★ Emphasis★ The related concepts and properties of real numbers, and the operation of real numbers.
☆ Summary ☆
I. Key concepts
The classification and concept of 1 figure
Digital series table:
Note: "Classification" principle: 1) Proportionality (no weight, no leakage)
2) There are standards
2. Non-negative number: the collective name of positive real number and zero. (Table: x≥0)
Common non-negative numbers are:
Property: If the sum of several non-negative numbers is 0, then every unburdened number is 0.
3. Reciprocity: ① Definition and characterization
② attribute: a.a ≠1/a (a ≠1); B. 1/a，a≠0； c . 0 1； a & gt 1， 1/a
4. Reciprocal: ① Definition and representation
② Properties: A.a when A.a≠0; The position of a and -a on the number axis; The sum of c is 0 and the quotient is-1.
5. Number axis: ① Definition ("three elements")
② Function: a. Visually compare real numbers; B. clearly reflect the absolute value; C. establish a one-to-one correspondence between points and real numbers.
6. Odd number, even number, prime number and composite number (positive integer-natural number)
Definition and expression:
Odd number: 2n- 1
Even number: 2n(n is a natural number)
7. Absolute value: ① Definition (two kinds):
Algebraic definition:
Geometric definition: the geometric meaning of the absolute value top of the number A is the distance from the point corresponding to the real number A on the number axis to the origin.
② A ≥ 0, and the symbol "│ │" is a sign of "non-negative number"; ③ There is only one absolute value of number A; ④ When dealing with any type of topic, as long as "│ │" appears, the key step is to remove the "│ │" symbol.
Second, the operation of real numbers
1. Arithmetic (addition, subtraction, multiplication, division, power and root)
2. Algorithm (five plus [multiplication] commutative law and associative law; [Multiplication versus Addition]
Distribution law)
3. Operation sequence: a. Advanced operation to low-level operation; B. (Operation at the same level) From "Left"
To the "right" (such as 5 ÷ 5); C (when there are brackets) from "small" to "medium" to "large".
Third, the application examples (omitted)
Attachment: Typical examples
1. It is known that the positions of a, b and x on the number axis are as follows. Please verify: │x-a│+│x-b│.
=b-a。
2. a-b=-2 and AB are known.
Mathematics knowledge points in Grade Three Chapter 2 Algebraic expressions
★ Key points ★ Related concepts, properties and operations of algebraic expressions.
☆ Summary ☆
I. Key concepts
Classification:
1. Algebras and Rational Expressions
Formulas that associate numbers or letters representing numbers with operational symbols are called algebraic expressions. independent
Numbers or letters are also algebraic.
Algebraic expressions and fractions are collectively called rational forms.
2. Algebraic expressions and fractions
Algebraic expressions involving addition, subtraction, multiplication, division and multiplication are called rational expressions.
Rational expressions without division or division but without letters are called algebraic expressions.
Rational number formula has division, and there are letters in division, which is called fraction.
3. Monomial and Polynomial
Algebraic expressions without addition and subtraction are called monomials. (product of numbers and letters-including single numbers or letters)
The sum of several monomials is called polynomial.
Note: ① According to whether there are letters in the division formula, algebraic expressions and fractions are distinguished; According to whether there are addition and subtraction operations in algebraic expressions, monomial and polynomial can be distinguished. ② When classifying algebraic expressions, the given algebraic expressions are taken as the object, not the deformed algebraic expressions. When we divide the category of algebra, we start from the representation. For example,
=x, =│x│ and so on.
4. Coefficients and indices
Difference and connection: ① from the position; (2) In the sense of representation.
5. Similar projects and their combinations
Conditions: ① The letters are the same; ② The indexes of the same letters are the same.
Basis of merger: law of multiplication and distribution
6. Radical form
The algebraic expression of square root is called radical.
Algebraic expressions that involve square root operations on letters are called irrational expressions.
Note: ① Judging from the appearance; ② Difference: It is a radical, but it is not an irrational number (it is an irrational number).
7. Arithmetic square root
(1) The positive square root of a positive number ([the difference between a ≥ 0-and "square root"]);
⑵ Arithmetic square root and absolute value
① Contact: all are non-negative, =│a│.
② Difference: │a│, where A is all real numbers; Where a is a non-negative number.
8. Similar quadratic root, simplest quadratic root, denominator of rational number.
After being transformed into the simplest quadratic root, the quadratic roots with the same number of roots are called similar quadratic roots.
The following conditions are satisfied: ① the factor of the root sign is an integer and the factor is an algebraic expression; (2) The number of roots does not include exhausted factors or factors.
Crossing out the root sign in the denominator is called denominator rationalization.
9. Index
(1)(- power supply, power supply operation)
①a & gt； 0, > 0; ②a & lt； 0, > 0(n is even), < 0(n is odd)
(2) Zero index: = 1(a≠0)
Negative integer index: = 1/ (a≠0, p is a positive integer)
Second, the law of operation and the law of nature
1. The law of addition, subtraction, multiplication, division, power and root of fractions.
2. The nature of the score
(1) Basic properties: = (m≠0)
(2) Symbolic law:
⑶ Complex fraction: ① Definition; ② Simplified methods (two kinds)
3. Algebraic expression algorithm (bracket deletion and bracket addition)
4. The essence of power operation: ① =; ② ÷ = ; ③ = ; ④ = ; ⑤
Skills:
5. Multiplication rule: (1) single× single; (2) single × many; 3 more x more.
6. Multiplication formula: (plus or minus)
(a+b)(a-b)= 1
(a b) =
7. Division rules: (1) single-single; (2) Too many orders.
8. Factorization: (1) definition; ⑵ Methods: A. Common factor method; B. formula method; C. cross multiplication; D. group decomposition method; E. find the root formula method.
9. The nature of arithmetic roots: =; ; (a≥0，b≥0)； (a≥0, b>0) (positive and negative)
10. radical algorithm: (1) addition rule (merging similar quadratic roots); (2) multiplication and division; (3) The denominator is reasonable: a; b； c。
1 1. Scientific notation: (1 ≤ A
Third, the application examples (omitted)
Four, comprehensive operands (omitted)