What algebraic expression is called an integer. Algebraic expressions
Numeric and algebraic expressions. Expression conversion.
What is an expression in mathematics? Why are expression conversions necessary?
The question, as they say, is interesting... The fact is that these concepts are the basis of all mathematics. All mathematics consists of expressions and their transformations. Not very clear? Let me explain.
Let's say you have an evil example. Very large and very complex. Let's say you're good at math and you're not afraid of anything! Can you answer right away?
You'll have to decide this example. Sequentially, step by step, this example simplify. According to certain rules, of course. Those. do expression conversion. How successfully you carry out these transformations, so you are strong in mathematics. If you don't know how to do the right transformations, in mathematics you can't do Nothing...
In order to avoid such an uncomfortable future (or present ...), it does not hurt to understand this topic.)
To begin with, let's find out what is an expression in math. What's happened numeric expression and what is algebraic expression.
What is an expression in mathematics?
Expression in mathematics is a very broad concept. Almost everything we deal with in mathematics is a set of mathematical expressions. Any examples, formulas, fractions, equations, and so on - it all consists of mathematical expressions.
3+2 is a mathematical expression. c 2 - d 2 is also a mathematical expression. And a healthy fraction, and even one number - these are all mathematical expressions. The equation, for example, is:
5x + 2 = 12
consists of two mathematical expressions connected by an equals sign. One expression is on the left, the other is on the right.
In general terms, the term mathematical expression" is used, most often, in order not to mumble. They will ask you what an ordinary fraction is, for example? And how to answer ?!
Answer 1: "It's... m-m-m-m... such a thing ... in which ... Can I write a fraction better? Which one do you want?"
The second answer option: "An ordinary fraction is (cheerfully and joyfully!) mathematical expression , which consists of a numerator and a denominator!"
The second option is somehow more impressive, right?)
For this purpose, the phrase " mathematical expression "very good. Both correct and solid. But for practical application, you need to be well versed in specific kinds of expressions in mathematics .
The specific type is another matter. This quite another thing! Each type of mathematical expression has mine a set of rules and techniques that must be used in the decision. To work with fractions - one set. For working with trigonometric expressions - the second. For working with logarithms - the third. And so on. Somewhere these rules coincide, somewhere they differ sharply. But do not be afraid of these terrible words. Logarithms, trigonometry and other mysterious things we will master in the relevant sections.
Here we will master (or - repeat, as you like ...) two main types of mathematical expressions. Numeric expressions and algebraic expressions.
Numeric expressions.
What's happened numeric expression? This is a very simple concept. The name itself hints that this is an expression with numbers. That is how it is. Mathematical expression made up of numbers, brackets and signs arithmetic operations is called a numeric expression.
7-3 is a numeric expression.
(8+3.2) 5.4 is also a numeric expression.
And this monster:
also a numeric expression, yes...
An ordinary number, a fraction, any calculation example without x's and other letters - all these are numerical expressions.
main feature numerical expressions in it no letters. None. Only numbers and mathematical icons (if necessary). It's simple, right?
And what can be done with numerical expressions? Numeric expressions can usually be counted. To do this, sometimes you have to open brackets, change signs, abbreviate, swap terms - i.e. do expression conversions. But more on that below.
Here we will deal with such a funny case when with a numerical expression you don't have to do anything. Well, nothing at all! This nice operation To do nothing)- is executed when the expression doesn't make sense.
When does a numeric expression not make sense?
Of course, if we see some kind of abracadabra in front of us, such as
then we won't do anything. Since it is not clear what to do with it. Some nonsense. Unless, to count the number of pluses ...
But there are outwardly quite decent expressions. For example this:
(2+3) : (16 - 2 8)
However, this expression is also doesn't make sense! For the simple reason that in the second brackets - if you count - you get zero. You can't divide by zero! This is a forbidden operation in mathematics. Therefore, there is no need to do anything with this expression either. For any task with such an expression, the answer will always be the same: "The expression doesn't make sense!"
To give such an answer, of course, I had to calculate what would be in brackets. And sometimes in brackets such a twist ... Well, there's nothing to be done about it.
There are not so many forbidden operations in mathematics. There is only one in this thread. Division by zero. Additional prohibitions arising in roots and logarithms are discussed in the relevant topics.
So, an idea of what is numeric expression- got. concept numeric expression doesn't make sense- realized. Let's go further.
Algebraic expressions.
If letters appear in a numerical expression, this expression becomes... The expression becomes... Yes! It becomes algebraic expression. For example:
5a 2 ; 3x-2y; 3(z-2); 3.4m/n; x 2 +4x-4; (a + b) 2; ...
Such expressions are also called literal expressions. Or expressions with variables. It's practically the same thing. Expression 5a +c, for example - both literal and algebraic, and expression with variables.
concept algebraic expression - wider than numerical. It includes and all numeric expressions. Those. a numeric expression is also an algebraic expression, only without the letters. Every herring is a fish, but not every fish is a herring...)
Why literal- It's clear. Well, since there are letters ... Phrase expression with variables also not very perplexing. If you understand that numbers are hidden under the letters. All sorts of numbers can be hidden under the letters ... And 5, and -18, and whatever you like. That is, a letter can replace for different numbers. That's why the letters are called variables.
In the expression y+5, For example, at - variable. Or just say " variable", without the word "value". Unlike the five, which is a constant value. Or simply - constant.
