An integer expression is a mathematical expression made up of numbers and literal variables using the operations of addition, subtraction and multiplication. Integers also include expressions that involve division by any number other than zero.
Whole expression examples
Below are some examples of integer expressions:
1. 12*a^3 + 5*(2*a -1);
3. 4*y- ((5*y+3)/5) -1;
Fractional Expressions
If an expression contains division by a variable or by another expression containing a variable, then such an expression is not an integer. This expression is called a fractional expression. Let us give a complete definition of a fractional expression.
A fractional expression is a mathematical expression that, in addition to the operations of addition, subtraction and multiplication performed with numbers and letter variables, as well as division by a number not equal to zero, also contains division into expressions with letter variables.
Examples of fractional expressions:
1. (12*a^3 +4)/a
3. 4*x- ((5*y+3)/(5-y)) +1;
Fractional and integer expressions make up two large sets of mathematical expressions. If we combine these sets, we get a new set called rational expressions. That is, rational expressions are all integer and fractional expressions.
We know that entire expressions make sense for any values of the variables that are included in it. This follows from the fact that to find the value of an entire expression it is necessary to perform actions that are always possible: addition, subtraction, multiplication, division by a number other than zero.
Fractional expressions, unlike whole ones, may not make sense. Since there is an operation of dividing by a variable or an expression containing variables, and this expression can become zero, but dividing by zero is impossible. The values of the variables at which the fractional expression will make sense are called acceptable values variables.
Rational fraction
One of the special cases rational expressions will be a fraction whose numerator and denominator are polynomials. For such a fraction in mathematics there is also a name - a rational fraction.
A rational fraction will make sense if its denominator is not zero. That is, all values of variables for which the denominator of the fraction is different from zero will be acceptable.
In the distant past, when the number system had not yet been invented, people counted everything on their fingers. With the advent of arithmetic and the basics of mathematics, it became much easier and more practical to keep records of goods, products, as well as household items. However, what does it look like? modern system calculus: what types are existing numbers divided into and what does “rational form of numbers” mean? Let's figure it out.
How many types of numbers are there in mathematics?
The very concept of “number” denotes a certain unit of any object that characterizes its quantitative, comparative or ordinal indicators. In order to correctly calculate the number of certain things or carry out certain mathematical operations with numbers (add, multiply, etc.), you should first of all become familiar with the varieties of these same numbers.
So, existing numbers can be divided into the following categories:
- Natural numbers are those numbers with which we count the number of objects (at least natural number equals 1, it is logical that the series of natural numbers is infinite, i.e. there is no largest natural number). The set of natural numbers is usually denoted by the letter N.
- Whole numbers. This set includes everyone and adds to it negative values, including the number "zero". The designation of a set of integers is written as the Latin letter Z.
- Rational numbers are those that we can mentally transform into a fraction, the numerator of which will belong to the set of integers, and the denominator will belong to the set of natural numbers. Below we will look in more detail at what a “rational number” means and give some examples.
- - a set that includes all rational and This set is denoted by the letter R.
- Complex numbers contain part of a real number and part of a variable number. Used in solving various cubic equations, which, in turn, can have a negative expression in the formulas (i 2 = -1).
What does “rational” mean: let’s look at examples
If rational numbers are considered to be those that we can represent in the form common fraction, then it turns out that all positive and negative integers are also included in the set of rationals. After all, any whole number, for example 3 or 15, can be represented as a fraction, where the denominator is one.
Fractions: -9/3; 7/5, 6/55 - here are examples rational numbers.
What does "rational expression" mean?
Go ahead. We have already discussed what the rational form of numbers means. Let's now imagine a mathematical expression that consists of a sum, difference, product or quotient different numbers and variables. Here's an example: a fraction in which the numerator is the sum of two or more integers, and the denominator contains both an integer and some variable. It is this kind of expression that is called rational. Based on the rule “you cannot divide by zero,” you can guess that the value of this variable cannot be such that the denominator value becomes zero. Therefore, when solving a rational expression, you must first determine the range of the variable. For example, if the denominator has the following expression: x+5-2, then it turns out that “x” cannot be equal to -3. Indeed, in this case, the entire expression turns to zero, so when solving it is necessary to exclude the integer -3 for this variable.
How to solve rational equations correctly?
Rational expressions can contain quite a lot a large number of numbers and even 2 variables, so sometimes solving them becomes difficult. To facilitate the solution of such an expression, it is recommended to perform certain operations in a rational way. So, what does “in a rational way” mean and what rules should be applied when making a decision?
