How to convert expression to identically equal. Identities, definition, notation, examples

Subject "Identity proofs» Grade 7 (KRO)

Textbook Makarychev Yu.N., Mindyuk N.G.

Lesson Objectives

Educational:

to acquaint and initially consolidate the concepts of "identically equal expressions", "identity", "identical transformations";

to consider ways of proving identities, to promote the development of skills for proving identities;

to check the assimilation of the studied material by students, to form the skills of applying the studied for the perception of the new.

Developing:

Develop competent mathematical speech of students (enrich and complicate vocabulary when using special mathematical terms),

develop thinking,

Educational: to cultivate industriousness, accuracy, correctness of recording the solution of exercises.

Lesson type: learning new material

During the classes

1 . Organizing time.

Checking homework.

Questions on homework.

Debriefing on the board.

Math needed
It's impossible without her
We teach, we teach, friends,
What do we remember in the morning?

2 . Let's do a workout.

Addition result. (Sum)

How many numbers do you know? (Ten)

Hundredth of a number. (Percent)

division result? (Private)

The smallest natural number? (one)

Is it possible when dividing natural numbers get zero? (No)

What is the largest negative integer. (-one)

What number cannot be divided by? (0)

Multiplication result? (Work)

The result of the subtraction. (Difference)

Commutative property of addition. (The sum does not change from the rearrangement of the places of the terms)

Commutative property of multiplication. (The product does not change from the permutation of the places of factors)

Study of new topic(definition with a note in a notebook)

Find the value of the expressions at x=5 and y=4

3(x+y)=3(5+4)=3*9=27

3x+3y=3*5+3*4=27

We got the same result. It follows from the distributive property that, in general, for any values ​​of the variables, the values ​​of the expressions 3(x + y) and 3x + 3y are equal.

Consider now the expressions 2x + y and 2xy. For x=1 and y=2 they take equal values:

However, you can specify x and y values ​​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 said to be 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 met with identities. Identities are equalities that express the basic properties of actions on numbers (Students comment on each property by 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: The replacement of one expression by another, identically equal to it, 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.

Identity transformations of expressions are widely used in calculating the values ​​of expressions and solving other problems. You already had to perform some identical transformations, for example, reduction of similar terms, expansion of brackets.

5 . No. 691, No. 692 (with pronunciation of the rules for opening brackets, multiplying negative and positive numbers)

Identities for choosing a rational solution:(front work)

6 . Summing up the lesson.

The teacher asks questions, and the students answer them as they wish.

What two expressions are called identically equal? Give examples.

What equality is called identity? Give an example.

What identical transformations do you know?

7. Homework. Learn definitions, Give examples of identical expressions (at least 5), write them in a notebook

This article provides an initial notion of identities. Here we define the identity, introduce the notation used, and, of course, give various examples identities

Page navigation.

What is identity?

It is logical to start the presentation of the material with identity definitions. In Yu. N. Makarychev's textbook, algebra for 7 classes, the definition of identity is given as follows:

Definition.

Identity is an equality true for any values ​​of the variables; any true numerical equality is also an identity.

At the same time, the author immediately stipulates that in the future this definition will be clarified. This clarification takes place in the 8th grade, after getting acquainted with the definition of acceptable values ​​​​of variables and ODZ. The definition becomes:

Definition.

Identities are true numerical equalities, as well as equalities that are true for all admissible values ​​of the variables included in them.

So why, when defining an identity, in the 7th grade we talk about any values ​​of variables, and in the 8th grade we start talking about the values ​​of variables from their DPV? Up to grade 8, work is carried out exclusively with integer expressions (in particular, with monomials and polynomials), and they make sense for any values ​​​​of the variables included in them. Therefore, in the 7th grade, we say that an identity is an equality that is true for any values ​​​​of the variables. And in the 8th grade, expressions appear that already make sense not for all values ​​of variables, but only for values ​​from their ODZ. Therefore, by identities, we begin to call equalities that are true for all admissible values ​​of the variables.

