Chapter 1 Rational Numbers
I. Positive and negative numbers
The concepts of positive and negative numbers
Negative number: a number less than 0. Positive number: a number greater than 0. 0 is neither positive nor negative.
Note: ① The letter A can represent any number. When a represents a positive number, -a is a negative number; When a is negative, -a is positive; When a represents 0, -a is still 0. (If it is judged that the number with a positive sign is positive and the number with a negative sign is negative, this statement is wrong. For example, +a and -a cannot make simple judgments. )
② Sometimes "+"can be added before positive numbers, and sometimes "+"can be omitted. Therefore, the positive sign omitting "+"is a positive sign.
2. Quantities with opposite meanings
If a positive number means a quantity with a certain meaning, a negative number can mean a quantity opposite to a positive number, such as:
8℃ above zero means:+8℃; 8 degrees below zero means 8 degrees below zero.
Expenditure and income; Increase or decrease; Profits and losses; North and south; East and west; Up and down; Growth and decrease, etc. Is the opposite number, they calculate:
Exceeding the original number, the number of increased growth is generally recorded as a positive number; Conversely, numbers less than the original number are generally recorded as negative numbers.
The meaning of 3.0
(1) 0 means "no", if there are 0 people in the classroom, that is to say, there is no one in the classroom;
0 is the dividing line between positive and negative numbers, and 0 is neither positive nor negative.
2. Rational number
The concept of 1. rational number
(1) Positive integers, 0 and negative integers are collectively called integers (0 and positive integers are collectively called natural numbers).
⑵ Positive score and negative score are collectively called scores.
(3) Positive integers, 0, negative integers, positive fractions and negative fractions can all be written in the form of fractions, and such numbers are called rational numbers.
Understanding: Only numbers that can change the number of components are rational numbers. ① π is an infinite acyclic decimal, which cannot be written in fractional form and is not a rational number. (2) Finite decimal and infinite cyclic decimal can be converted into component numbers, both of which are rational numbers.
Note: After the introduction of negative numbers, the range of odd and even numbers is also expanded. For example, -2, -4, -6, -8… are even numbers, and-1, -3, -5… are also odd numbers.
2.( 1) Any number that can be written in form is a rational number. Positive integers, 0 and negative integers are collectively referred to as integers; Positive and negative scores are collectively called scores; Integers and fractions are collectively called rational numbers. Note: 0 is neither positive nor negative; -a is not necessarily negative, and +a is not necessarily positive; P is not a rational number;
(2) Classification of rational numbers: ① Classification by positive and negative:
(2) According to the meaning of rational number:
Summary: ① Positive integers and 0 are collectively called non-negative integers (also called natural numbers).
② Negative integers and 0 are collectively referred to as non-positive integers.
③ Positive rational numbers and 0 are collectively called non-negative rational numbers.
④ Negative rational numbers and 0 are collectively called non-positive rational numbers.
(3) Note: among rational numbers, 1, 0 and-1 are three special numbers with their own characteristics; These three numbers divide the numbers on the number axis into four areas, and the numbers in these four areas also have their own characteristics;
(4) Natural numbers? 0 and positive integer; a & gt0 ? A is a positive number; a & lt0 ? A is a negative number;
a≥0？ Is it a positive number or 0? A is negative; a≤ 0？ Is it negative or 0? A is a non-positive number.
3. Counting axes
The concept of number axis
The straight line that defines the origin, positive direction and unit length is called the number axis.
Note: the (1) axis is a straight line extending infinitely to both ends; (2) Origin, positive direction and unit length are the three elements of the number axis, which are indispensable; (3) The unit length on the same axis should be unified; (4) The three elements of the number axis are specified according to actual needs.
2. The relationship between points on the number axis and rational numbers.
(1) All rational numbers can be represented by points on the number axis, positive rational numbers can be represented by points on the right of the origin, negative rational numbers can be represented by points on the left of the origin, and 0 can be represented by the origin.
⑵ All rational numbers can be represented by points on the number axis, but not all points on the number axis represent rational numbers, that is, there is no one-to-one correspondence between rational numbers and points on the number axis. (For example, point π on the number axis is not a rational number)
3. Use the number axis to represent the size of two numbers.
(1) Comparing the numbers on the number axis, the number on the right is always greater than the number on the left;
2 positive numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than negative numbers;
(3) Comparing two negative numbers, the number far from the origin is less than the number close to the origin.