Term algebraic expression means that to work with this expression, you need to use the laws and rules algebra. If arithmetic works with specific numbers, then algebra- with all the numbers at once. A simple example for clarification.
In arithmetic, one can write that
But if we write a similar equality through algebraic expressions:
a + b = b + a
we will decide immediately All questions. For all numbers stroke. For an infinite number of things. Because under the letters A And b implied All numbers. And not only numbers, but even other mathematical expressions. This is how algebra works.
When does an algebraic expression make no sense?
Everything is clear about the numerical expression. You can't divide by zero. And with letters, is it possible to find out what we are dividing by ?!
Let's take the following variable expression as an example:
2: (A - 5)
Does it make sense? But who knows him? A- any number...
Any, any... But there is one meaning A, for which this expression exactly doesn't make sense! And what is that number? Yes! It's 5! If the variable A replace (they say - "substitute") with the number 5, in parentheses, zero will turn out. which cannot be divided. So it turns out that our expression doesn't make sense, If a = 5. But for other values A does it make sense? Can you substitute other numbers?
Certainly. In such cases, it is simply said that the expression
2: (A - 5)
makes sense for any value A, except a = 5 .
The entire set of numbers Can substitute into the given expression is called valid range this expression.
As you can see, there is nothing tricky. We look at the expression with variables, and think: at what value of the variable is the forbidden operation obtained (division by zero)?
And then be sure to look at the question of the assignment. What are they asking?
doesn't make sense, our forbidden value will be the answer.
If they ask at what value of the variable the expression has the meaning(feel the difference!), the answer will be all other numbers except for the forbidden.
Why do we need the meaning of the expression? He is there, he is not... What's the difference?! The fact is that this concept becomes very important in high school. Extremely important! This is the basis for such solid concepts as the range of valid values or the scope of a function. Without this, you will not be able to solve serious equations or inequalities at all. Like this.
Expression conversion. Identity transformations.
We got acquainted with numerical and algebraic expressions. Understand what the phrase "the expression does not make sense" means. Now we need to figure out what expression conversion. The answer is simple, outrageously.) This is any action with an expression. And that's it. You have been doing these transformations since the first class.
Take the cool numerical expression 3+5. How can it be converted? Yes, very easy! Calculate:
This calculation will be the transformation of the expression. You can write the same expression in a different way:
We didn't count anything here. Just write down the expression in a different form. This will also be a transformation of the expression. It can be written like this:
And this, too, is the transformation of an expression. You can make as many of these transformations as you like.
Any action on an expression any writing it in a different form is called an expression transformation. And all things. Everything is very simple. But there is one thing here very important rule. So important that it can safely be called main rule all mathematics. Breaking this rule inevitably leads to errors. Do we understand?)
Let's say we've transformed our expression arbitrarily, like this:
Transformation? Certainly. We wrote the expression in a different form, what is wrong here?
It's not like that.) The fact is that the transformations "whatever" mathematics is not interested at all.) All mathematics is built on transformations in which the appearance, but the essence of the expression does not change. Three plus five can be written in any form, but it must be eight.
transformations, expressions that do not change the essence called identical.
Exactly identical transformations and allow us, step by step, to transform complex example into a simple expression, keeping essence of the example. If we make a mistake in the chain of transformations, we will make a NOT identical transformation, then we will decide another example. With other answers that are not related to the correct ones.)
Here it is the main rule for solving any tasks: compliance with the identity of transformations.
I gave an example with a numerical expression 3 + 5 for clarity. In algebraic expressions, identical transformations are given by formulas and rules. Let's say there is a formula in algebra:
a(b+c) = ab + ac
So, in any example, we can instead of the expression a(b+c) feel free to write an expression ab+ac. And vice versa. This identical transformation. Mathematics gives us a choice of these two expressions. And which one to write - from case study depends.
Another example. One of the most important and necessary transformations is the basic property of a fraction. You can see more details at the link, but here I just remind the rule: if the numerator and denominator of a fraction are multiplied (divided) by the same number, or an expression that is not equal to zero, the fraction will not change. Here is an example of identical transformations for this property:
As you probably guessed, this chain can be continued indefinitely...) Very important property. It is it that allows you to turn all sorts of example monsters into white and fluffy.)
There are many formulas defining identical transformations. But the most important - quite a reasonable amount. One of the basic transformations is factorization. It is used in all mathematics - from elementary to advanced. Let's start with him. in the next lesson.)
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you can get acquainted with functions and derivatives.
Algebraic expression an expression made up of letters and numbers connected by the signs of the operations of addition, subtraction, multiplication, division, raising to an integer power and extracting the root (the exponents and the root must be constant numbers). A. in. is called rational with respect to some of the letters included in it if it does not contain them under the root extraction sign, for example rational with respect to a, b and c. A. in. is called an integer with respect to some letters if it does not contain division by expressions containing these letters, for example 3a / c + bc 2 - 3ac / 4
is integer with respect to a and b. If some of the letters (or all) are considered variables, then A. c. is an algebraic function.