So identity is special case equality. That is, any identity is an equality. But not every equality is an identity, but only an equality that is true for any values ​​of variables from their range of acceptable values.

Identity sign

It is known that in writing equalities, an equal sign of the form “=” is used, to the left and to the right of which there are some numbers or expressions. If we add one more horizontal line to this sign, we get identity sign"≡", or as it is also called equal sign.

The sign of identity is usually used only when it is necessary to emphasize that we have before us not just equality, but precisely identity. In other cases, the representations of identities do not differ in form from equalities.

Identity Examples

It's time to bring examples of identities. The definition of identity given in the first paragraph will help us with this.

The numerical equalities 2=2 are examples of identities, since these equalities are true, and any true numerical equality is, by definition, an identity. They can be written as 2≡2 and .

Numerical equalities of the form 2+3=5 and 7−1=2·3 are also identities, since these equalities are true. That is, 2+3≡5 and 7−1≡2 3 .

Let's move on to examples of identities that contain not only numbers, but also variables in their notation.

Consider the equality 3·(x+1)=3·x+3 . For any value of the variable x, the written equality is true due to the distributive property of multiplication with respect to addition, therefore, the original equality is an example of an identity. Here is another example of an identity: y (x−1)≡(x−1)x:x y 2:y, here the range of acceptable values ​​for the variables x and y is all pairs (x, y) , where x and y are any numbers except zero.

But the equalities x+1=x−1 and a+2 b=b+2 a are not identities, since there are values ​​of the variables for which these equalities will be incorrect. For example, for x=2, the equality x+1=x−1 turns into the wrong equality 2+1=2−1 . Moreover, the equality x+1=x−1 is not achieved at all for any values ​​of the variable x . And the equality a+2 b=b+2 a turns into an incorrect equality if we take any various meanings variables a and b . For example, with a=0 and b=1, we will come to the wrong equality 0+2 1=1+2 0 . Equality |x|=x , where |x| - variable x , is also not an identity, since it is not true for negative values x .

Examples of the most famous identities are sin 2 α+cos 2 α=1 and a log a b =b .

In conclusion of this article, I would like to note that when studying mathematics, we constantly encounter identities. Number action property records are identities, for example, a+b=b+a , 1 a=a , 0 a=0 and a+(−a)=0 . Also, the identities are

Basic properties of addition and multiplication of numbers.

Commutative property of addition: when the terms are rearranged, the value of the sum does not change. For any numbers a and b, the equality is true

The associative property of addition: in order to add a third number to the sum of two numbers, you can add the sum of the second and third to the first number. For any numbers a, b and c the equality is true

Commutative property of multiplication: permutation of factors does not change the value of the product. For any numbers a, b and c, the equality is true

The associative property of multiplication: in order to multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third.

For any numbers a, b and c, the equality is true

Distributive property: To multiply a number by a sum, you can multiply that number by each term and add the results. For any numbers a, b and c the equality is true

It follows from the commutative and associative properties of addition that in any sum you can rearrange the terms as you like and combine them in groups in an arbitrary way.

Example 1 Let's calculate the sum 1.23+13.5+4.27.

To do this, it is convenient to combine the first term with the third. We get:

1,23+13,5+4,27=(1,23+4,27)+13,5=5,5+13,5=19.

It follows from the commutative and associative properties of multiplication: in any product, you can rearrange the factors in any way and arbitrarily combine them into groups.

Example 2 Let's find the value of the product 1.8 0.25 64 0.5.

Combining the first factor with the fourth, and the second with the third, we will have:

1.8 0.25 64 0.5 \u003d (1.8 0.5) (0.25 64) \u003d 0.9 16 \u003d 14.4.

The distribution property is also valid when the number is multiplied by the sum of three or more terms.

For example, for any numbers a, b, c and d, the equality is true

a(b+c+d)=ab+ac+ad.

