduction of similar terms, can change the permissible values of the variable.
Techniques used to prove identities
Bring the left side of the identity to the right or vice versa using identity transformations
Reduce both sides to the same expression using identical transformations
Transfer the expressions in one part of the expression to another and prove that the resulting difference is equal to $0$
Which of the above techniques to use to prove a given identity depends on the original identity.
Example 2
Prove the identity $((a+b+c))^2- 2(ab+ac+bc)=a^2+b^2+c^2$
Solution: To prove this identity, we use the first of the above methods, namely, we will transform the left side of the identity until it is equal to the right.
Let's consider the left side of the identity: $\ ((a+b+c))^2- 2(ab+ac+bc)$ - it represents the difference of two polynomials. In this case, the first polynomial is the square of the sum of three terms. To square the sum of several terms, we use the formula:
\[((a+b+c))^2=a^2+b^2+c^2+2ab+2ac+2bc\]
To do this, we need to multiply a number by a polynomial. Remember that for this we need to multiply the common factor behind the brackets by each term of the polynomial in the brackets. Then we get:
$2(ab+ac+bc)=2ab+2ac+2bc$
Now let's return to the original polynomial, it will take the form:
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)$
Please note that before the bracket there is a “-” sign, which means that when the brackets are opened, all the signs that were in the brackets change to the opposite.
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)= a ^2+b^2+c^2+2ab+2ac+2bc-2ab-2ac-2bc$
Let us present similar terms, then we obtain that the monomials $2ab$, $2ac$,$\ 2bc$ and $-2ab$,$-2ac$, $-2bc$ cancel each other out, i.e. their sum is $0$.
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)= a ^2+b^2+c^2+2ab+2ac+2bc-2ab-2ac-2bc=a^2+b^2+c^2$
This means that by means of identical transformations we have obtained an identical expression on the left side of the original identity
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2$
Note that the resulting expression shows that the original identity is true.
Please note that in the original identity all values of the variable are allowed, which means we proved the identity using identity transformations, and it is true for all possible values of the variable.
Having gained an idea of identities, it’s 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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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 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. Numerical expressions that have identical values are also called identically equal.
This definition is used up to grade 8 and is valid for whole 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 definitions of valid and invalid variable values, as well as range of permissible values of the permissible value variable, 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.
Examples of identically equal expressions
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.
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.
The concept of identity
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.
Examples of identities
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.
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:
Develop students’ competent mathematical speech (enrich and complicate vocabulary 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 a new topic (definition with writing in a notebook)
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 distributive property it follows that in general, for any values of the variables, the values of the 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.
Identical transformations of 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