Key points of rational number in junior one mathematics
First, the main points of knowledge
The main content of this chapter can be summarized as the concept of rational number and the operation of rational number. The concept of rational numbers can be recognized and understood by using the number axis, and these concepts can be strung together by using the number axis. The operation of rational numbers is the focus of the whole chapter. We should pay attention to four aspects in the specific operation, one is the algorithm, the other is the algorithm, the third is the operation sequence, and the fourth is the approximate calculation.
Basic knowledge:
1, positionnumber: a number greater than 0 is called a positive number.
2. Negative number: A number preceded by a negative sign "-"is called a negative number.
3,0 is neither positive nor negative.
4.rationalnumber: Positive integers, negative integers, 0, positive fractions and negative fractions can all be written in the form of fractions, and such numbers are called rational numbers.
5. numberaxis: Numbers are usually represented by points on a straight line, which is called number axis.
The number axis meets the following requirements:
(1) Take any point on a straight line to represent the number 0, and this point is called the origin;
(2) Generally, from the origin to the right (or upward) is the positive direction, and from the origin to the left (or downward) is the negative direction;
(3) Select the appropriate length as the unit length.
6.oppositenumber: Two numbers with equal absolute values but different negative signs are called opposite numbers.
7.absolutevalue Generally speaking, the distance between the point representing the number A on the number axis and the origin is called the absolute value of the number A, and write down |a|. From the definition of absolute value: |a-b| represents the distance from point A to point B on the number axis. The absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of 0 is 0. Positive number is greater than 0, 0 is greater than negative number, and positive number is greater than negative number; Two negative numbers, the larger one has the smaller absolute value.
8, rational number addition rule
(1) Add two numbers with the same sign, take the same sign, and add the absolute values.
(2) Add two numbers with different absolute values, take the sign of the addend with larger absolute value, and subtract the one with smaller absolute value from the one with larger absolute value. Two opposite numbers add up to 0.
(3) When a number is added to 0, the number is still obtained.
Additive commutative law: In addition of rational numbers, two numbers are added, the positions of addends are exchanged, and the sum remains the same. Expression: a+b = b+a.
Law of addition and association: rational number addition, when three numbers are added, the first two numbers or the last two numbers are added first, and the sum is unchanged.
Expression: (a+b)+c=a+(b+c)
9, rational number subtraction rule: subtracting a number is equal to adding the reciprocal of this number. Expression: a-b=a+(-b)
10, rational number multiplication rule
Multiply two numbers, the same sign is positive, the different sign is negative, and then multiply by the absolute value.
Any number multiplied by 0 is 0.
Multiplication commutative law: Generally speaking, in rational number multiplication, two numbers are multiplied, and the position of the commutative factor and the product are equal. Expression: ab=ba
Multiplication and association law: when three numbers are multiplied, the first two numbers are multiplied first, or the last two numbers are multiplied first, and the products are equal. Expression: (ab)c=a(bc)
Multiplication and distribution law: generally speaking, a number multiplied by the sum of two numbers is equivalent to multiplying this number by these two numbers respectively, and then adding the products.
Expression: a(b+c)=ab+ac
1 1, countdown
The quotient of 1 divided by a number (except zero) is called the reciprocal of this number. If two numbers are reciprocal, the product of these two numbers is equal to 1.
12, rational number division rule: divide two numbers, the same sign is negative, the different sign is positive, and divide by the absolute value. Divide 0 by any number that is not equal to 0 to get 0.
13. Power of rational numbers: The operation of finding the product of n identical factors is called power, and the result of power is called power. In, a is called radix and n is called exponent.
According to the multiplication rule of rational numbers, it can be concluded that the odd power of negative numbers is negative and the even power of negative numbers is positive. Any power of a positive number is a positive number, and any power of a positive integer is 0.
14, mixed operation order of rational numbers
(1) "Power first, then multiply and divide, and finally add and subtract" order;
(2) the same layer operation, from left to right;
(3) If there are brackets, do the operation in brackets first, and then follow the brackets, brackets and braces in turn.
15, science and technology method: express numbers greater than 10 as a- 10n (where a is a number with only one integer digit (i.e. 0 16, about $ NUMBER);
17, rational number can be written as m/n(m, n are integers, n? 0) Form. On the other hand, the shape is m/n(m, n is an integer, n? 0) is a rational number. So the rational number can be m/n(m, n is an integer, n? 0) means.
Expand knowledge:
1, number set: put some numbers together to form a number set, which is called number set for short.
(1) The number set composed of all rational numbers is called the rational number set;
(2) A number set consisting of all integers is called an integer set.
2. Any rational number can be represented by a point on the number axis, which embodies the mathematical idea of combining numbers with shapes.
3. According to the geometric meaning of absolute value, we know: |a|? 0, that is, for any rational number A, its absolute value is non-negative.