Big soviet encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
See what "Algebraic Expression" is in other dictionaries:
An expression made up of letters and numbers connected by signs of algebraic operations: addition, subtraction, multiplication, division, raising to a power, extracting a root ... Big encyclopedic Dictionary
algebraic expression- - Topics oil and gas industry EN algebraic expression ... Technical Translator's Handbook
An algebraic expression is one or more algebraic quantities (numbers and letters) interconnected by signs of algebraic operations: addition, subtraction, multiplication and division, as well as extracting the root and raising to an integer ... ... Wikipedia
An expression made up of letters and numbers connected by signs of algebraic operations: addition, subtraction, multiplication, division, raising to a power, extracting a root. * * * ALGEBRAIC EXPRESSION ALGEBRAIC EXPRESSION, expression, ... ... encyclopedic Dictionary
algebraic expression- algebrinė išraiška statusas T sritis fizika atitikmenys: engl. algebraic expression vok. algebraischer Ausdruck, m rus. algebraic expression, n pranc. expression algebrique, f … Fizikos terminų žodynas
An expression made up of letters and numbers connected by the signs of algebras. actions: addition, subtraction, multiplication, division, exponentiation, root extraction ... Natural science. encyclopedic Dictionary
An algebraic expression with respect to a given variable, in contrast to a transcendental one, is an expression that does not contain other functions of a given quantity, except for sums, products or powers of this quantity, moreover, terms ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
EXPRESSION, expressions, cf. 1. Action according to Ch. express express. I can't find words to express my gratitude. 2. more often than not The embodiment of an idea in the forms of some kind of art (philosophical). Only a great artist is able to create such an expression, ... ... Dictionary Ushakov
An equation obtained by equating two algebraic expressions (See Algebraic Expression). A. y. with one unknown is called fractional if the unknown is included in the denominator, and irrational if the unknown is included under ... ... Great Soviet Encyclopedia
EXPRESSION- the primary mathematical concept, which means a record of letters and numbers connected by signs of arithmetic operations, while brackets, function symbols, etc. can be used; usually B is the formula million part of it. Distinguish In (1) ... ... Great Polytechnic Encyclopedia
Articles on natural sciences and mathematics
What is a numeric and algebraic expression?
Numeric expression- this is any record made up of numbers and signs of arithmetic operations and written according to known rules, as a result of which it has a certain meaning. For example, the following entries are numeric expressions: 4 + 5; -1.05 × 22.5 - 34. On the other hand, the notation × 16 - × 0.5 is not numeric, since, although it consists of numbers and signs of arithmetic operations, it is not written according to the rules for compiling numerical expressions.
If a numerical expression contains letters instead of numbers (all or only some), then this expression is already algebraic.
The meaning of using letters is approximately as follows. Instead of letters, different numbers can be substituted, which means that the expression can have different meanings. Algebra as a science studies the principles of simplifying expressions, searching for and using various rules, laws, formulas. Algebra studies the most rational ways performing calculations, and just for this, generalizations are needed, that is, the use of variables (letters) instead of specific numbers.
Algebraic facts include the laws of addition and multiplication, the concepts of a negative number, ordinary and decimal fractions and the rules for arithmetic operations with them, the properties of ordinary fractions. Algebra is called upon to understand all this variety of facts, to teach them how to use them, to see the applicability of laws in specific numerical and algebraic expressions.
When a numeric expression is evaluated, the result is its value. The value of an algebraic expression can only be calculated if certain numerical values are substituted for letters. For example, the expression a ÷ b with a = 3 and b = 5 has a value of 3 ÷ 5 or 0.6. However, an algebraic expression may be such that, for some values of the variables (letters), it may not make sense at all. For the same example (a ÷ b), the expression does not make sense for b = 0, since you cannot divide by zero.
Therefore, they talk about permissible and not allowed values variables for this or that algebraic expression.
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Algebraic expressions
- Concept definition
- Expression value
- Identity expressions
- Problem solving
- What have we learned?
Concept definition
What expressions are called algebraic? This is a mathematical notation made up of numbers, letters and signs of arithmetic operations. The presence of letters is the main difference between numerical and algebraic expressions. Examples:
A letter in algebraic expressions represents a number. Therefore, it is called a variable - in the first example it is the letter a, in the second - b, and in the third - c. The algebraic expression itself is also called variable expression.
Expression value
Meaning of an algebraic expression is the number obtained as a result of performing all the arithmetic operations that are specified in this expression. But in order to get it, the letters must be replaced by numbers. Therefore, the examples always indicate which number corresponds to the letter. Consider how to find the value of the expression 8a-14*(5-a) if a=3.
Let's substitute the number 3 instead of the letter a. We get the following entry: 8*3-14*(5-3).
As in numerical expressions, the solution of an algebraic expression is carried out according to the rules for performing arithmetic operations. Let's solve everything in order.
Thus, the value of the expression 8a-14*(5-a) for a=3 is -4.
The value of a variable is called valid if the expression makes sense for it, that is, it is possible to find its solution.
An example of a valid variable for the expression 5:2a is the number 1. Substituting it into the expression, we get 5:2*1=2.5. The invalid variable for this expression is 0. If we substitute zero into the expression, we get 5:2*0, that is, 5:0. You can't divide by zero, so the expression doesn't make sense.
Identity expressions
If two expressions are equal for any values of their constituent variables, they are called identical.
Example identical expressions
:
4(a+c) and 4a+4c.
Whatever values the letters a and c take, the expressions will always be equal. Any expression can be replaced by another, identical to it. This process is called identity transformation.
An example of an identical transformation
.
4 * (5a + 14c) - this expression can be replaced by an identical one by applying the mathematical law of multiplication. To multiply a number by the sum of two numbers, you need to multiply this number by each term and add the results.
Thus, the expression 4*(5a+14c) is identical to 20a+64c.
