Decision principle, monotone bounded theorem, interval set theorem, finite covering theorem, convergence point theorem, Cauchy convergence criterion.

Real number is a general term for rational number and irrational number. Real and imaginary numbers * * * together form a complex number. Real numbers can be divided into rational and irrational numbers, or algebraic and transcendental numbers. Real number sets are usually represented by the black letter r, and real numbers are uncountable.

Real number is the core research object of real number theory. The set of all real numbers can be called real number system or real number continuum. Any complete Archimedean ordered field can be called a real number system. It is unique in the sense of order-preserving isomorphism, and is often expressed by R. Because R is an arithmetic system that defines arithmetic operations, it is called the real number system.

Real numbers can be used to measure continuous quantities. Theoretically, any real number can be expressed as an infinite decimal, and to the right of the decimal point is an infinite series (cyclic or acyclic).

In practice, real numbers are often approximate to a finite decimal (n digits are reserved after the decimal point, and n is a positive integer). In the computer field, because computers can only store a limited number of decimal places, real numbers are often represented by floating-point numbers.

Development history

Around 500 BC, Greek mathematicians headed by Pythagoras realized that rational numbers could not meet the needs of geometry, but Pythagoras himself did not admit the existence of irrational numbers.

It was not until17th century that real numbers were widely accepted in Europe. 18th century, calculus was developed on the basis of real numbers. 187 1 year, German mathematician Cantor first put forward a strict definition of real numbers.

Elementary operation

The basic operations that real numbers can realize are addition, subtraction, multiplication, division, multiplication and so on. For non-negative numbers (that is, positive numbers and 0), you can also perform a root operation. The result of addition, subtraction, multiplication, division (divisor is not zero) and square of real numbers is still real numbers. Any real number can be raised to an odd power, and the result is still a real number. Only non-negative real numbers can be raised to even powers, and the result is still real numbers.