Identical transformations of expressions. Identities, definition, designation, examples
Consider two equalities:
1. A 12 * A 3 \u003d A 7 * A 8
This equality will be performed at any values \u200b\u200bof the variable a. The area of \u200b\u200bpermissible values \u200b\u200bfor that equality will be all many real numbers.
2. A 12: A 3 \u003d A 2 * A 7.
This inequality will be performed for all values \u200b\u200bof the variable A, except for zero. The area of \u200b\u200bpermissible values \u200b\u200bfor this inequality will be all many real numbers, except for zero.
Each of these equalities can be argued that it will be true for any permissible values \u200b\u200bof variables a. Such equalities in mathematics are called identities.
The concept of identity
The identity is equality faithful in any permissible values \u200b\u200bof variables. If in this equality to substitute, instead of variables, any valid values, then the correct numerical equality should be obtained.
It is worth noting that faithful numeric equality are also identities. Identities, for example, will be the properties of actions on numbers.
3. A + B \u003d B + A;
4. A + (B + C) \u003d (A + B) + C;
6. A * (B * C) \u003d (A * B) * C;
7. A * (B + C) \u003d A * B + A * C;
11. A * (- 1) \u003d -a.
If two expressions with any valid variables are respectively equal, then such expressions are called identically equal. Below are several examples of identically equal expressions:
1. (A 2) 4 and A 8;
2. A * b * (- a ^ 2 * b) and - 3 * b 2;
3. ((x 3 * x 8) / x) and x 10.
We can always replace one expression by any other expression, identically equal to the first. Such a replacement will be identical conversion.
Examples of identities
Example 1: The following equalities are identities:
1. A + 5 \u003d 5 + A;
2. A * (- b) \u003d -a * b;
3. 3 * A * 3 * B \u003d 9 * A * B;
Not all the above expressions will be identities. Of these equalities, the identities are only 1.2 and 3 equality. Whatever numbers we do not put in them, instead of variables a and b, we still have faithful numeric equality.
But 4 equality is no longer identity. Because not with all permissible values, this equality will be performed. For example, at values \u200b\u200bA \u003d 5 and B \u003d 2, the following result will be:
This equality is not true, since the number 3 is not equal to the number -3.
Subject "Proof of identity»Grade 7 (CRO)
Tutorial Makarychev Yu.N., Mindyuk N.G.
Objectives lesson
Educational:
to acquaint and primarily consolidate the concepts "identically equal expressions", "identity", "identical transformations";
consider ways of evidence of identities, to promote the creation of evidence of identities;
check the assimilation of the student traveled material, form the ability to apply the new application learned to perceive the new.
Developing:
Develop a competent mathematical speech of students (enrich and complicate vocabulary when using special mathematical terms),
develop thinking
Educational: to educate hard work, accuracy, correctness of recording exercise solutions.
Type of lesson: learning a new material
During the classes
1 . Organizing time.
Check your homework.
Home Questions.
Place the decision at the board.
Mathematics need
Without it impossible
Learn, we learn, friends,
What do we remember in the morning?
2 . We will warm up.
Result of addition. (Sum)
How many numbers do you know? (Ten)
Cottage part of the number. (Percent)
The result of division? (Private)
The smallest natural number? (one)
Is it possible when dividing natural numbers to get zero? (not)
Name the greatest integer negative number. (-one)
What number cannot be divided into? (0)
Multiplication result? (Composition)
The result of subtraction. (Difference)
Move the property of addition. (From the permutation of places of the terms the amount does not change)
Move the multiplication property. (From the permutation of places of multipliers, the work does not change)
Studying a new topic (definition with a notebook)
Find the value of expressions at x \u003d 5 and y \u003d 4
3 (x + y) \u003d 3 (5 + 4) \u003d 3 * 9 \u003d 27
3 + 3U \u003d 3 * 5 + 3 * 4 \u003d 27
We got the same result. From the distribution property it follows that in general, with any values \u200b\u200bof variables, the values \u200b\u200bof expressions 3 (x + y) and 3x + 3u are equal.