The number that precedes the literal variable in an algebraic expression is called a coefficient. Coefficient and variable are multipliers.
Problem solving
Algebraic expressions are used to solve problems and equations.
Let's consider the task. Petya came up with a number. In order for classmate Sasha to guess it, Petya told him: first I added 7 to the number, then subtracted 5 from it and multiplied by 2. As a result, I got the number 28. What number did I guess?
To solve the problem, you need to designate the hidden number with the letter a, and then perform all the indicated actions with it.
Now let's solve the resulting equation.
Petya guessed the number 12.
What have we learned?
An algebraic expression is a record made up of letters, numbers and signs of arithmetic operations. Each expression has a value that is found by doing all the arithmetic in the expression. The letter in an algebraic expression is called a variable, and the number in front of it is called a coefficient. Algebraic expressions are used to solve problems.
6.4.1. Algebraic expression
I. Expressions in which numbers, signs of arithmetic operations and brackets can be used along with letters are called algebraic expressions.
Examples of algebraic expressions:
2m-n; 3 · (2a+b); 0.24x; 0.3a-b · (4a + 2b); a 2 - 2ab;
Since a letter in an algebraic expression can be replaced by some different numbers, the letter is called a variable, and the algebraic expression itself is called an expression with a variable.
II. If in an algebraic expression the letters (variables) are replaced by their values and the specified actions are performed, then the resulting number is called the value of the algebraic expression.
Examples. Find the value of an expression:
1) a + 2b -c for a = -2; b = 10; c = -3.5.
2) |x| + |y| -|z| at x = -8; y=-5; z = 6.
1) a + 2b -c for a = -2; b = 10; c = -3.5. Instead of variables, we substitute their values. We get:
2+ 2 · 10- (-3,5) = -2 + 20 +3,5 = 18 + 3,5 = 21,5.
2) |x| + |y| -|z| at x = -8; y=-5; z = 6. Substitute indicated values. Remember that the modulus of a negative number is equal to its opposite number, and the modulus positive number equal to that number. We get:
|-8| + |-5| -|6| = 8 + 5 -6 = 7.
III. The values of a letter (variable) for which the algebraic expression makes sense are called valid values of the letter (variable).
Examples. At what values of the variable the expression does not make sense?
Solution. We know that it is impossible to divide by zero, therefore, each of these expressions will not make sense with the value of the letter (variable) that turns the denominator of the fraction to zero!
In example 1), this is the value a = 0. Indeed, if instead of a we substitute 0, then the number 6 will need to be divided by 0, but this cannot be done. Answer: expression 1) does not make sense when a = 0.
In example 2) the denominator x - 4 = 0 at x = 4, therefore, this value x = 4 and cannot be taken. Answer: expression 2) does not make sense for x = 4.
In example 3) the denominator is x + 2 = 0 for x = -2. Answer: expression 3) does not make sense at x = -2.
In example 4) the denominator is 5 -|x| = 0 for |x| = 5. And since |5| = 5 and |-5| \u003d 5, then you cannot take x \u003d 5 and x \u003d -5. Answer: expression 4) does not make sense for x = -5 and for x = 5.
IV. Two expressions are said to be identically equal if, for any admissible values of the variables, the corresponding values of these expressions are equal.
Example: 5 (a - b) and 5a - 5b are identical, since the equality 5 (a - b) = 5a - 5b will be true for any values of a and b. Equality 5 (a - b) = 5a - 5b is an identity.
Identity is an equality that is valid for all admissible values of the variables included in it. Examples of identities already known to you are, for example, the properties of addition and multiplication, the distribution property.
The replacement of one expression by another, identically equal to it, is called an identical transformation or simply a transformation of an expression. Identity transformations expressions with variables are executed based on the properties of operations on numbers.
a) convert the expression to identically equal using the distributive property of multiplication:
1) 10 (1.2x + 2.3y); 2) 1.5 (a -2b + 4c); 3) a·(6m -2n + k).
Solution. Recall the distributive property (law) of multiplication:
(a+b) c=a c+b c(distributive law of multiplication with respect to addition: in order to multiply the sum of two numbers by a third number, you can multiply each term by this number and add the results).
(a-b) c=a c-b c(distributive law of multiplication with respect to subtraction: in order to multiply the difference of two numbers by a third number, you can multiply by this number reduced and subtracted separately and subtract the second from the first result).
1) 10 (1.2x + 2.3y) \u003d 10 1.2x + 10 2.3y \u003d 12x + 23y.
2) 1.5 (a -2b + 4c) = 1.5a -3b + 6c.
3) a (6m -2n + k) = 6am -2an +ak.
b) convert the expression to identically equal using the commutative and associative property(laws of) addition:
4) x + 4.5 + 2x + 6.5; 5) (3a + 2.1) + 7.8; 6) 5.4s -3 -2.5 -2.3s.
Solution. We apply the laws (properties) of addition:
a+b=b+a(displacement: the sum does not change from the rearrangement of the terms).
(a+b)+c=a+(b+c)(associative: in order to add a third number to the sum of two terms, you can add the sum of the second and third to the first number).
4) x + 4.5 + 2x + 6.5 = (x + 2x) + (4.5 + 6.5) = 3x + 11.
5) (3a + 2.1) + 7.8 = 3a + (2.1 + 7.8) = 3a + 9.9.
6) 6) 5.4s -3 -2.5 -2.3s = (5.4s -2.3s) + (-3 -2.5) = 3.1s -5.5.
