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.
Analysis of the concept of rational number
Shandongli
Example 1 The age of the seventh-grade freshmen is 12 years and 6 months. The following are the actual ages of the four students when they entered school. Please use positive and negative numbers to indicate the difference of their suitable school age.
Xiao Ming: 12 years and 5 months; Xiaohong: 12 years old 10 months;
Xiaogang: 13 years old; Xiao Bin: 12 years old.
When using the above method to record everyone's age and school age, if "Xiao Fei is 0 months old", what does it mean?
Analysis: Although this question has no definite opposite quantity, it implies the size of age, which is actually an opposite quantity. Of course, it can be expressed by positive numbers and negative numbers.
Solution: Xiao Ming-one month; Xiaohong +4 months; Xiaogang +6 months; Xiao Bin -6 months.
0 months in Xiao Fei means that Xiao Fei is exactly 12 years and 6 months.
Example 2 Fill in the following numbers in the corresponding number groups.
,0,-2,8, , ,-3.5, .
Analysis: Don't confuse positive integers with natural number set, 0 is a natural number, and 0 is neither positive nor negative.
Solution:
Example 3 Draw points representing the following numbers on the number axis and compare their sizes.
-5, ,0,3.5,- .
Analysis: first mark these numbers on the number axis, and then compare "the number on the right is greater than the number on the left"
Solution:
Namely:
-5