4. The methods to compare the sizes of two rational numbers are:
(1) directly compare the positions of points corresponding to rational numbers on the number axis;
(2) Comparison according to regulations: two positive numbers; Positive numbers and zeros; Negative numbers and zeros; Positive and negative numbers; Two negative numbers reflect the mathematical idea of classified discussion;
(3) Difference method: A-B >; 0a & gtb；
(4) Business Law: A/B > 1，b & gt0a & gtb.
Key points of rational number in junior one mathematics
(1) positive and negative numbers
1. positive number: a number greater than 0.
2. Negative number: a number less than 0.
3.0 neither positive nor negative.
4. Positive numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than negative numbers.
(2) rational number
1. rational number: a number consisting of integers and fractions. Include positive integer, 0, negative integer, positive fraction and negative fraction. Can be written as the ratio of two integers. Irrational numbers cannot be written as the ratio of two integers. It is written in decimal form, and the numbers after the decimal point are infinite. For example:? )
2. Integer: positive integer, 0, negative integer, collectively referred to as integer.
3. Score: positive score and negative score.
(3) Number axis
1. Number axis: Numbers are represented by points on a straight line, which is called number axis. Draw a straight line and take any point on the straight line to represent the number 0. This zero point is called the origin, which specifies that the right or upward direction of the straight line is positive; Select the appropriate length as the unit length, so as to take points on the number axis. )
2. Three elements of the number axis: origin, positive direction and unit length.
3. Antiquities: Only two numbers with different symbols are called reciprocal. The antonym of 0 is still 0.
4. Absolute value: the absolute value of a positive number is itself, and the absolute value of a negative number is its inverse; The absolute value of 0 is 0, two negative numbers, the larger absolute value is smaller.
Addition and subtraction of rational numbers
1. Sign first, then calculate the absolute value.
2. Addition algorithm: the same sign is added, and the absolute value is added. For the addition of different symbols, take the sign of the addend with large absolute value, and subtract the sign with small absolute value from the sign with large absolute value. Two opposite numbers add up to 0. Add and subtract a number with 0, and you still get this number.
3. additive commutative law: a+b=b+a is added, the position of the addend is exchanged, and the sum is unchanged.
4. The law of addition and association: (a+b)+c=a+(b+c) three numbers are added, the first two numbers are added first, or the last two numbers are added first, and the sum is unchanged.
5.a-b=a+(-b) Subtracting a number is equal to adding the reciprocal of this number.
(5) rational number multiplication (first determine the sign of the product, and then determine the size of the product)
1. The same symbol is positive, different symbols are negative, and the absolute values are multiplied. Any number multiplied by 0 is 0.
2. Two numbers whose product is 1 are reciprocal.
3. Multiplicative commutative law: ab=ba
4. Multiplicative associative law: (ab)c=a(bc)
5. Multiplication and distribution law: a(b+c)=ab+ac.
(6) rational number division
1. First divide and multiply, then sign, and finally find the result.
2. dividing by a number that is not equal to 0 is equal to multiplying the reciprocal of this number.
3. Divide two numbers, the same sign is positive and the different sign is negative, and divide by the absolute value. Divide 0 by any number that is not equal to 0, and you will get 0.
(7) Stand aside
1. The operation of finding the product of n identical factors is called power. Write one. The result of multiplication is called power, a is called base, and n is called exponent. )
2. The odd power of a negative number is negative and the even power of a negative number is positive; Any positive integer power of 0 is 0.
3. Multiplication with the same base, constant base and exponential addition.
4. Divided by the same base, the base is constant, minus the exponent.
(8) Mixed operations of addition, subtraction, multiplication and division of rational numbers.
1. Multiply first, then multiply and divide, and finally add and subtract.
2. Operate at the same level, from left to right.
3. If there are brackets, do the operation in brackets first, and then follow the brackets, brackets and braces in turn.
(9) Scientific notation, divisor and significant figures.
Chapter II Algebraic Expressions (1) Algebraic Expressions
1. Algebraic expression: monomials and polynomials are collectively called algebraic expressions.
2. Monomial: The formula consisting of the product of numbers and letters is called monomial. A single number or letter is also a monomial.
3. coefficient; In the monomial, the numerical factor is called the coefficient of the monomial.
4。 Times: The sum of the indices of all the letters in a monomial is called the times of this monomial.
5. Polynomial: The sum of several monomials is called polynomial.
6. Term: Each monomial that constitutes a polynomial is called a polynomial term.
7. Constant term: the term without letters is called constant term.
8. Degree of Polynomial: In a polynomial, the degree of the term with the highest degree is called the degree of this polynomial.
9. Similar terms: In polynomials, terms with the same letters and the same index of the same letters are called similar terms.