Consider now the expressions 2x + y and 2h. At x \u003d 1 and y \u003d 2, they take equal values:
However, you can specify such values \u200b\u200bof x and y, in which the values \u200b\u200bof these expressions are not equal. For example, if x \u003d 3, y \u003d 4, then
Definition: Two expressions, the values \u200b\u200bof which are equal for any values \u200b\u200bof variables, are called identically equal.
Expressions 3 (x + y) and 3x + 3ow are identically equal, and the expressions 2x + y and 2h are not identically equal.
Equality 3 (x + y) and 3x + 3a is true for any values \u200b\u200bof x and y. Such equalities are called identities.
Definition: Equality, faithful for any values \u200b\u200bof variables, is called the identity.
The identities of the faithful numerical equality are also considered. With the identities we have already met. The identities are equalities expressing the basic properties of actions above the numbers (students comment on each property, pronouncing it).
a + b \u003d b + a
aB \u003d BA.
(A + B) + C \u003d A + (B + C)
(AB) C \u003d A (BC)
a (B + C) \u003d AB + AC
Give other examples of identities
Definition: Replace one expression to another, identically equal to the expression, is called identical conversion or simply by the transformation of the expression.
The identity transformations of expressions with variables are based on the properties of the number of numbers.
The identity transformations of expressions are widely used when calculating the values \u200b\u200bof expressions and solving other tasks. Some identical conversions you have had to perform, for example, bringing such components, disclosing brackets.
5 . № 691, № 692 (with pronouncing the rules of disclosing brackets, multiplication of negative and positive numbers)
Identities to choose a rational solution:(Front work)
6 . Summing up the lesson.
The teacher asks questions, and students respond to them at will.
What two expressions are called identically equal? Give examples.
What equality is called identity? Lead an example.
What identity conversions are you known?
7. Homework. Learn definitions, bring examples of identical expressions (at least 5), write them down in a notebook
The main properties of addition and multiplication of numbers.
Moving property of addition: the amount of the sum is not changed from the permutation of the terms. For any numbers a and b is true equality
The combination property of addition: to add the third number to the sum of two numbers, you can add the second and third amount to the first number. For any numbers a, b and c are true equality
Multiplication Property: From the rearrangement of multipliers, the product value does not change. For any numbers A, B and C are true equality
The combination property of multiplication: To multiply the work of two numbers to multiply by the third number, you can multiply the first number to the work of the second and third.
For any numbers A, B and C are true equality
Distribution Property: To multiply a number of amount, you can multiply this number to each aligned and folded the results. For any numbers a, b and c are true equality
From the recessive and combination properties of the addition, it follows: in any amount you can somehow rearrange the components and randomly combine them into groups.
Example 1 Calculate the amount of 1.23 + 13.5 + 4.27.
To do this, it is convenient to merge 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.
From the Movement and Fashion Properties of Multiplication, it is necessary: \u200b\u200bin any product, you can quickly rearrange the multipliers and randomly combine them into groups.
Example 2 Find the value of the product 1.8 · 0.25 · 64 · 0.5.
By combining the first factor with the fourth, and the second on 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 valid and in the case when the number is multiplied by the amount of three and more components.
For example, for any numbers a, b, c and d, equality is true
a (B + C + D) \u003d AB + AC + AD.
We know that subtraction can be replaced by adding, adding to a reduced number opposite to subtractable:
This allows a numeric expression of the type A-B to consider the sum of the numbers a and -b, the numerical expression of the form A + B-C-D is considered the sum of the number a, b, -c, -d, etc. The considered properties of actions are fair and for such sums.
Example 3 Find the expression value of 3.27-6.5-2.5 + 1.73.
This expression is the sum of numbers 3.27, -6.5, -2.5 and 1.73. Applying the addition properties, we obtain: 3,27-6.5-2.5 + 1.73 \u003d (3.27 + 1.73) + (- 6.5-2.5) \u003d 5 + (- 9) \u003d -four.
Example 4 Calculate the work 36 · ().
The multiplier can be considered as the sum of numbers and -. Using the distribution property of multiplication, we get:
36 () \u003d 36 · -36 · \u003d 9-10 \u003d -1.
