1. Definition and characteristics
Real numbers refer to all numbers that can be represented by points on the number axis. They have no restrictions or specific forms. Real numbers include rational numbers and irrational numbers. Rational numbers can be expressed in fractional form, while irrational numbers cannot be expressed in fractional form, such as π and √ 2.
2. Rational number
A rational number is a number that can be expressed as the ratio of two integers. Rational numbers include integers, fractions and decimals. Integers are positive numbers, negative numbers and zeros without decimal parts; A fraction is the ratio of two integers, where the denominator is not zero; Decimals are numbers consisting of integers and numbers after the decimal point.
3. Irrational numbers
Irrational number refers to a number that cannot be expressed by finite decimal or cyclic decimal. An irrational number cannot be expressed by the ratio of two integers, and its acyclic decimal part is infinitely acyclic. Common irrational numbers are π, √2, e, etc.
4. Properties of real numbers
Real numbers have a series of important properties, such as closeness, comparability and continuity. The closeness of real numbers means that any two real numbers still get a real number after four operations of addition, subtraction, multiplication and division.
Comparability of real numbers means that the sizes of different real numbers can be compared through the size relationship. The continuity of real numbers means that there are infinitely many real numbers between any two real numbers on the real number axis.
5. Representation and representation method of real numbers
Real numbers can be expressed in decimal form, fractional form and radical form. The decimal form represents real numbers as integer parts and decimal parts, such as 3.14; Fractional form represents real numbers as the ratio of two integers, such as1/2; The radical formula indicates that the real number is the square root or cube root of a number, such as √2.
6. Application fields of real numbers
Real number is one of the most basic concepts in mathematics, which is widely used in various fields. In physics, real numbers are used to describe physical quantities such as position, velocity and acceleration of objects. In economics, real numbers are used to represent monetary quantity and economic indicators; In computer science, real numbers are used to simulate and calculate continuous variables.
7. Further research on real numbers
The study of real numbers is an important subject in the field of mathematics, which involves the accuracy, continuity and properties of irrational numbers. Further research on real numbers includes approximate representation of real numbers, de-division of real numbers, completeness of real numbers and so on.
summary
Real number is a number including rational number and irrational number, which can be expressed as a point on the number axis. Rational numbers can be expressed as the ratio of two integers, while irrational numbers cannot be expressed as fractions. Real numbers are widely used in various fields because of their important properties such as closeness, comparability and continuity. The study of real numbers also involves the further approximate representation and completeness of real numbers.