A rational expression in algebraic expression. If there is no division or fraction, if there is a division and fraction, but there is no variable in the division or denominator, it is called an algebraic expression.
Algebraic expressions can be divided into definitions and operations, definitions can be divided into monomials and polynomials, and operations can be divided into addition, subtraction, multiplication and division.
Addition and subtraction involve merging similar items. Multiplication and division include basic operations, rules and formulas. Basic operations can be divided into power operations. Rules can be divided into algebra and division, and formulas can be divided into multiplication formula, zero exponential power and negative integer exponential power.
Algebraic expressions and similar items
1. Single item
(1) The concept of monomial: Algebraic expressions such as the product of numbers and letters are called monomials, and a single number or letter is also a monomial.
Note: Numbers and letters have a product relationship.
(2) Single coefficient: the numerical factors and property symbols in a single item are called single coefficient.
If a single item contains only numerical factors, the positive single item coefficient is 1 and the negative single item coefficient is-1.
(3) The number of monomials: The sum of the indices of all the letters in the monomials is called the number of monomials.
2.polynomial
The concept of (1) polynomial: The sum of several monomials is called polynomial. In polynomials, each monomial is called a polynomial term, and the term without letters is called a constant term. Polynomials with several terms are called polynomials. The symbols in polynomials are regarded as the natural symbols of each term. A univariate polynomial of degree n can have at most N+ 1 terms.
(2) Degree of Polynomial: The degree of the term with the highest degree in the polynomial is the degree of the polynomial.
(3) the arrangement of polynomials:
1. Sorting polynomials in descending alphabetical order is called sorting polynomials in descending alphabetical order.
2. Arranging a polynomial according to the exponent of a letter from small to large is called arranging polynomials according to the ascending power of this letter.
Since a polynomial is the sum of several monomials, the position of each term can be exchanged by the addition algorithm, while keeping the value of the original polynomial unchanged.
In order to facilitate the calculation of polynomials, a polynomial is usually arranged in a neat and simple form in a certain order, which is the arrangement of polynomials.
Pay attention to when doing polynomial arrangement problems:
(1) Since a single item contains its preceding attribute symbol, the attribute symbol of each item should still be regarded as a part of the item and moved together.
(2) The arrangement of polynomials with two or more letters should pay attention to:
A. first of all, it must be arranged according to the index of which letter.
B. decide whether to arrange according to this letter.
(3) Algebraic expression:
Monomial and polynomial are collectively called algebraic expressions.
(4) the concept of similar items:
Items with the same letters and times are called similar items, and several constant items are also called similar items.
When mastering the concept of similar items, we should pay attention to:
1. To judge whether several monomials or terms are similar, two conditions must be mastered:
(1) contains the same letters.
The same letter has the same number of times.
2. Similar items have nothing to do with coefficient or alphabetical order.
3. Several constant terms are similar.
(5) Merge similar items:
1. The concept of merging similar projects:
Merging similar terms in polynomials into one term is called merging similar terms.
2. Rules for merging similar projects:
The coefficients of similar items are added together, and the results are taken as coefficients, and the indexes of letters and letters remain unchanged.
3. To merge similar projects:
(1). Find similar projects accurately.
(2) Reverse the distribution law, add the coefficients of similar items together (enclosed in brackets), and keep the letters and their indices unchanged.
(3) Write the merged result.
When mastering the merger of similar projects, we should pay attention to:
1. If the coefficients of two similar items are opposite, the result after merging similar items is 0.
2. Don't leave out items that can't be merged.
3. As long as there are no more similar items, it is the result (either a single item or a polynomial).
The key to merging similar items: correctly judging similar items.
Algebraic expression and multiplication of algebraic expression
Algebraic expressions can be divided into definitions and operations, definitions can be divided into monomials and polynomials, and operations can be divided into addition, subtraction, multiplication and division.
Addition and subtraction involve merging similar items. Multiplication and division include basic operations, rules and formulas. Basic operations can be divided into power operations. Rules can be divided into algebra and division, and formulas can be divided into multiplication formula, zero exponential power and negative integer exponential power.
The power rule of the same base: multiply with the power of the same base and add with the index of the same base.
Power law: power, constant basis, exponential multiplication.
Power law of product: the power of product is equal to the power obtained by multiplying the factors of product respectively and then multiplying them.
The multiplication of monomials and monomials has the following rules: the monomials are multiplied by their coefficients and the same base respectively, and other letters and their exponents are kept as the factorial of the product.
There are the following rules for the multiplication of monomial and polynomial: the multiplication of monomial and polynomial is to multiply each term of polynomial with monomial, and then add the products.
Polynomial and polynomial multiplication have the following rules: polynomial and polynomial multiplication, first multiply each term of one polynomial with each term of another polynomial, and then add the obtained products.
Square difference formula: the product of the sum of two numbers and the difference between these two numbers is equal to the square difference between these two numbers.