V) transform the expression into identically equal using the commutative and associative properties (laws) of multiplication:
7) 4 · X · (-2,5); 8) -3,5 · 2y · (-1); 9) 3a · (-3) · 2s.
Solution. Let's apply the laws (properties) of multiplication:
a b=b a(displacement: permutation of factors does not change the product).
(a b) c=a (b c)(combinative: 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).
7) 4 · X · (-2,5) = -4 · 2,5 · x = -10x.
8) -3,5 · 2y · (-1) = 7y.
9) 3a · (-3) · 2s = -18as.
If an algebraic expression is given as a reducible fraction, then using the fraction reduction rule, it can be simplified, i.e. replace identically equal to it by a simpler expression.
Examples. Simplify by using fraction reduction.
Solution. To reduce a fraction is to divide its numerator and denominator by the same number (expression) other than zero. Fraction 10) will be reduced by 3b; fraction 11) reduce by A and fraction 12) reduce by 7n. We get:
Algebraic expressions are used to formulate formulas.
A formula is an algebraic expression written as an equality that expresses the relationship between two or more variables. Example: the path formula you know s=v t(s - distance traveled, v - speed, t - time). Remember what other formulas you know.
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Rule value of an algebraic expression
Numeric and Algebraic Expressions
In elementary school, you learned how to do calculations with whole and fractional numbers solve equations, get to know geometric shapes, With coordinate plane. All this was the content of one school subject "Mathematics". In fact, such an important field of science as mathematics is subdivided into a huge number of independent disciplines: algebra, geometry, probability theory, mathematical analysis, mathematical logic, mathematical statistics, game theory, etc. Each discipline has its own objects of study, its own methods of cognition of reality.
Algebra, which we are about to study, gives a person the opportunity not only to perform various calculations but also teaches him to do it as quickly and rationally as possible. The person who owns algebraic methods, has an advantage over those who do not own these methods: he calculates faster, more successfully orients himself in life situations makes decisions more clearly, thinks better. Our task is to help you master algebraic methods, your task is not to resist learning, to be ready to follow us, overcoming difficulties.
In fact, in the lower grades, a window has already been opened for you in Magic world algebra, because algebra primarily studies numerical and algebraic expressions.
Recall that a numeric expression is any record made up of numbers and signs of arithmetic operations (composed, of course, with meaning: for example, 3 + 57 is a numeric expression, while 3 + : is not a numeric expression, but a meaningless set of characters). For some reason (we will talk about them later), letters are often used instead of specific numbers (mainly from the Latin alphabet); then an algebraic expression is obtained. These expressions can be very cumbersome. Algebra teaches to simplify them using different rules, laws, properties, algorithms, formulas, theorems.
Example 1. Simplify the numeric expression:
Solution. Now we will remember something together with you, and you will see how many algebraic facts you already know. First of all, you need to develop a plan for the implementation of calculations. To do this, you will have to use the conventions accepted in mathematics about the order of actions. Procedure in this example will be like this:
1) find the value A of the expression in the first brackets:
A \u003d 2.73 + 4.81 + 3.27 - 2.81;
2) find the value of the expression in the second brackets:
3) we divide A by B - then we will know what number C is contained in the numerator (i.e., above the horizontal line);
4) find the value D of the denominator (i.e., the expression contained under the horizontal bar):
D \u003d 25 - 37-0.4;
5) we divide C by D - this will be the desired result. So, there is a calculation plan (and the presence of a plan is half
success!), let's start its implementation.
1) Find A \u003d 2.73 + 4.81 + 3.27 - 2.81. Of course, you can count in a row or, as they say, “on the forehead”: 2.73 + 4.81, then add to this number
3.27, then subtract 2.81. But a cultured person will not calculate like that. He will remember the commutative and associative laws of addition (however, he does not need to remember them, they are always in his head) and will calculate as follows:
(2,73 + 3,27) + 4,81 — 2,81) = 6 + 2 = 8.
And now once again we will analyze together what mathematical facts we had to remember in the process of solving the example (and not just remember, but also use it).
1. The order of arithmetic operations.
2. Commutative law of addition: a + b = b + a.
4. Combination law of addition:
a + b + c = (a + b) + c = a + (b + c).
5. Associative law of multiplication: abc = (ab)c = a(bc).
6. Concepts common fraction, decimal fraction , a negative number.
7. Arithmetic operations with decimal fractions.
8. Arithmetic operations with ordinary fractions.
10. Action rules with positive and negative numbers. You know all this, but all this is algebraic facts. Thus, you already had some familiarity with algebra in the lower grades. The main difficulty, as can already be seen from Example 1, is that there are quite a lot of such facts, and they must not only be known, but also be able to use, as they say, "at the right time and in the right place." This is what we will learn.
Since the letters that make up the algebraic expression can be given different numerical values (i.e., you can change the values of the letters), these letters are called variables.
b) Similarly, following the order of operations, we successively find:
You can't divide by zero! What does this mean in this case (and in other similar cases)? This means that when : the given algebraic expression does not make sense.
The following terminology is used: if specific values letters (variables) the algebraic expression has a numerical value, then the specified values of the variables are called admissible; if, for specific values of the letters (variables), the algebraic expression does not make sense, then the indicated values of the variables are called invalid.
So, in example 2, the values a = 1 and b = 2, a = 3.7 and b = -1.7 are valid, while the values
invalid (more precisely: the first two pairs of values are valid, and the third pair of values is invalid).