4. Special maximum (minimum) number on the number axis
(1) The minimum natural number is 0, and there is no maximum natural number;
(2) The minimum positive integer is 1, and there is no maximum positive integer;
(3) The largest negative integer is-1, and there is no smallest negative integer.
5. What number does A stand for?
⑴a & gt； 0 means that a is a positive number; On the other hand, if a is a positive number, then a >；; 0;
⑵a & lt； 0 means that a is a negative number; On the other hand, if a is negative, then a
(3) A = 0 means that A is 0; On the other hand, if a is 0, then a=0.
6. The law of motion of points on the number axis
According to the movement of the point, if you move a few unit lengths to the left, you will subtract a few, and if you move a few unit lengths to the right, you will add a few, so that you can get the required point position.
Four. antique
1. Inverse
Only two numbers with different signs are called reciprocal, one of which is the reciprocal of the other, and the reciprocal of 0 is 0.
Note: (1) Antiquities appear in pairs; (2) The opposite numbers only have different symbols. If one is positive and the other is negative;
(3) The inverse of 0 is itself; The inverse itself is a number of 0.
2. The nature and judgment of reciprocal
(1) Any number has a reciprocal and only one;
The reciprocal of 0 is 0;
(3) If the sum of the two opposites is 0 and the two opposites are 0, that is, A and B are opposites, then a+b=0.
3. The geometric meaning of reciprocal
Two points on the number axis with the same distance from the origin represent two opposite numbers; Two mutually opposite numbers, the corresponding points (except 0) on the number axis are on both sides of the origin, and the distance from the origin is equal. The reciprocal of 0 corresponds to the origin; The origin represents the reciprocal of 0.
Note: On the number axis, two points representing opposite numbers are symmetrical about the origin.
4. The solution of reciprocal
(1) To find the reciprocal of a number, just add and subtract the sign "-"before it (for example, the reciprocal of 5 is-5); The antonym of 0 is still 0;
(2) Find the reciprocal of the sum or difference of multiple numbers, add "-"in brackets, and then simplify (for example; The reciprocal of 5a+b is -(5a+b). Simplified to-5a-b); Note: the inverse of a-b+c is-a+b-c; The inverse of a-b is b-a; The inverse of a+b is-a -a-b；;
(3) To find the single number with "-"in front, you should also enclose it in brackets before adding "-",and then simplify it (for example, the inverse of -5 is -(-5) and simplify it to 5); ) The sum of opposites is 0? a+b=0？ A and b are reciprocal.
5. Representation method of reciprocal
(1) Generally speaking, the inverse of the number A is -a, where A is any rational number, which can be positive, negative or 0.
When a>0, -a
When a<0,-a > 0 (the reciprocal of a negative number is a positive number)
When a=0, -a=0, (the reciprocal of 0 is 0)
6. Simplification of multiple symbols
Simplification law of multiple symbols: the number of "+"signs does not affect the simplification result, and can be omitted directly; The number of "-"determines the final simplification result; That is, when the number of "-"is odd, the result is negative, and when the number of "-"is even, the result is positive.
Absolute value of verb (abbreviation of verb)
Geometric definition of absolute value
Generally speaking, the distance between the point representing the number A on the number axis and the origin is called the absolute value of A, which is denoted as |a|.
2. Algebraic definition of absolute value
(1) The absolute value of a positive number is itself; (2) The absolute value of a negative number is its inverse; The absolute value of 0 is 0.
Can be expressed in letters as follows:
① If a>0, then | a | = a② If A
It can be summarized as ①: a ≥ 0,
②a≤0，& lt═>； |a|=-a (the absolute value of a non-positive number is equal to its inverse; A number whose absolute value is equal to its opposite number is not positive. )
3. The essence of absolute value
The absolute value of any rational number is non-negative, that is, the absolute value is non-negative. Therefore, if a takes any rational number, there is |a|≥0. That is, the absolute value of (1) positive number is itself, the absolute value of 0 is 0, and the absolute value of negative number is its inverse; Note: the absolute value means the distance between the point representing a number on the number axis and the origin; A number with an absolute value of 0 is 0. That is, a = 0.
