The values of the following expressions are identically equal. Identity transformations
Topic "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 to prove identities, to contribute to the development of skills to prove identities;
to check the students' assimilation of the material covered, 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 to get zero when dividing natural numbers? (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)
Studying a 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 will define an identity, introduce the notation used, and, of course, give various examples of 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 a special case of 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 will turn into an incorrect equality if we take any different values of the 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 of 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 into 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 numerical expression of the form a-b to be considered the sum of numbers a and -b, a numerical expression of the form a + b-c-d to be considered 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) = -four.
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.
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.
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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 acceptable x values will be the entire set of real numbers, and for the second, the set of all real numbers, except for zero, because then we will 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 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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