I. Introduction to Real Number Set

In mathematics, r stands for a set of real numbers. Because the English word of real number is real number, the set of real numbers is represented by R; Real numbers can be intuitively regarded as a one-to-one correspondence between finite decimals and infinite decimals, and between real numbers and points on the number axis, but the whole of real numbers can not be described only by enumeration.

Real number set is a set of real numbers, which usually contains all rational numbers and irrational numbers. Calculus was developed on the basis of real numbers in the18th century. But there was no precise definition of real number set at that time. It was not until 187 1 that German mathematician Cantor put forward the strict definition of real numbers for the first time. Any nonempty set with an upper bound (contained in R) must have an upper bound.

Second, the real number set classification?

Real numbers can be subdivided into two types in two different ways: one is based on rational numbers and irrational numbers. Second, according to algebraic number and transcendental number. Cantor proved that even algebraic numbers (which are far more common than rational numbers) still have the same potential as integers. There may be a mistake here: the set of algebraic numbers is not a countable set.

Addition theorem, multiplication theorem and completeness axiom of real number set;

1, addition theorem

For any elements A and B belonging to the set R, their addition a+b can be defined, and a+b belongs to R; Addition has a constant of 0, a+0=0+a=a (so there is an opposite number); Addition has commutative law, a+b = b+a; Addition has an associative law, (a+b)+c=a+(b+c).

2. Multiplication theorem

For any element A and B belonging to the set R, their products A and B can be defined, and A and B belong to R; Multiplication has constants 1, a 1 = 1 A = A (so there is a reciprocal besides 0); Multiplication has commutative law, a b = b a multiplication has associative law, (ab) c = a (b c); Multiplication has a distribution law for addition, that is, A (B+C) = (B+C) A = AB+A C.

3. Complete axioms

Any set that is not empty and has an upper bound (included in R) must have an upper supremum. Let A and B be two sets contained in R. For any X belonging to A and Y belonging to B, there is X.