In general, in example 2, such values of the variables a, b will be invalid, for which either a + b = 0, or a - b = 0. For example, a = 7, b = - 7 or a = 28.3, b = 28 ,3 - invalid pairs of values; in the first case, a + b = 0, and in the second case, a - b = 0. In both cases, the denominator of the expression given in this example vanishes, and, we repeat, it is impossible to divide by zero. Now, probably, you yourself will be able to come up with both valid pairs of values for the variables a, b, and invalid pairs of values for these variables in Example 2. Try it!
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Algebra lessons introduce us to various types expressions. As new material arrives, the expressions become more complex. When you get acquainted with the powers, they are gradually added to the expression, complicating it. It also happens with fractions and other expressions.
To make the study of the material as convenient as possible, this is done by certain names in order to be able to highlight them. This article will give a complete overview of all the basic school algebraic expressions.
Monomials and polynomials
Expressions monomials and polynomials are studied in school curriculum starting from 7th grade. Textbooks have given definitions of this kind.
Definition 1
monomials are numbers, variables, their degrees with natural indicator, any works made with their help.
Definition 2
polynomials is called the sum of monomials.
If we take, for example, the number 5, the variable x, the degree z 7, then the products of the form 5 x And 7 x 2 7 z 7 are considered single members. When the sum of monomials of the form is taken 5+x or z 7 + 7 + 7 x 2 7 z 7, then we get a polynomial.
To distinguish a monomial from a polynomial, pay attention to the degrees and their definitions. The concept of coefficient is important. When reducing similar terms, they are divided into the free term of the polynomial or the leading coefficient.
Most often, some actions are performed on monomials and polynomials, after which the expression is reduced to see a monomial. Addition, subtraction, multiplication, and division are performed, relying on an algorithm to perform operations on polynomials.
When there is one variable, it is possible to divide the polynomial into a polynomial, which are represented as a product. This action is called the factorization of a polynomial.
Rational (algebraic) fractions
The concept of rational fractions is studied in grade 8 high school. Some authors call them algebraic fractions.
Definition 3
Rational algebraic fraction They call a fraction in which polynomials or monomials, numbers, take the place of the numerator and denominator.
Consider the example of the record rational fractions of type 3 x + 2 , 2 a + 3 b 4 , x 2 + 1 x 2 - 2 and 2 2 x + - 5 1 5 y 3 x x 2 + 4 . Based on the definition, we can say that every fraction is considered a rational fraction.
Algebraic fractions can be added, subtracted, multiplied, divided, raised to a power. This is discussed in more detail in the section on operations with algebraic fractions. If it is necessary to convert a fraction, they often use the property of reduction and reduction to common denominator.
Rational Expressions
IN school course the concept of irrational fractions is being studied, since it is necessary to work with rational expressions.
Definition 4
Rational Expressions are considered numeric and alphabetic expressions where rational numbers and letters with addition, subtraction, multiplication, division, raising to an integer power.
Rational expressions may not have signs belonging to the function that lead to irrationality. Rational expressions do not contain roots, degrees with fractional irrational exponents, degrees with variables in the exponent, logarithmic expressions, trigonometric functions and so on.
Based on the rule above, we will give examples of rational expressions. From the above definition, we have that both a numerical expression of the form 1 2 + 3 4, and 5, 2 + (- 0, 1) 2 2 - 3 5 - 4 3 4 + 2: 12 7 - 1 + 7 - 2 2 3 3 - 2 1 + 0 , 3 are considered rational. Expressions containing letters are also referred to as rational a 2 + b 2 3 a - 0, 5 b , with variables of the form a x 2 + b x + c and x 2 + x y - y 2 1 2 x - 1 .
All rational expressions are divided into integer and fractional.
Integer rational expressions
Definition 5Integer rational expressions are such expressions that do not contain division into expressions with variables of negative degree.
From the definition, we have that an integer rational expression is also an expression containing letters, for example, a + 1 , an expression containing several variables, for example, x 2 y 3 − z + 3 2 and a + b 3 .
Expressions like x: (y − 1) and 2 x + 1 x 2 - 2 x + 7 - 4 cannot be rational integers, since they have division by an expression with variables.
Fractional rational expressions
Definition 6Fractional rational expression is an expression that contains division by an expression with negative degree variables.
It follows from the definition that fractional rational expressions can be 1: x, 5 x 3 - y 3 + x + x 2 and 3 5 7 - a - 1 + a 2 - (a + 1) (a - 2) 2 .
If we consider expressions of this type (2 x - x 2): 4 and a 2 2 - b 3 3 + c 4 + 1 4, 2, then they are not considered fractional rational, since they do not have expressions with variables in the denominator.
Expressions with powers
Definition 7Expressions that contain powers in any part of the notation are called power expressions or power expressions.
For the concept, we give an example of such an expression. They may not contain variables, for example, 2 3 , 32 - 1 5 + 1 . 5 3 . 5 · 5 - 2 5 - 1 . 5 . Also characteristic power expressions of the form 3 x 3 x - 1 + 3 x , x y 2 1 3 . In order to solve them, it is necessary to perform some transformations.
Irrational expressions, expressions with roots
The root, which has a place in the expression, gives it a different name. They are called irrational.
Definition 8
Irrational expressions name expressions that have signs of roots in the record.