We know that subtraction can be replaced by addition by adding to the minuend the opposite number to the subtrahend:

This allows a numeric expression type a-b consider the sum of numbers a and -b, consider a numerical expression of the form a + b-c-d as the sum of numbers a, b, -c, -d, etc. The considered properties of actions are also valid for such sums.

Example 3 Let's find the value of the expression 3.27-6.5-2.5+1.73.

This expression is the sum of the numbers 3.27, -6.5, -2.5 and 1.73. Applying the addition properties, we get: 3.27-6.5-2.5+1.73=(3.27+1.73)+(-6.5-2.5)=5+(-9) = -4.

Example 4 Let's calculate the product 36·().

The multiplier can be thought of as the sum of the numbers and -. Using the distributive property of multiplication, we get:

36()=36-36=9-10=-1.

Identities

Definition. Two expressions whose corresponding values ​​are equal for any values ​​of the variables are said to be identically equal.

Definition. An equality that is true for any values ​​of the variables is called an identity.

Let's find the values ​​of the expressions 3(x+y) and 3x+3y for x=5, y=4:

3(x+y)=3(5+4)=3 9=27,

3x+3y=3 5+3 4=15+12=27.

We got the same result. It follows from the distributive property that, in general, for any values ​​of the variables, the corresponding values ​​of the expressions 3(x+y) and 3x+3y are equal.

Consider now the expressions 2x+y and 2xy. For x=1, y=2 they take equal values:

However, you can specify x and y values ​​such that the values ​​of these expressions are not equal. For example, if x=3, y=4, then

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)=x+3y, true for any values ​​of x and y, is an identity.

True numerical equalities are also considered identities.

So, identities are equalities expressing the main properties of actions on numbers:

a+b=b+a, (a+b)+c=a+(b+c),

ab=ba, (ab)c=a(bc), a(b+c)=ab+ac.

Other examples of identities can be given:

a+0=a, a+(-a)=0, a-b=a+(-b),

a 1=a, a (-b)=-ab, (-a)(-b)=ab.

Identity transformations of expressions

The replacement of one expression by another, identically equal to it, 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.

To find the value of the expression xy-xz given the values ​​x, y, z, you need to perform three steps. For example, with x=2.3, y=0.8, z=0.2 we get:

xy-xz=2.3 0.8-2.3 0.2=1.84-0.46=1.38.

This result can be obtained in only two steps, using the expression x(y-z), which is identically equal to the expression xy-xz:

xy-xz=2.3(0.8-0.2)=2.3 0.6=1.38.

We have simplified the calculations by replacing the expression xy-xz with the identically equal expression x(y-z).

Identity transformations of expressions are widely used in calculating the values ​​of expressions and solving other problems. Some identical transformations have already been performed, for example, the reduction of similar terms, the opening of brackets. Recall the rules for performing these transformations:

to bring like terms, you need to add their coefficients and multiply the result by the common letter part;

if there is a plus sign in front of the brackets, then the brackets can be omitted, retaining the sign of each term enclosed in brackets;

if there is a minus sign before the brackets, then the brackets can be omitted by changing the sign of each term enclosed in brackets.

Example 1 Let's add like terms in the sum 5x+2x-3x.

We use the rule for reducing like terms:

5x+2x-3x=(5+2-3)x=4x.

This transformation is based on the distributive property of multiplication.

Example 2 Let's expand the brackets in the expression 2a+(b-3c).

Applying the rule for opening brackets preceded by a plus sign:

2a+(b-3c)=2a+b-3c.

The performed transformation is based on the associative property of addition.

Example 3 Let's expand the brackets in the expression a-(4b-c).

Let's use the rule for expanding brackets preceded by a minus sign:

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

The performed transformation is based on the distributive property of multiplication and the associative property of addition. Let's show it. Let's represent the second term -(4b-c) in this expression as a product (-1)(4b-c):

a-(4b-c)=a+(-1)(4b-c).

