Understanding of rational numbers
Rational number is a general term for integer (positive integer, 0, negative integer) and fraction. Positive integers and fractions are collectively referred to as positive rational numbers, and negative integers and fractions are collectively referred to as negative rational numbers. Therefore, the number of rational number sets can be divided into positive rational numbers, negative rational numbers and zero. Since any integer or fraction can be converted into a cyclic decimal, and vice versa, every cyclic decimal can also be converted into an integer or fraction, so a rational number can also be defined as a cyclic decimal.
The set of rational numbers is an extension of the set of integers. In the set of rational numbers, the four operations of addition, subtraction, multiplication and division (divisor is not zero) are unimpeded.
Order of ab rational numbers: If a-b is a positive rational number, when A is greater than or less than B, remember that A > or B < A ... Any two unequal rational numbers can compare sizes.
An important difference between rational number set and integer set is that rational number set is dense while integer set is dense. After arranging rational numbers in order of size, there must be other rational numbers between any two rational numbers, which is density. Integer sets do not have this feature, so there are no other integers between two adjacent integers.
Rational numbers are compact subsets of real numbers: every real number has a rational number that is arbitrarily close. A related property is that only rational numbers can be transformed into finite continued fractions. Rational numbers have an ordered topology according to their sequences. Rational number is a (dense) subset of real number, so it also has subspace topology.
Rational Numbers and Their Classification
According to different standards, rational numbers can be divided into the following two types:
(1) Classification according to the definition of rational numbers;
(2) Classification according to the nature of rational numbers;
Extended data
Basic operations of rational numbers
Add operation
1, add two numbers with the same sign, take the same sign as the addend, and add the absolute values.
2. Add two numbers with different signs. If the absolute values are equal, the sum of two numbers with opposite numbers is 0; If the absolute values are not equal, take the sign of the addend with the larger absolute value and subtract the smaller absolute value from the larger absolute value.
3. Add two numbers with opposite numbers to get 0.
4. Adding a number to 0 still gets this number.
You can add two opposite numbers first.
6. Numbers with the same sign can be added first.
7. Numbers with the same denominator can be added first.
8. If several numbers can be added to get an integer, they can be added first.
subtraction
Subtracting a number is equivalent to adding the reciprocal of this number, that is, the subtraction of rational numbers is converted into addition.
multiply operation
1, the same sign is positive, the different sign is negative, and the absolute value is multiplied.
Any number multiplied by zero will get zero.
3. Multiply several numbers that are not equal to zero. The sign of the product is determined by the number of negative factors. When there are odd negative factors, the product is negative, and when there are even negative factors, the product is positive.
When several numbers are multiplied, if one factor is zero, the product is zero.
5. Multiply several numbers that are not equal to zero, first determine the sign of the product, and then multiply it by the absolute value.
Division operation
1 divided by a number that is not equal to zero is equal to the reciprocal of this number.
2. Divide two numbers, the same sign is positive and the different sign is negative, and divide by the absolute value. Divide zero by any number that is not equal to zero to get zero.
note:
Zero can't be a divisor and denominator.
Division and multiplication of rational numbers are reciprocal operations.
In the division operation, according to the law that the same sign is positive and the different sign is negative, the sign is determined first and then divided by the absolute value. If there is a fraction in the formula, it is usually calculated by turning it into a false fraction first. If it is not divisible, all division operations are converted into multiplication operations.
reference data
Baidu encyclopedia rational number