(1) mathematical terms. The floorboard of rational number and irrational number.

(2) the exact number. How much does the company have? Please tell me the real number!

[Edit this paragraph] Mathematical terminology

[Edit this paragraph] 1, basic concepts

Real numbers include rational numbers and irrational numbers. Among them, irrational number is infinite acyclic decimal, and rational number includes integer and fraction.

Mathematically, real numbers are intuitively defined as numbers corresponding to points on the number axis. Originally, real numbers were just numbers, but later the concept of imaginary numbers was introduced. The original numbers were called "real numbers"-meaning "real numbers".

Real numbers can be divided into rational numbers and irrational numbers, or algebraic numbers and transcendental numbers, or positive numbers, negative numbers and zero. A set of real numbers is usually represented by the letter r or r n, and r n represents an n-dimensional real number space. Real numbers are uncountable. Real number is the core research object of real analysis.

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.

(1) inverse number (only two numbers with different signs, we will say that one of them is the inverse number of the other) The inverse number of the real number A is-a.

(2) Absolute value (the distance between a point corresponding to a number and the origin 0 on the number axis) The absolute value of the real number A is:

| a | a | = 1aWhen a is a positive number, | a | = a

② When a is 0, |a|=0.

③ When a is negative, | a | =-a.

③ Reciprocal (the product of two real numbers is 1, so these two numbers are reciprocal) The reciprocal of real number A is: 1/a (a≠0).

[Edit this paragraph] 2. Historical data

Egyptians began to use fractions as early as around 1000 BC. Around 500 BC, Greek mathematicians headed by Pythagoras realized the necessity of irrational numbers. Indians invented negative numbers around 600 AD. It is said that China also invented negative numbers, but it was a little later than Indian.

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

[Edit this paragraph] 3. Related definition

Constructing real numbers from rational numbers

Real numbers can be constructed as the complement of rational numbers by converging to the decimal or binary expansion of a unique real number, such as {3,3. 1 3. 14,3.1413.14/kloc. Real numbers can be constructed from rational numbers in different ways. Here is one of them. Please refer to the construction of real numbers for other methods.

axiomatic method

Let r be the set of all real numbers, then:

Set R is a field: it can add, subtract, multiply and Divison, and it has some common properties such as commutative law and associative law.

The field r is an ordered field, that is, for all real numbers x, y and z, there is a total order relation ≥:

If x ≥ y, then x+z ≥ y+z; ;

If x ≥ 0, y ≥ 0, xy ≥ 0.

The set R satisfies Dai Dejin completeness, that is, the nonempty set S (S∈R, S ≠ φ) of any R. If S has an upper bound in R, then S has an upper supremum in R.

The last one is the key to distinguish real numbers from rational numbers. For example, the set of all rational numbers whose square is less than 2 has an upper bound of rational numbers, such as1.5; But there is no supremum for rational numbers (because √2 is not a rational number).

Real numbers are uniquely determined by the above properties. More precisely, given any two Dai Dejin complete ordered domains R 1 and R2, there is a unique domain isomorphism from R 1 to R2, that is, they can be regarded as the same algebraically.

[Edit this paragraph] 4. Related attributes

Elementary operation

The basic operations that can be realized by real numbers are addition, subtraction, multiplication, division, square and so on. For non-negative numbers, 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.

Categorization

As a metric space or a uniform space, a real number set is a complete space, which has the following properties:

All Cauchy sequences of real numbers have a real limit.

A set of rational numbers is not a complete space. For example, (1, 1.4,1.41.4,1.4,1.4. In fact, it has a real limit √2. Real numbers are the completion of rational numbers-this is also a way to construct real number sets.

The existence of limit is the basis of calculus. The completeness of real numbers is equivalent to the fact that there is no "gap" in the straight line in Euclidean geometry.

"complete ordered domain"

Real number sets are usually described as "completely ordered fields", which can be explained in several ways.

First, an ordered domain can be a complete lattice. However, it is easy to find that no ordered field can be a complete lattice. This is because the ordered domain has no maximum element (for any element z, z+ 1 will be larger). Therefore, "complete" here does not mean complete lattice.

In addition, the ordered domain satisfies Dai Dejin completeness, which has been defined in the above axiom. The uniqueness also shows that the "completeness" here is Dai Dejin completeness. The meaning of this completeness is very close to the method of constructing real numbers by Dai Dejin division, that is, starting from the ordered domain of rational numbers, Dai Dejin completeness is established by standard methods.