Identities
Definition. Two expressions, the corresponding values \u200b\u200bof which are equal for any values \u200b\u200bof variables, are called identically equal.
Definition. Equality, faithful for any values \u200b\u200bof variables, is called the identity.
Find the values \u200b\u200bof expressions 3 (x + y) and 3x + 3y at x \u003d 5, y \u003d 4:
3 (x + y) \u003d 3 (5 + 4) \u003d 3 · 9 \u003d 27,
3X + 3Y \u003d 3 · 5 + 3 · 4 \u003d 15 + 12 \u003d 27.
We got the same result. From the distribution property it follows that in general, with any values \u200b\u200bof variables, the corresponding values \u200b\u200bof expressions 3 (x + y) and 3x + 3y are equal.
Consider now the expressions 2x + y and 2xy. At x \u003d 1, y \u003d 2, they take equal values:
However, you can specify such values \u200b\u200bX and Y, in which the values \u200b\u200bof these expressions are not equal. For example, if x \u003d 3, y \u003d 4, then
Expressions 3 (x + y) and 3x + 3y are identically equal, and the expressions 2x + y and 2xy are not identically equal.
Equality 3 (x + y) \u003d x + 3y, faithful for any x and y values, is a identity.
The identities of the faithful numerical equality are also considered.
So, the identities are equalities expressing the main properties of actions above the numbers:
a + b \u003d b + a, (a + b) + c \u003d a + (b + c),
aB \u003d BA, (AB) C \u003d A (BC), A (B + C) \u003d AB + AC.
Other examples of identities can also be given:
a + 0 \u003d a, a + (- a) \u003d 0, a - b \u003d a + (- b),
a · 1 \u003d a, a · (-b) \u003d - AB, (-a) (- b) \u003d AB.
Identical transformations of expressions
The replacement of one expression to another, identically equal to it is called the identical conversion or simply by the transformation of the expression.
The identity transformations of expressions with variables are based on the properties of the number of numbers.
To find the value of the XY-XZ expression at the specified x, y, z values, three actions must be performed. For example, at x \u003d 2.3, y \u003d 0.8, z \u003d 0,2 we get:
xy - xz \u003d 2.3 · 0.8-2.3 · 0.2 \u003d 1.84-0.46 \u003d 1.38.
This result can be obtained by performing only two actions, if you use the expression X (Y-Z), identically equal to the expression XY-XZ:
xy - xz \u003d 2.3 (0.8-0.2) \u003d 2.3 · 0.6 \u003d 1.38.
We simplified the calculation, replacing the expression XY-XZ identically equal to the expression X (Y-Z).
The identity transformations of expressions are widely used when calculating the values \u200b\u200bof expressions and solving other tasks. Some identical transformations have already had to be performed, for example, bringing such terms, disclosing brackets. Recall the rules for performing these transformations:
to bring similar terms, it is necessary to fold their coefficients and the result is multiplied by a common letterway;
if the "Plus" sign is standing in front of the brackets, then the brackets can be omitted, while retaining the sign of each term enclosed in the bracket;
if there is a "minus" sign in front of the brackets, then the brackets can be omitted by changing the sign of each term enclosed in the bracket.
Example 1 We present similar terms in the amount of 5x + 2x-3x.
We use the rule of bringing similar terms:
5x + 2x-3x \u003d (5 + 2-3) x \u003d 4x.
This conversion is based on the distribution property of multiplication.
Example 2 Recall brackets in the expression 2a + (B-3C).
Applying the disclosure rule of the brackets, in front of which the "Plus" sign is:
2a + (B-3C) \u003d 2A + B-3C.
The transformation is based on a combination property of addition.
Example 3 Recall brackets in the expression A- (4B-C).
We use the rules for disclosing brackets, facing the minus sign:
a- (4B-C) \u003d A-4B + C.
The conversion performed is based on the distribution property of multiplication and the combination property of addition. Show it. Imagine in this expression the second term - (4B-C) as a work (-1) (4B-C):
a- (4B-C) \u003d A + (- 1) (4B-C).
Applying the specified properties of actions, we get:
a- (4B-C) \u003d a + (- 1) (4B-C) \u003d A + (- 4B + C) \u003d A-4B + C.