Applying these properties of actions, we get:

a-(4b-c)=a+(-1)(4b-c)=a+(-4b+c)=a-4b+c.

In the course of studying algebra, we came across the concepts of polynomial (for example ($y-x$ ,$\ 2x^2-2x$ and so on) 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 in polynomials and in algebraic fractions there are variables and numerical values, arithmetic operations: addition, subtraction, multiplication, exponentiation. The difference between these concepts is that division by a variable is not performed in polynomials, while division by a variable can be performed in algebraic fractions.

Both polynomials and algebraic fractions are called rational algebraic expressions in mathematics. But polynomials are integer rational expressions, and algebraic fractions are fractionally rational expressions.

Can be obtained from fractional --rational expression whole algebraic expression using the identical transformation, which in this case will be the main property of a fraction - reduction of fractions. Let's check it out in practice:

Example 1

Transform:$\ \frac(x^2-4x+4)(x-2)$

Decision: Convert Given fractional rational equation possible by using the main property fractions - abbreviations, i.e. dividing the numerator and denominator by the same number or expression other than $0$.

This fraction cannot be reduced immediately, it is necessary to convert the numerator.

We 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 has the form

\[\frac(x^2-4x+4)(x-2)=\frac(x^2-4x+4)(x-2)=\frac(((x-2))^2)( x-2)=\frac(\left(x-2\right)(x-2))(x-2)\]

Now we see that there is a common factor in the numerator and denominator - this is the expression $x-2$, on 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 have obtained that the original fractional-rational expression $\frac(x^2-4x+4)(x-2)$ has become a polynomial $x-2$, i.e. whole rational.

Now let's 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 for the reduction by the polynomial $x-2$ to be possible, the denominator of the fraction must not be equal to $0$ (as well as the factor by which we reduce. In this example the denominator and the multiplier are the same, but this is not always the case).

Variable values ​​for which the algebraic fraction will exist are called valid variable values.

We put a condition on the denominator of the fraction: $x-2≠0$, then $x≠2$.

So 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 possible values ​​of the variable.

An identical transformation is any replacement of the original expression with an identically equal one. Such transformations include the following actions: addition, subtraction, multiplication, parentheses, algebraic fractions to a common denominator, reduction of algebraic fractions, reduction of similar terms, etc. It must be taken into account that a number of transformations, such as reduction, reduction of similar terms, can change the allowable values ​​of the variable.

Techniques used to prove identities

Convert the left side of the identity to the right side or vice versa using identity transformations

Reduce both parts 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 methods to use to prove a given identity depends on the original identity.

Example 2

Prove the identity $((a+b+c))^2- 2(ab+ac+bc)=a^2+b^2+c^2$

Decision: 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 side.

Consider the left side of the identity: $\ ((a+b+c))^2- 2(ab+ac+bc)$- it is 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. Recall that for this we need to multiply the common factor outside the brackets by each term of the polynomial in brackets. Then we get:

$2(ab+ac+bc)=2ab+2ac+2bc$

Now back to the original polynomial, it will take the form:

$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)$

Note that there is a “-” sign in front of the bracket, which means that when the brackets are opened, all the signs that were in the brackets are reversed.

$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)= a ^2+b^2+c^2+2ab+2ac+2bc-2ab-2ac-2bc$

If we bring similar terms, then we get that the monomials $2ab$, $2ac$,$\ 2bc$ and $-2ab$,$-2ac$, $-2bc$ cancel each other out, i.e. their sum is equal to $0$.

$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)= a ^2+b^2+c^2+2ab+2ac+2bc-2ab-2ac-2bc=a^2+b^2+c^2$

So, by identical transformations, we have obtained 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.

Note that in the original identity, all values ​​of the variable are allowed, which means that we have proved the identity using identical transformations, and it is true for all allowed values ​​of the variable.