The absolute value of a number is non-negative, and the number with the smallest absolute value is 0. The absolute value can be expressed as: or; Namely: | a | ≥ 0; The problem of absolute value is often discussed in categories;
(3) The absolute value of any number is not less than the original number. Namely: | a | ≥ a; ; ;
(4) The absolute values of two numbers are the same positive number, and they are opposite. That is, if | x | = a(a >；; 0), then x = a；;
5] The absolute values of two opposite numbers are equal. That is: |-a|=|a| or |a|=|b| If a+b = 0; |a| is an important non-negative number, that is | a | ≥ 0; Note: | a || b | = | a b |,
[6] Two numbers with equal absolute values are equal or opposite. That is: |a|=|b|, then a=b or a =-b;
Once, if the sum of the absolute values of several numbers is equal to 0, then these numbers are simultaneously 0. That is |a|+|b|=0, then a=0 and b=0.
(Common properties of non-negative numbers: if the sum of several non-negative numbers is 0, then only these non-negative numbers are 0 at the same time)
4. Comparison of rational numbers
⑴ Compare the size of two numbers by using the number axis: when two numbers on the number axis are compared, the number on the left is always smaller than the number on the right, or the number on the right is always larger than the number on the left.
⑵ Compare the size of two negative numbers with absolute values: two negative numbers compare the size, and the absolute value is larger than the small one; Compare the sizes of two numbers with different signs, and the positive number is greater than the negative number.
(3) The greater the absolute value of a positive number, the greater the number;
(4) Positive numbers are always greater than 0 and negative numbers are always less than 0;
(5) Positive numbers are greater than all negative numbers;
(6) large number-decimal number >; 0, decimal-large number < 0.
5. Simplification of absolute value
(1) when a≥0, | a | =-a② when a≤0, | a | =-a.
6. Know the absolute value of a number and find it.
The absolute value of the number A is the distance from the point representing the number A on the number axis to the origin. Generally speaking, there are two rational numbers with the same positive absolute value, which are opposite to each other. A number with an absolute value of 0 is 0, and there is no number with a negative absolute value.
6. Addition and subtraction of rational numbers.
1. rational number addition rule
(1) Add two numbers with the same symbol, take the same symbol, and add the absolute values;
⑵ Add two numbers with different symbols whose absolute values are not equal, 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;
(3) The sum of two mutually opposite numbers is zero;
(4) Add a number to 0 and still get this number.
2. Arithmetic of rational number addition
(1) additive commutative law: a+b = b+a.
⑵ law of additive combination: (a+b)+c=a+(b+c)
When using the algorithm, we must use it flexibly according to the needs to achieve the purpose of simplification. Usually, there are the following laws:
(1) first add two opposites-"combination of opposites";
2. First add two numbers with the same symbol-"the combination of the same symbol";
③ The numbers with the same denominator are added first-"the combination method with the same denominator";
(4) Add several numbers to get an integer, and add them first-"rounding method";
⑤ Addition of integers to integers and decimals to decimals-"isomorphic combination method".
3. Additional nature
The sum of a number plus a positive number is greater than the original number; The sum after adding negative number is less than the original number; The sum after adding 0 is equal to the original number. Namely:
(1) when b>0, a+b > When b, A 2
4. Rational number subtraction rules
Subtracting a number is equal to adding the reciprocal of this number. Expressed in letters: a-b=a+(-b).
5. The significance of adding and subtracting rational numbers into addition.
In the mixed operation of rational number addition and subtraction, according to the law of rational number subtraction, subtraction can be converted into addition, and then calculation can be made according to the law of addition.
In the summation formula, the parentheses of each addend and the preceding plus sign are usually omitted and written as the sum of the omitted plus signs. For example:
(-8)+(-7)+(-6)+(+5)=-8-7-6+5.
How to read the sum formula: ① According to the meaning expressed by this formula, it is read as "the sum of negative 8, negative 7, negative 6 and positive 5"
② Read it as "minus 8 minus 7 minus 6 plus 5" according to the operation meaning.
6. Some skills of applying associative law in rational number addition and subtraction mixed operation:
Seven. Multiplication and division of rational numbers
1. Multiplication Rule of Rational Numbers
Rule 1: multiply two numbers, the same sign is positive and the different sign is negative, and the multiplication takes the absolute value; ("The same sign is positive and the different sign is negative" refers to the situation of "multiplying two numbers". If there are more than two factors, Rule 3 must be applied. )
Rule 2: any number multiplied by 0 will get 0;
Rule 3: Multiply several numbers that are not 0. When the number of negative factors is even, the product is positive. When the number of negative factors is odd, the product is negative;
Rule 4: Multiply several numbers. If one of the factors is 0, the product is equal to 0.