This article gives the initial presentation of identities. Here we will define identity, we introduce the designation used, and, of course, we give various examples of identities.
Navigating page.
What is identity?
Logically start the presentation of material with definitions of identity. In the textbook Makarychev Yu. N. Algebra for 7 classes The definition of identity is given like this:
Definition.
Identity - this is equality faithful in any values \u200b\u200bof variables; Any faithful numerical equality is also identity.
At the same time, the author immediately stipulates that in the future this definition will be clarified. This clarification occurs in grade 8, after acquaintance with the definition of permissible values \u200b\u200bof variables and OTZ. The definition becomes like this:
Definition.
Identities - These are faithful numeric equality, as well as equality that are true for all permissible values \u200b\u200bof the variables included in them.
So why, determining the identity, in the 7th grade we are talking about any values \u200b\u200bof variables, and in the 8th grade begin to talk about the values \u200b\u200bof the variables from the OTZ? Up to grade 8, work is carried out exclusively with overall expressions (in particular, with single-wing and polynomials), and they make sense for any values \u200b\u200bof the variables included in them. Therefore, in grade 7, we say that the identity is equality faithful in any values \u200b\u200bof variables. And in the 8th grade, expressions appear, which already make sense not for all values \u200b\u200bof variables, but only for values \u200b\u200bof their OTZ. Therefore, we begin to call the equalities faithful with all valid values \u200b\u200bof variables.
So identity is a special case of equality. That is, any identity is equality. But not any equality is a identity, but only such equality, which is true for any values \u200b\u200bof variables from their area of \u200b\u200bpermissible values.
Sign of identity
It is known that the equalization of the equality of the form "\u003d", on the left and right of which there are some numbers or expressions. If this sign add another horizontal line, then it turns out sign of identity "≡", or as it is also called sign of identical equality.
The sign of identity is usually used only when it is necessary to emphasize that we are not just equality, namely the identity. In other cases, the recording of identities does not differ from equalities.
Examples of identities
It's time to bring examples of identities. This will help us determine the identity given in the first paragraph.
Numeric equalities 2 \u003d 2 are examples of identities, since these equalities are correct, and any correct numeric equality by definition is a identity. They can be written as 2≡2 and.
The identities are the numerical equality of the form 2 + 3 \u003d 5 and 7-1 \u003d 2 · 3, since these equalities are correct. That is, 2 + 3≡5 and 7-1≡2 · 3.
Go to examples of identities containing not only numbers in your record, but also variables.
Consider the equality 3 · (x + 1) \u003d 3 · x + 3. With any value of the variable x, the recorded equality is correct by virtue of the distribution properties of multiplication relative to the addition, therefore, the initial equality is an example of identity. Here is another example of identity: y · (x-1) ≡ (x - 1) · x: x · y 2: yHere the area of \u200b\u200bpermissible values \u200b\u200bof the variables X and Y constitute all pairs (x, y), where X and Y are any numbers, except for zero.
But the equalities x + 1 \u003d x - 1 and a + 2 · b \u003d b + 2 · a are not identities, since there are values \u200b\u200bof variables in which these equalities will be incorrect. For example, at x \u003d 2, the equality X + 1 \u003d X - 1 appeals to the incorrect equality 2 + 1 \u003d 2-1. Moreover, the equality X + 1 \u003d X - 1 is not at all reached at no matter the values \u200b\u200bof the variable x. And the equality A + 2 · B \u003d B + 2 · A will turn into incorrect equality if you take any different values \u200b\u200bof variables a and b. For example, at a \u003d 0 and b \u003d 1, we will come to the incorrect equality 0 + 2 · 1 \u003d 1 + 2 · 0. Equality | x | \u003d x, where | x | - The variable x is also not a identity, as it is incorrect for negative values \u200b\u200bx.
Examples of the most known identities are the species SIN 2 α + COS 2 α \u003d 1 and A log a B \u003d b.