- The first type, when it is enough just to simplify the expression. To do this, you can resort to the operation of reducing the numerator and denominator to an irreducible value. For example, if the numerator contains the expression 18x, and the denominator 9x, then, by reducing both exponents by 9x, we simply get an integer equal to 2.
- The second method is practical when we have a monomial in the numerator and a polynomial in the denominator. Let's look at an example: in the numerator we have 5x, and in the denominator - 5x + 20x 2. In this case, it is best to take the variable in the denominator out of brackets, we get the following form of the denominator: 5x(1+4x). Now you can use the first rule and simplify the expression by canceling 5x in the numerator and denominator. As a result, we get a fraction of the form 1/1+4x.
What operations can you perform with rational numbers?
The set of rational numbers has a number of its own characteristics. Many of them are very similar to the characteristics present in integers and natural numbers, due to the fact that the latter are always included in the set of rationals. Here are a few properties of rational numbers, knowing which you can easily solve any rational expression.
- The commutative property allows you to sum two or more numbers, regardless of their order. Simply put, changing the places of the terms does not change the sum.
- The distributive property allows you to solve problems using the distribution law.
- And finally, the operations of addition and subtraction.
Even schoolchildren know what “rational form of numbers” means and how to solve problems based on such expressions, so an educated adult simply needs to remember at least the basics of the set of rational numbers.
Any fractional expression (clause 48) can be written in the form , where P and Q are rational expressions, and Q necessarily contains variables. Such a fraction is called a rational fraction.
Examples of rational fractions:
The main property of a fraction is expressed by an identity that is fair under the conditions here - a whole rational expression. This means that the numerator and denominator of a rational fraction can be multiplied or divided by the same non-zero number, monomial or polynomial.
For example, the property of a fraction can be used to change the signs of members of a fraction. If the numerator and denominator of a fraction are multiplied by -1, we get Thus, the value of the fraction will not change if the signs of the numerator and denominator are simultaneously changed. If you change the sign of only the numerator or only the denominator, then the fraction will change its sign:
For example,
60. Reducing rational fractions.
To reduce a fraction means to divide the numerator and denominator of the fraction by a common factor. The possibility of such a reduction is due to the basic property of the fraction.
In order to reduce a rational fraction, you need to factor the numerator and denominator. If it turns out that the numerator and denominator have common factors, then the fraction can be reduced. If there are no common factors, then converting a fraction through reduction is impossible.
Example. Reduce fraction
Solution. We have
The reduction of a fraction is carried out under the condition .
61. Reducing rational fractions to a common denominator.
The common denominator of several rational fractions is a whole rational expression that is divided by the denominator of each fraction (see paragraph 54).
For example, the common denominator of fractions is a polynomial since it is divisible by both and by and polynomial and polynomial and polynomial, etc. Usually they take such a common denominator that any other common denominator is divisible by Echosen. This simplest denominator is sometimes called the lowest common denominator.
In the example discussed above, the common denominator is We have
Reducing these fractions to a common denominator is achieved by multiplying the numerator and denominator of the first fraction by 2. and the numerator and denominator of the second fraction by Polynomials are called additional factors for the first and second fractions, respectively. The additional factor for a given fraction is equal to the quotient of dividing the common denominator by the denominator of the given fraction.
To reduce several rational fractions to a common denominator, you need:
1) factor the denominator of each fraction;
2) create a common denominator by including as factors all the factors obtained in step 1) of the expansions; if a certain factor is present in several expansions, then it is taken with an exponent equal to the largest of the available ones;
3) find additional factors for each of the fractions (for this, the common denominator is divided by the denominator of the fraction);
4) by multiplying the numerator and denominator of each fraction by an additional factor, bring the fraction to a common denominator.
Example. Reduce a fraction to a common denominator
Solution. Let's factorize the denominators:
The following factors must be included in the common denominator: and the least common multiple of the numbers 12, 18, 24, i.e. This means that the common denominator has the form
Additional factors: for the first fraction for the second for the third. So, we get:
62. Addition and subtraction of rational fractions.
The sum of two (and in general any finite number) rational fractions with the same denominators is identically equal to a fraction with the same denominator and with a numerator equal to the sum of the numerators of the fractions being added:
The situation is similar in the case of subtracting fractions with like denominators:
Example 1: Simplify an expression
Solution.
To add or subtract rational fractions with different denominators, you must first reduce the fractions to a common denominator, and then perform operations on the resulting fractions with the same denominators.