Complete square formula: the square of the sum of two numbers is equal to the sum of the squares of these two numbers, plus twice the product of these two numbers. The square of the difference between two numbers is equal to the sum of the squares of these two numbers, MINUS twice the product of these two numbers.
Same base powers divides, the base remains the same, and the exponent is subtracted.
On the learning points of algebraic expressions
Tu Xinmin
Algebraic formula is the most basic formula in algebra, so it is necessary to introduce algebraic formula and learn the following contents (such as fractions, quadratic equations with one variable, etc.). ). On the basis of studying rational number operations, simple algebraic expressions, linear equations and inequalities, algebraic expressions are introduced. In fact, the relevant contents of algebraic expressions have been learned in the sixth grade, but now the contents of algebraic expressions are more applicable than in the past, which increases the background of practical application.
Block diagram of knowledge structure in this chapter:
There are many knowledge points in this chapter that are important or difficult. The key points and difficulties are as follows.
First, the four operations of algebraic expressions
Addition and subtraction of 1. algebraic expressions
Merging similar items is the key and difficult point. When merging similar items, we should pay attention to the following three points: ① Only by mastering the concept of similar items can we distinguish similar items and accurately grasp the two standard letters and letter indexes for judging similar items; (2) The definition of merging similar terms means merging similar terms in polynomials into one term. After merging similar terms, the number of terms in the formula will be reduced, thus simplifying the polynomial; (3) "Merging" refers to adding the coefficients of similar items, and the obtained results are used as new coefficients, and the letters and letter indexes of similar items should remain unchanged.
2. Multiplication and division of algebraic expressions
The emphasis is on multiplication and division of algebraic expressions, especially multiplication formulas. It is difficult for students to master the structural characteristics of multiplication formula and the broad meaning of letters in the formula. Therefore, the flexible application of multiplication formula is difficult, and the handling of symbols in brackets is another difficulty when adding (or removing) brackets. Parentheses (or brackets) are the deformation of polynomials, which should be carried out according to the law of parenthesis (or brackets). In the multiplication and division of algebraic expressions, the single multiplication and division is the key, because the multiplication and division of general polynomials should be "transformed" into the single multiplication and division.
The main problems of the four operations of algebraic expressions are:
Four operations of (1) monomial
This kind of questions mostly appear in the form of multiple-choice questions and application questions, which are characterized by examining four operations of monomials.
(2) Operation of monomial and polynomial
This kind of problems mostly appear in the form of solving problems, which are highly skilled and characterized by examining the four operations of monomials and polynomials.
Second, factorization.
The difficulty is the four basic methods of factorization (raising common factor, using formula, grouping factorization and cross multiplication). Factorization is the reverse deformation of algebraic expression multiplication, and the introduction of factorization should firmly grasp this point.
Power refers to the result of power operation. N m means n times m times. The result of taking n m as a power is called the m power of n.
Where n is called the base and m is called the exponent (written as superscript). When the superscript cannot be used, for example, in programming languages or e-mails, it is usually written as n m or n**m, or it is written as n↑m through Gartner arrow notation, which is read as "m power of n".
When the index is 1, it is generally not written because it is the same as the bottom value; When the index is 2 or 3, it can be read as "the square of n" and "the cube of n"
The meaning of n m can also be regarded as that the initial value of 1× n× n...: 1 (the unit element of multiplication) is multiplied by the radix index so many times. With this definition, it is easy to think of how to generalize the exponent 0 and negative numbers: the zeroth degree of all numbers except 0 is 1, that is, n0 =1; When the exponent of the power is negative, it is equal to1/n m.
The power of exponential fraction is defined as x^m/n = n√x^m √ x m √ x m.
Power does not conform to the associative law and commutative law.
Because the power of ten is easy to calculate, just add zero after it, so scientific notation simplifies the way of recording numbers; The power of two is very useful in computer science.
Rational number (rational number)
Integers and fractions are collectively called rational numbers, and any rational number can be written in the form of fraction m/n, where m and n are integers, and n≠0, m and n are coprime.
Decimals with infinite cycles and numbers with infinite roots are called irrational numbers, such as π, 3.1414.50080.00000000006 ......
Rational numbers are just the opposite. Integer, fraction and 0 are collectively called rational numbers.
Include integers and fractions, and can also be expressed as finite decimals or infinite cyclic decimals.
This definition applies to decimal and other decimal (such as binary) numbers.
Mathematically, a rational number is the ratio of an integer a to a nonzero integer b, which is usually written as a/b, so it is also called a fraction. The Greek name is λ ο γ ο? 0? 9, the original meaning is "rational number", but the Chinese translation is not appropriate, and it gradually becomes "rational number". Real numbers that are not rational numbers are called irrational numbers.
The set of all rational numbers is expressed as q, and the fractional part of rational numbers is finite or cyclic.
Rational numbers are divided into integers and fractions.
Integers are divided into positive integers, negative integers and 0.
Scores are divided into positive scores and negative scores.
Positive integers and 0 are also called natural numbers.
For example, 3, -98. 1 1, 5.272 and 7/22 are all rational numbers.