Countdown
Two numbers whose product is 1 are reciprocal, and one of them is called reciprocal of the other, which means a = 1 (a ≠ 0), that is to say, A is reciprocal, that is, A is reciprocal.
Reciprocal: Two numbers whose product is 1 are reciprocal; Note: 0 has no reciprocal; If a≠0, the reciprocal is; The reciprocal itself is a number of 1; If ab= 1? A and b are reciprocal; If ab=- 1? A and b are negative reciprocal.
Note: ①0 has no reciprocal;
To find the reciprocal of a false score or a true score, just reverse the numerator and denominator of this score; When calculating the reciprocal of a fraction, first turn the fraction into a false fraction, and then reverse the positions of the numerator and denominator;
③ The reciprocal of a positive number is a positive number, and the reciprocal of a negative number is a negative number. Find the reciprocal of a number without changing its properties;
④ The number whose reciprocal equals itself is 1 or-1, excluding 0.
3. Multiplication and arithmetic of rational numbers
(1) Multiplication commutative law: Generally speaking, in rational number multiplication, two numbers are multiplied, and the position of the commutative factor is equal to the product. Namely ab=ba
⑵ Law of Multiplication: When three numbers are multiplied, the first two numbers are multiplied or the last two numbers are multiplied, and the products are equal. That is, (ab)c=a(bc).
(3) Multiplication and distribution law: Generally speaking, the multiplication of a number with the sum of two numbers is equivalent to the multiplication of this number with these two numbers respectively, and the products are added. That is, a(b+c)=ab+ac.
4. Division rule of rational numbers
(1) divided by a number that is not equal to 0 is equal to multiplying the reciprocal of this number; Note: Zero cannot be divisible.
(2) Divide two numbers, the one with the same sign is positive, and the one with different signs is negative, and divide by the absolute value. Divide 0 by any number that is not equal to 0 to get 0.
5. Mixed operation of rational number multiplication and division
(1) The mixed operation of multiplication and division usually decomposes division into multiplication first, then determines the sign of the product, and finally obtains the result.
(2) The mixed operation of addition, subtraction, multiplication and division of rational numbers, if there is no bracket to indicate what operation to do first, will be carried out in the order of' multiplication and division first, then addition and subtraction'.
Eight. Power of rational number
1. The concept of power
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 the base and n is called the exponent.
(1)a2 is an important non-negative number, that is, A2 ≥ 0; If a2+|b|=0? a=0，b = 0；
(2) According to the law, the decimal point of the cardinal number moves by one place and the decimal point of the square number moves by two places.
2. The nature of power
(1) The odd power of a negative number is negative, and the even power of a negative number is positive; Note: When n is positive odd number: (-a)n=-an or (a -b)n=-(b-a)n, when n is positive even number: (-a)n =an or (a-b) n = (b-a) n. 。
(2) Any power of a positive number is a positive number, and any power of a positive integer is 0.
9. Mixed operation of rational numbers
When doing the mixed operation of rational numbers, pay attention to the following operation order:
1. Multiply first, then multiply and divide, and finally add and subtract;
2. Operation at the same level, from left to right;
3. If there are brackets, do the operation in brackets first, and then press brackets, middle brackets and braces in turn.
X. scientific symbols
It is a scientific notation to represent numbers greater than 10 (where n is a positive integer).
Accuracy of divisor: a divisor rounded to that bit, that is, the divisor is accurate to that bit.
Significant digit: all digits from the first non-zero digit on the left to the exact digit are called the significant digits of this approximation.
Hybrid algorithm: multiply first, then multiply and divide, and finally add and subtract; Note: How to calculate simply and accurately is the most important principle of mathematical calculation.
Special value method: it is a method of substituting numbers that meet the requirements of the topic into speculation to verify the establishment of the topic, but it cannot be used for proof.
Sum of numbers equal to itself:
The opponent's number is equal to his own number: 0
The reciprocal equals its own number: 1,-1.
Numbers whose absolute values are equal to themselves: positive numbers and 0.
Square equals its own number: 0, 1.
A cube is equal to its own number: 0, 1,-1.
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