It can be seen from the definition that these are expressions of the form 64 , x - 1 4 3 + 3 3 , 2 + 1 2 - 1 - 2 + 3 2 , a + 1 a 1 2 + 2 , x y , 3 x + 1 + 6 x 2 + 5 x and x + 6 + x - 2 3 + 1 4 x 2 3 + 3 - 1 1 3 . Each of them has at least one root icon. The roots and degrees are connected, so you can see expressions such as x 7 3 - 2 5, n 4 8 · m 3 5: 4 · m 2 n + 3.
Trigonometric expressions
Definition 9trigonometric expression are expressions containing sin , cos , tg and ctg and their inverses - arcsin , arccos , arctg and arcctg .
Examples of trigonometric functions are obvious: sin π 4 cos π 6 cos 6 x - 1 and 2 sin x t g 2 x + 3 , 4 3 t g π - arcsin - 3 5 .
To work with such functions, it is necessary to use the properties, basic formulas of lines and inverse functions. The article transformation of trigonometric functions will reveal this issue in more detail.
Logarithmic Expressions
After getting acquainted with logarithms, we can talk about complex logarithmic expressions.
Definition 10
Expressions that have logarithms are called logarithmic.
An example of such functions would be log 3 9 + ln e , log 2 (4 a b) , log 7 2 (x 7 3) log 3 2 x - 3 5 + log x 2 + 1 (x 4 + 2) .
You can find such expressions where there are degrees and logarithms. This is understandable, since from the definition of the logarithm it follows that this is an exponent. Then we get expressions like x l g x - 10 , log 3 3 x 2 + 2 x - 3 , log x + 1 (x 2 + 2 x + 1) 5 x - 2 .
To deepen the study of the material, you should refer to the material on the transformation of logarithmic expressions.
Fractions
There are expressions of a special kind, which are called fractions. Since they have a numerator and a denominator, they can contain not just numeric values, but also expressions of any type. Consider the definition of a fraction.
Definition 11
Shot they call such an expression that has a numerator and a denominator, in which there are both numerical and alphabetic designations or expressions.
Examples of fractions that have numbers in the numerator and denominator look like this 1 4 , 2 , 2 - 6 2 7 , π 2 , - e π , (− 15) (− 2) . The numerator and denominator can contain both numerical and alphabetic expressions of the form (a + 1) 3 , (a + b + c) (a 2 + b 2) , 1 3 + 1 - 1 3 - 1 1 1 + 1 1 + 1 5 , cos 2 α - sin 2 α 1 + 3 t g α , 2 + ln 5 ln x .
Although expressions such as 2 5 − 3 7 , x x 2 + 1: 5 are not fractions, however, they do have a fraction in their notation.
General expression
Senior classes consider tasks of increased difficulty, which contains all the combined tasks of group C in the USE. These expressions are particularly complex and have various combinations of roots, logarithms, powers, and trigonometric functions. These are jobs like x 2 - 1 sin x + π 3 or sin a r c t g x - a x 1 + x 2 .
Their appearance indicates that it can be attributed to any of the above species. Most often they are not classified as any, since they have a specific combined solution. They are considered as expressions of a general form, and no additional clarifications or expressions are used for description.
When solving such an algebraic expression, it is always necessary to pay attention to its notation, the presence of fractions, powers, or additional expressions. This is necessary in order to accurately determine the way to solve it. If you are not sure about its name, then it is recommended to call it an expression general type and decide according to the above algorithm.
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The arithmetic operation that is performed last when calculating the value of the expression is the "main".
That is, if you substitute some (any) numbers instead of letters, and try to calculate the value of the expression, then if last action there will be a multiplication - it means that we have a product (the expression is decomposed into factors).
If the last action is addition or subtraction, this means that the expression is not factored (and therefore cannot be reduced).
To fix it yourself, a few examples:
Examples:
Solutions:
1. I hope you did not immediately rush to cut and? It was still not enough to “reduce” units like this:
The first step should be to factorize:
4. Addition and subtraction of fractions. Bringing fractions to a common denominator.
Adding and subtracting ordinary fractions is a well-known operation: we look for a common denominator, multiply each fraction by the missing factor and add / subtract the numerators.
Let's remember:
Answers:
1. The denominators and are coprime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:
2. Here the common denominator is:
3. First thing here mixed fractions turn them into wrong ones, and then - according to the usual scheme:
It is quite another matter if the fractions contain letters, for example:
Let's start simple:
a) Denominators do not contain letters
Here everything is the same as with ordinary numerical fractions: we find a common denominator, multiply each fraction by the missing factor and add / subtract the numerators:
now in the numerator you can bring similar ones, if any, and factor them:
Try it yourself:
Answers:
b) Denominators contain letters
Let's remember the principle of finding a common denominator without letters:
First of all, we determine the common factors;
Then we write out all the common factors once;
and multiply them by all other factors, not common ones.
To determine the common factors of the denominators, we first decompose them into simple factors:
We emphasize the common factors:
Now we write out the common factors once and add to them all non-common (not underlined) factors:
This is the common denominator.
Let's get back to the letters. The denominators are given in exactly the same way:
We decompose the denominators into factors;
determine common (identical) multipliers;
write out all the common factors once;
We multiply them by all other factors, not common ones.
So, in order:
1) decompose the denominators into factors:
2) determine the common (identical) factors:
3) write out all the common factors once and multiply them by all the other (not underlined) factors:
So the common denominator is here. The first fraction must be multiplied by, the second - by:
By the way, there is one trick:
For example: .