All rational numbers form a set, that is, rational number set, which is represented by bold letter Q, while some modern math books are represented by hollow letter Q.
The rational number set is a subset of the real number set. For related contents, see number system expansion.
A set of rational numbers is a field, that is, four operations can be performed in it (except that 0 is a divisor). For these operations, the following algorithms hold (a, b, c, etc. Represents any rational number):
① the commutative law of addition A+B = B+A;
② the associative law of addition A+(B+C) = (A+B)+C;
(3) There is a number 0, so that 0+a = a+0 = a;
(4) For any rational number A, there is an addition inverse element, which is denoted as -a, so that a+(-a) = (-a)+a = 0;
⑤ The commutative law of multiplication AB = BA
⑥ Multiplicative associative law A (BC) = (AB) C;
⑦ Distribution law A (B+C) = AB+AC;
⑧ Multiplication has a unit 1≠0, so that for any rational number A,1a = a1= a;
Pet-name ruby For rational number A which is not 0, there is a multiplication inverse 1/a, so a (1/a) = (1/a) A =1.
⑩ 0a = 0 Text explanation: whether a number is multiplied by 0 or 0.
In addition, rational number is an ordered domain, that is, there is an ordered relation ≤
Rational number is also an Archimedes field, that is, rational numbers a and b, a≥0, B >；; 0, we can find a natural number n, which makes nb >；; Answer: It is not difficult to infer that there is no maximum rational number.
The name rational number is worth mentioning. The name "rational number" is puzzling, and rational numbers are no more "reasonable" than other numbers. In fact, this seems to be a mistake in translation. The word rational number comes from the west and is rational in English. Rational usually means "rational". China translated western scientific works in modern times into "rational numbers" according to Japanese translation methods. However, this word comes from ancient Greece, and its English root is ratio, which means ratio (the root here is English and the Greek meaning is the same). So the meaning of this word is also very clear, that is, the "ratio" of integers. In contrast, "irrational number" is a number that cannot be accurately expressed as the ratio of two integers, but it is not unreasonable.
Rational number addition and subtraction mixed operation
1. The significance of unifying addition and subtraction of rational numbers into addition;
For subtraction in addition and subtraction mixed operation, we can convert subtraction into addition according to the rational number subtraction rule, thus unifying the mixed operation into addition operation. The unified formula is the sum of several positive numbers or negative numbers. We call this formula algebraic sum.
2. The method and steps of rational number addition and subtraction mixed operation:
(1) The subtraction in rational number mixing operation is converted into addition by using the subtraction rule.
(2) Use the law of addition, additive commutative law and the law of addition combination to perform simple operations.
The concepts of absolute value and reciprocal in rational number range have the same meaning in real number range.
Generally speaking, rational numbers are classified as follows:
Integer, fraction; Positive numbers, negative numbers and zero; Negative rational number
Integer and fraction are collectively called rational numbers, which can be expressed in the form of a/b, where a and b are both integers and prime numbers. We often use rational numbers in our daily life. Such as how much money, how many kilograms, etc.
Any real number that can't be expressed in a/b form is irrational, also called infinite acyclic decimal.
A difficult problem.
Where is the boundary of rational number?
According to the definition, infinite cyclic decimals and finite decimals (integers can be considered as decimals with 0 after decimal point) are collectively called rational numbers, and infinite cyclic decimals are irrational numbers.
However, it is impossible for human beings to write rational numbers with the largest number of digits. It is a rational number for all human beings on the earth, or for creatures smarter than the earth. For everyone on the earth, it may be impossible to know whether it is rational or irrational. Therefore, the boundary between rational numbers and irrational numbers is actually close to irrational numbers, and between any two very close irrational numbers, infinite rational numbers can be added, and vice versa.
No one knows the boundary of rational number, or the boundary of rational number is infinitely close to irrational number.
Theorem: It is impossible to write a non-infinite cyclic rational number with the largest number of digits. Although its definition has a limited number of digits, it is infinitely close to irrational numbers, so that there is no way to judge.
Proof: Suppose we write a non-infinite circular rational number with the largest number of digits, and we add one more bit at the end of this number. This number is still a finite rational number, but it is one more than the written rational number, which proves that the original written non-infinite circular rational number is not the largest number. So it is impossible to write a non-infinite cyclic rational number with the largest number of digits.
Comparison between irrational numbers and rational numbers;
For the definition of infinite acyclic decimals, they are irrational numbers and other irrational numbers are rational numbers. Irrational numbers are difficult to prove. Every irrational number, no matter how many digits you know, has a rational number, while rational numbers with shorter digits have no irrational numbers, so there are many rational numbers.
Decimals defined as finite digits and infinite cycles are rational numbers, and infinite acyclic numbers are irrational numbers. Only those who think that numbers greater than a finite number are irrational can prove that there are more irrational numbers than rational numbers, but that is obviously the result of classifying many rational numbers as irrational numbers. Under this definition, because the boundary is unclear, it can't be compared unless someone can prove it forcefully.