In conclusion of this article, I would like to note that when studying mathematics, we are constantly confronted with identities. The records of the properties of actions with numbers are identities, for example, a + b \u003d b + a, 1 · a \u003d a, 0 · a \u003d 0 and a + (- a) \u003d 0. Also identities are
Having received an idea of \u200b\u200bidentities, it is logical to go to acquaintance with. In this article, we will answer the question that such identically equal expressions, as well as in the examples we will understand which expressions are identically equal, and which - no.
Navigating page.
What is identically equal expressions?
The definition of identically equal expressions is given in parallel with the definition of the identity. This happens on the lessons of algebra in grade 7. In the textbook on algebra for the 7 classes of the author Yu. N. Makarychev, this wording is given:
Definition.
- These are the expressions whose values \u200b\u200bare equal with any values \u200b\u200bof the variables included in them. Numerical expressions that correspond to the same values \u200b\u200bare also called identically equal.
This definition is used up to grade 8, it is valid for integer expressions, as they make sense for any values \u200b\u200bof the variables included in them. And in grade 8, the definition of identically equal expressions is specified. Let us explain what it is connected with.
In grade 8, the study of other types of expressions, which, in contrast to the whole expressions, may not make sense at some values \u200b\u200bof variables. This forces it to impose the definitions of permissible and unacceptable values \u200b\u200bof variables, as well as the area of \u200b\u200bpermissible values \u200b\u200bof the OTZ variable, and as a result - to clarify the definition of identically equal expressions.
Definition.
Two expressions whose values \u200b\u200bare equal with all valid values \u200b\u200bof the variables included in them are called identically equal expressions. Two numeric expressions that have the same values \u200b\u200bare also called identically equal.
In this definition of identically equal expressions, it is worth clarifying the meaning of the phrase "with all the permissible values \u200b\u200bof the variables that are included in them." It implies all such values \u200b\u200bof variables in which both identically equal expressions simultaneously make sense. This idea is explained in the next paragraph, considering examples.
The definition of identically equal expressions in the textbook Mordkovich A. G. is a little different:
Definition.
Identically equal expressions - These are the expressions in the left and right parts of the identity.
In meaning, this and the previous definition coincide.
Examples of identically equal expressions
The definitions entered in the previous paragraph allow examples of identically equal expressions.
Let's start with identically equal numerical expressions. Numeric expressions 1 + 2 and 2 + 1 are identically equal, since they correspond to equal values \u200b\u200bof 3 and 3. Also identically equal to expressions 5 and 30: 6, as well as expressions (2 2) 3 and 2 6 (the values \u200b\u200bof the latest expressions are equal to force). But the numeric expressions 3 + 2 and 3-2 are not identically equal, since it corresponds to values \u200b\u200b5 and 1, respectively, and they are not equal.
Now we will give examples of identically equal expressions with variables. Such are the expressions A + B and B + A. Indeed, with any values \u200b\u200bof variables a and b, recorded expressions take the same values \u200b\u200b(which follows from numbers). For example, at a \u003d 1 and b \u003d 2, we have a + b \u003d 1 + 2 \u003d 3 and b + a \u003d 2 + 1 \u003d 3. For any other values \u200b\u200bof variables A and B, we also obtain equal values \u200b\u200bof these expressions. Expressions 0 · X · y · z and 0 are also identically equal for any values \u200b\u200bof the variables x, y and z. But the expressions 2 · x and 3 · x are not identically equal, since, for example, with x \u003d 1, their values \u200b\u200bare not equal. Indeed, with x \u003d 1, the expression 2 · x is 2 · 1 \u003d 2, and the expression 3 · x is 3 · 1 \u003d 3.
When the values \u200b\u200bof permissible values \u200b\u200bof variables in expressions coincide, as, for example, in expressions A + 1 and 1 + A, or A · b · 0 and 0, or, and the values \u200b\u200bof these expressions are equal to all values \u200b\u200bof the variables from these areas, then Everything is clear - these expressions are identically equal with all the permissible values \u200b\u200bof the variables included in them. So A + 1≡1 + A for any A, expressions A · b · 0 and 0 are identically equal for any values \u200b\u200bof variables a and b, and expressions are identically equal for all x; Ed. S. A. Telikovsky. - 17th ed. - M.: Enlightenment, 2008. - 240 s. : IL. - ISBN 978-5-09-019315-3.
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