We see the same factors in the denominators, only everything with different indicators. The common denominator will be:
to the extent
to the extent
to the extent
in degree.
Let's complicate the task:
How to make fractions have the same denominator?
Let's remember the basic property of a fraction:
Nowhere is it said that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!
See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What has been learned?
So, another unshakable rule:
When you bring fractions to a common denominator, use only the multiplication operation!
But what do you need to multiply to get?
Here on and multiply. And multiply by:
Expressions that cannot be factorized will be called "elementary factors".
For example, is an elementary factor. - Same. But - no: it is decomposed into factors.
What about expression? Is it elementary?
No, because it can be factorized:
(you already read about factorization in the topic "").
So, the elementary factors into which you decompose the expression with letters is an analogue prime factors into which you decompose the numbers. And we will do the same with them.
We see that both denominators have a factor. It will go to the common denominator in the power (remember why?).
The multiplier is elementary, and they do not have it in common, which means that the first fraction will simply have to be multiplied by it:
Another example:
Solution:
Before multiplying these denominators in a panic, you need to think about how to factor them? Both of them represent:
Great! Then:
Another example:
Solution:
As usual, we factorize the denominators. In the first denominator, we simply put it out of brackets; in the second - the difference of squares:
It would seem that there are no common factors. But if you look closely, they are already so similar ... And the truth is:
So let's write:
That is, it turned out like this: inside the bracket, we swapped the terms, and at the same time, the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.
Now we bring to a common denominator:
Got it? Now let's check.
Tasks for independent solution:
Answers:
Here we must remember one more thing - the difference of cubes:
Please note that the denominator of the second fraction does not contain the formula "square of the sum"! The square of the sum would look like this:
A is the so-called incomplete square of the sum: the second term in it is the product of the first and last, and not their doubled product. The incomplete square of the sum is one of the factors in the expansion of the difference of cubes:
What if there are already three fractions?
Yes, the same! First of all, let's make it so that maximum amount factors in the denominators were the same:
Pay attention: if you change the signs inside one bracket, the sign in front of the fraction changes to the opposite. When we change the signs in the second bracket, the sign in front of the fraction is reversed again. As a result, he (the sign in front of the fraction) has not changed.
We write out the first denominator in full in the common denominator, and then we add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it goes like this:
Hmm ... With fractions, it’s clear what to do. But what about the two?
It's simple: you know how to add fractions, right? So, you need to make sure that the deuce becomes a fraction! Remember: a fraction is a division operation (the numerator is divided by the denominator, in case you suddenly forgot). And there is nothing easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:
Exactly what is needed!
5. Multiplication and division of fractions.
Well, the hardest part is now over. And ahead of us is the simplest, but at the same time the most important:
Procedure
What is the procedure for calculating a numeric expression? Remember, considering the value of such an expression:
Did you count?
It should work.
So, I remind you.
The first step is to calculate the degree.
The second is multiplication and division. If there are several multiplications and divisions at the same time, you can do them in any order.
And finally, we perform addition and subtraction. Again, in any order.
But: the parenthesized expression is evaluated out of order!
If several brackets are multiplied or divided by each other, we first evaluate the expression in each of the brackets, and then multiply or divide them.
What if there are other parentheses inside the brackets? Well, let's think: some expression is written inside the brackets. What is the first thing to do when evaluating an expression? That's right, calculate brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.
So, the order of actions for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):
Okay, it's all simple.
But that's not the same as an expression with letters, is it?
No, it's the same! Only instead of arithmetic operations it is necessary to do algebraic operations, that is, the operations described in the previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use it when working with fractions). Most often, for factorization, you need to use i or simply take the common factor out of brackets.
Usually our goal is to represent an expression as a product or quotient.
For example:
Let's simplify the expression.
1) First we simplify the expression in brackets. There we have the difference of fractions, and our goal is to represent it as a product or quotient. So, we bring the fractions to a common denominator and add:
It is impossible to simplify this expression further, all factors here are elementary (do you still remember what this means?).
2) We get:
Multiplication of fractions: what could be easier.
3) Now you can shorten:
OK it's all over Now. Nothing complicated, right?
Another example:
Simplify the expression.
First, try to solve it yourself, and only then look at the solution.
Solution:
First of all, let's define the procedure.
First, let's add the fractions in brackets, instead of two fractions, one will turn out.
Then we will do the division of fractions. Well, we add the result with the last fraction.
I will schematically number the steps:
Now I will show the whole process, tinting the current action with red:
1. If there are similar ones, they must be brought immediately. At whatever moment we have similar ones, it is advisable to bring them right away.
2. The same goes for reducing fractions: as soon as an opportunity arises to reduce, it must be used. The exception is fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.
Here are some tasks for you to solve on your own:
And promised at the very beginning:
Answers:
Solutions (brief):
If you coped with at least the first three examples, then you, consider, have mastered the topic.
Now on to learning!
EXPRESSION CONVERSION. SUMMARY AND BASIC FORMULA
Basic simplification operations:
- Bringing similar: to add (bring) like terms, it is necessary to add their coefficients and assign the letter part.
- Factorization: taking the common factor out of brackets, applying, etc.
- Fraction reduction: the numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, from which the value of the fraction does not change.
1) numerator and denominator factorize
2) if there are common factors in the numerator and denominator, they can be crossed out.IMPORTANT: only multipliers can be reduced!
- Addition and subtraction of fractions:
; - Multiplication and division of fractions:
;