These two integrity concepts ignore the structure of the domain. Ordered groups (fields are special groups) can define uniform spaces, and uniform spaces have the concept of complete spaces. The above completeness describes only special cases. (The concept of completeness of uniform space is adopted here, instead of the well-known completeness of metric space, because the definition of metric space depends on the properties of real numbers. Of course, R is not the only uniformly complete ordered domain, but it is the only uniformly complete Archimedean domain. In fact, "completely Archimedean domain" is more common than "completely ordered domain". It can be proved that any uniformly complete Archimedes domain must be Dai Dejin complete (and vice versa, of course). The significance of this completeness is very close to the method of constructing real numbers by Cauchy series, that is, starting from Archimedes domain of rational numbers, uniform completeness is established by standard methods.

"Complete Archimedes domain" was first put forward by Hilbert, and he also wanted to express some meanings different from the above. He believes that real numbers constitute the largest Archimedean domain, that is, all other Archimedean domains are subdomains of R, so saying that R is "complete" means that adding any element to it will make it no longer an Archimedean domain. The significance of this completeness is very close to the method of constructing real numbers from hyperreal numbers, that is, starting from a pure class containing all (hyperreal) ordered fields, finding the largest Archimedes field from its subdomains.

advancement

The set of real numbers is uncountable, that is, the number of real numbers is strictly greater than the number of natural numbers (although both are infinite). This can be proved by cantor diagonal method. In fact, the potential of real number set is 2ω (see the potential of continuum), which is the potential of natural number set's power set. Because only countable elements in a real number set may be algebraic numbers, most real numbers are transcendental numbers. In the subset of real number set, there is no set whose potential is strictly greater than natural number set and strictly less than real number set, which is the continuum hypothesis. This assumption cannot be proved to be correct because it has nothing to do with the axioms of set theory.

The square root of all nonnegative real numbers belongs to R, but it is not true for negative numbers. This shows that the order on R is determined by its algebraic structure. Moreover, all odd polynomials have at least one root belonging to R. These two properties make R the most important example of real closed fields. Proving this is the first half of proving the basic theorem of algebra.

Real number sets have a canonical measure, that is, Lebesgue measure.

The supremum axiom of real number set is applicable to a subset of real number set and is a statement of second-order logic. Real number sets cannot be described by only first-order logic: 1. l？ Wenhai-Scholer theorem shows that the real number set has a countable dense subset, which satisfies the same proposition as the real number set itself in first-order logic. 2. The set of hyperreal numbers is much larger than R, but it also satisfies the same first-order logical proposition as R. The ordered domain satisfying the same first-order logical proposition as R is called the nonstandard model of R, which is the research content of nonstandard analysis. Prove first-order logical propositions with nonstandard models (which may be simpler than in R), so as to ensure that these propositions are also established in R. ..

Topological property

The set of real numbers constitutes a metric space: the distance between x and y is set as the absolute value |x-y|. As a totally ordered set, it also has an ordered topology. Here, the topology obtained from the metric and order relations is the same. Real number set is also a contractible space (so it is also a connected space), a locally compact space, a separable space, and a Bailey space of 1 dimension. But the set of real numbers is not a compact space. These can be determined by specific properties, for example, an infinitely continuous sortable topology must be homeomorphic to a real number set. The following is an overview of the topological properties of real numbers:

Let a be a real number. The neighborhood of is a subset of the real number set, which contains a line segment with a.

R is a separable space.

Q is dense everywhere in R.

The open set of R is the union of open intervals.

The compact subset of r is a bounded closed set. In particular, all finite line segments with endpoints are compact subsets.

Every bounded sequence in R has a convergent subsequence.

R is connected and simply connected.

The connected subsets in R are line segments, rays and R itself. From this property, the intermediate value theorem can be derived quickly.

[Edit this paragraph] 5. Extension and generalization

Real number sets can be extended and generalized in several different ways:

Perhaps the most natural extension is the complex number. A complex set contains the roots of all polynomials. However, complex sets are not ordered domains.

The ordered field extended by real number set is a group of super real numbers, including infinitesimal and infinitesimal. This is not Archimedes' domain.

Sometimes, the formal elements +∞ and -∞ are added to the real number set to form an extended real number axis. It is a compact space, not a domain, but it retains many properties of real numbers.

Self-adjoint operators in Hilbert space generalize real number sets in many aspects: they can be ordered (although not necessarily completely ordered) and complete; All their eigenvalues are real numbers; They form a real associative algebra.