According to daily experience, the set of rational numbers seems to be "dense" on the number axis, so the ancients always thought that rational numbers could meet the actual needs of measurement. Take a square with a side length of 1cm as an example. How long is its diagonal? Under the specified precision (for example, the error is less than 0.00 1 cm), the accurate measurement result can always be expressed by rational numbers (for example, 1.4 14 cm). However, the mathematicians of the Pythagorean school in ancient Greece found that the length of this diagonal line could not be completely and accurately expressed only by rational numbers, which completely dealt a blow to their mathematical thinking; They think:
The ratio of any two line segments can be expressed by the ratio of natural numbers.
For this reason, Pythagoras himself even has the belief that "everything is a number", where the number refers to natural numbers (1, 2, 3 ...), and all positive rational numbers are obtained by the ratio of natural numbers. The fact that there is a "gap" in the set of rational numbers is a great blow to many mathematicians at that time; See the first mathematical crisis.
From ancient Greece to the seventeenth century, mathematicians gradually accepted the existence of irrational numbers and regarded them as equal numbers with rational numbers. Later, the concept of imaginary number was introduced to show the difference, which means "real number". At that time, although imaginary number appeared and was widely used, the strict definition of real number was still a problem. Until the concepts of function, limit and convergence were clarified, Dai Dejin, Cantor and other talents at the end of19th century strictly dealt with real numbers. In the current elementary mathematics, there is no strict definition of real numbers, but it is generally considered that real numbers are decimals (finite or infinite). The complete definition of real number is in geometry, and the points on the straight line correspond to real numbers one by one; See the number axis.
Real numbers can be divided into rational numbers (such as 42,) and irrational numbers (such as π, √2), algebraic numbers and transcendental numbers (rational numbers are algebraic numbers), or positive numbers, negative numbers and zero. A set of real numbers is usually represented by the letter r or. Rn stands for 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 continuously changing 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.
[Edit] 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. It was not until 187 1 that German mathematician Cantor put forward the strict definition of real numbers for the first time.
[edit] definition
[Edit] 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 and y ≥ 0, then x'y ≥ 0.
Set r satisfies Dai Dejin completeness, that is, non-empty subset S (S? r，S≦？ ), if S has an upper bound in R, then S has an upper bound 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 they are not rational numbers).
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] Example
15 (integer)
2. 12 1 (limited decimal)
1.3333333 ... (infinite loop decimal)
π = 3. 14 15926 ... (infinite acyclic decimal)
(irrational number)
(score)
[Edit] property
[Edit] Basic operation
In the real number domain, the basic operations that can be realized 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 open even powers, and the result is still real numbers.
Completeness of editing
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. Real number is the completion of rational number: this is also a method to construct real number set.
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.
[Edit] Complete the ordered field
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.
[Edit] Advanced Properties
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 is independent of ZFS axiomatic system 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. ..
[Edit] Topological properties
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:
Make a real number. The neighborhood of is a subset of the set of real numbers containing line segments.
This is a separable space.
It's dense everywhere in the middle.
The open set of is the union of open intervals.
The compact subset of is a bounded closed set. In particular, all finite line segments with endpoints are compact subsets.
Every bounded sequence has a convergent subsequence.
Is connected and simply connected.
The connected subsets in are line segments, rays and themselves. From this property, the intermediate value theorem can be derived quickly.
Interval nested theorem: Let it be a bounded closed set sequence, and its intersection is not empty. The strict table method is as follows:
.
[Edit] 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.