10. Merging similar items: Merging similar items in polynomials into one item is called merging similar items.
(2) Algebraic expression addition and subtraction Algebraic expression addition and subtraction operation, if you encounter brackets, remove the brackets first, and then merge similar items.
1. bracket removal: Generally speaking, several algebraic expressions are added and subtracted. If there are brackets, remove them first, and then merge similar items. If the factor outside the brackets is positive, the symbols of the items in the original brackets are the same as the original symbols after the brackets are removed. If the factor outside the brackets is negative, the symbols of the items in the original brackets are opposite to those after the brackets are removed.
2. Merging similar items: Merging similar items in polynomials into one item is called merging similar items. After merging similar items, the coefficient of the obtained item is the sum of the coefficients of similar items before merging, and the letter part remains unchanged.
After combing the knowledge points, let's take a look at the related exercises. According to the situation of doing the problem, analyze what knowledge points you have not mastered.
1, from the number axis, 0 is ()
A, the smallest integer b, the largest negative number c, the smallest rational number d and the smallest nonnegative number.
2, the reciprocal of a number is less than itself, this number is ()
A, nonnegative B, positive C, 0D, negative
3. The highest temperatures in winter in three cities in China are-10℃, 1℃ and -7℃ respectively. Their correct arrangement from high to low is ().
a，- 10℃，-7℃， 1℃B，-7℃，- 10℃， 1℃C， 1℃，-7℃， 10℃D， 1℃，- 10℃，-7℃
4, the following statement is correct ()
A. Positive numbers and negative numbers are collectively called rational number B. Rational numbers refer to five types of C, including integer, fraction, positive rational number, negative rational number and 0. Rational numbers are either integers or fractions d, and integers include positive integers and negative integers.
5, if a and b are both rational numbers, a >；; 0, b<0, and | a | < |b|, the following statement is incorrect ().
A, if the numbers A and B are represented on the number axis, A is on the right side of the origin and B is on the left side of the origin.
B, because positive numbers are greater than all negative numbers, a> B.
C, if the numbers A and B are represented on the number axis, then the distance from the number A to the origin is smaller than the distance from the number A to the origin.
D on the number axis, the points representing a, |a| and b are a, b and |a| respectively from left to right.
6. In the following algebraic expressions: (1/2)ab, (a+b)/2, ab2+b+ 1, (3/x)+(2/y), x3+x2-3, there is () A.2 B.3 polynomial.
A, -3x2B, (5a-4b)/7C, (3a+2)/5xD, -2005
Key knowledge points in the first volume of junior one mathematics
Real number:
? Rational numbers and irrational numbers are collectively called real numbers.
Rational number:
Integers and fractions are collectively called rational numbers.
Irrational number:
Irrational numbers refer to infinite cyclic decimals.
Natural number:
The numbers 0, 1, 2, 3, 4 ~ (including 0) representing objects are all called natural numbers.
Number axis:
The straight line that defines the point, the positive direction and the unit length is called the number axis.
Countdown:
Two numbers with different symbols are opposite.
Countdown:
Two numbers whose product is 1 are reciprocal.
Absolute value:
The distance between the point representing the number A and the point on the number axis is called the absolute value of A, the absolute value of a positive number is itself, the absolute value of a negative number is its inverse, and the absolute value of 0 is 0.
Mathematical theorem formula
Rational number arithmetic
(1) addition rule: add two numbers with the same symbol, take the same symbol, and add the absolute values; Add two numbers with different signs, take the sign of the addend with larger absolute value, subtract the smaller absolute value from the larger absolute value, and add two numbers with opposite numbers to get 0.
(2) Law of subtraction: subtracting a number is equal to adding the reciprocal of this number.
(3) Multiplication rule: two numbers are multiplied, the same sign is positive and the different sign is negative, and the multiplication takes the absolute value; Multiply any number by 0 to get 0.
(4) Division rule: dividing by a number is equal to multiplying the reciprocal of this number; Divide two numbers, the same sign is positive and the different sign is negative, and divide by the absolute value; Divide 0 by any number that is not equal to 0 to get 0.
Angular bisector: a ray drawn from the vertex of an angle, which can be divided into two parts. This ray is called the bisector of this angle.
Chapter 1 intersection of mathematics
1. Adjacent complementary angle: Among the four angles formed by the intersection of two straight lines, there is a common vertex and a common edge. Such an angle is called an adjacent complementary angle. Adjacent complementary angle is an angle with special positional and quantitative relationship, that is, it must be complementary angle, but it is not necessarily adjacent complementary angle.
Second, right angle: formed by the intersection of two straight lines. The two sides of the two corners are opposite extension lines, so the vertex angle can also be said to be? Two angles formed by two sides of an angle extending in opposite directions are called antipodal angles? .
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