Example 2: Simplify an expression
Solution. We have
63. Multiplication and division of rational fractions.
The product of two (and in general any finite number) rational fractions is identically equal to a fraction whose numerator is equal to the product of the numerators, and the denominator is equal to the product of the denominators of the fractions being multiplied:
The quotient of dividing two rational fractions is identically equal to a fraction whose numerator is equal to the product of the numerator of the first fraction and the denominator of the second fraction, and the denominator is the product of the denominator of the first fraction and the numerator of the second fraction:
The formulated rules of multiplication and division also apply to the case of multiplication or division by a polynomial: it is enough to write this polynomial in the form of a fraction with a denominator of 1.
Given the possibility of reducing a rational fraction obtained as a result of multiplying or dividing rational fractions, they usually strive to factorize the numerators and denominators of the original fractions before performing these operations.
Example 1: Perform multiplication
Solution. We have
Using the rule for multiplying fractions, we get:
Example 2: Perform division
Solution. We have
Using the division rule, we get:
64. Raising a rational fraction to a whole power.
To raise a rational fraction to a natural power, you need to raise the numerator and denominator of the fraction separately to this power; the first expression is the numerator, and the second expression is the denominator of the result:
Example 1: Convert to a fraction of power 3.
Solution Solution.
When raising a fraction to a negative integer power, an identity is used that is valid for all values of the variables for which .
Example 2: Convert an expression to a fraction
65. Transformation of rational expressions.
Transforming any rational expression comes down to adding, subtracting, multiplying and dividing rational fractions, as well as raising a fraction to a natural power. Any rational expression can be converted into a fraction, the numerator and denominator of which are whole rational expressions; this is usually the goal identity transformations rational expressions.
Example. Simplify an expression
66. The simplest transformations of arithmetic roots (radicals).
When converting arithmetic korias, their properties are used (see paragraph 35).
Let's look at a few examples of using properties arithmetic roots for the simplest transformations of radicals. In this case, we will consider all variables to take only non-negative values.
Example 1. Extract the root of a product
Solution. Applying the 1° property, we get:
Example 2. Remove the multiplier from under the root sign
Solution.
This transformation is called removing the factor from under the root sign. The purpose of the transformation is to simplify the radical expression.
Example 3: Simplify.
Solution. By the property of 3° we have. Usually they try to simplify the radical expression, for which they take the factors out of the corium sign. We have
Example 4: Simplify
Solution. Let's transform the expression by introducing a factor under the sign of the root: By property 4° we have
Example 5: Simplify
Solution. By the property of 5°, we have the right to divide the exponent of the root and the exponent of the radical expression by the same natural number. If in the example under consideration we divide the indicated indicators by 3, we get .
Example 6. Simplify expressions:
Solution, a) By property 1° we find that to multiply roots of the same degree, it is enough to multiply the radical expressions and extract the root of the same degree from the result obtained. Means,
b) First of all, we must reduce the radicals to one indicator. According to the property of 5°, we can multiply the exponent of the root and the exponent of the radical expression by the same natural number. Therefore, Next, we now have in the resulting result dividing the exponents of the root and the degree of the radical expression by 3, we obtain.
This lesson will cover basic information about rational expressions and their transformations, as well as examples of transformations of rational expressions. This topic kind of summarizes the topics we have studied so far. Transformations of rational expressions involve addition, subtraction, multiplication, division, exponentiation algebraic fractions, reduction, factorization, etc. As part of the lesson, we will look at what a rational expression is, and also analyze examples of their transformation.
Subject:Algebraic fractions. Arithmetic operations on algebraic fractions
Lesson:Basic information about rational expressions and their transformations
Definition
Rational expression is an expression consisting of numbers, variables, arithmetic operations and the operation of exponentiation.
Let's look at an example of a rational expression:
Special cases of rational expressions:
1st degree: 
;
2. monomial: ;
3. fraction: .
Converting a rational expression is a simplification of a rational expression. The order of actions when transforming rational expressions: first there are operations in brackets, then multiplication (division) operations, and then addition (subtraction) operations.
Let's look at several examples of transforming rational expressions.
Example 1
Solution:
Let's solve this example step by step. The action in parentheses is executed first.
Answer:
Example 2
Solution:
Answer:
Example 3
Solution:
Answer: .
Note: perhaps when you see this example An idea arose: reduce the fraction before reducing it to a common denominator. Indeed, it is absolutely correct: first it is advisable to simplify the expression as much as possible, and then transform it. Let's try to solve this same example in the second way.
As you can see, the answer turned out to be absolutely similar, but the solution turned out to be somewhat simpler.
In this lesson we looked at rational expressions and their transformations, as well as several specific examples transformation data.
Bibliography
1. Bashmakov M.I. Algebra 8th grade. - M.: Education, 2004.
2. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. and others. Algebra 8. - 5th ed. - M.: Education, 2010.