In mathematics, intervals usually refer to such a classreal numberSet: If x and y are twoaggregateSo, any number between x and ybelong toThis collection.For example, a set consisting of real numbers conforming to 0 ≤ x ≤ 1 is an interval, which contains 0, 1, and all real numbers between 0 and 1.Other examples include:Set of real numbers, a set of negative real numbers, etc.
Interval atintegralTheory plays an important role because they are the most "simple"Set of real numbersThey can be easily defined as "length", or“measure". Then, the concept of" measure "can be extended to extend the Borrell measure, andLebesgue measure。
Interval is also the core concept of interval arithmetic.Interval arithmetic is a kind ofnumerical analysisMethod is used to calculate the rounding error.
The concept of interval can also be extended to anyTotally ordered setT'ssubsetS，So that if x and y belong to S, and x<z<y, z also belongs to S.for exampleintegerThe interval [- 1... 2] refers to the set {- 1,0,1,2}.
In general interval notation, parentheses indicate "exclusion" and square brackets indicate "inclusion".For example, the interval (10, 20) represents all real numbers between 10 and 20, but does not include 10 and 20.On the other hand, [10, 20] indicates all real numbers between 10 and 20, as well as 10 and 20.When we arbitrarily refer to an interval, we usually write it with the capital letter I.
Some countries use commas to represent decimal points. To avoid confusion, semicolons should be used instead of commas to separate two numbers.[1-2]for example[1, 2.3]It will be written as[1; 2,3]。Otherwise, if only the decimal point is written as a comma, the previous example will become[1,2,3]Has.At this time, it is impossible to know whether it is between 1.2 and 3, or between 1 and 2.3.
In France and other European countries
And
replace
And
。such as
finish writing sth.
，
finish writing sth.
。This writing was originally included inInternational Organization for StandardizationISO 31-11 prepared.ISO 31-11 is a set of specifications for mathematical symbols used in physical science and technology.In 2009, it has been replaced by the newly formulated ISO 80000-2, no longer including
All representempty set，Single element setCannot be expressed by interval, for example, the set {0} cannot be expressed as [0] or [0,0].When a>b, the four symbols above are generally regarded as representing empty sets.When the interval is not an empty set, a and b are called intervalEndpoint。Generally, b - a is defined as the length of the interval.Intervalmidpoint(a+b)/2.
The interval [a, b] is sometimes calledline segment。(If it is not an empty set or a single element set)
In addition to indicating intervals, parentheses and square brackets have other uses, depending on the context.for example
It is also occasionally used to express ordered pairs, especially in the field of computer science.Also in number theory, we use
Represents an integer
The least common multiple of.
Some authors
To represent the interval
In the set of real numbersComplement, that is, it contains real numbers less than or equal to a, and real numbers greater than or equal to b.
Infinite interval
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We can use
Sign to indicate that the interval is unbounded in a certain direction.Specific definitions are as follows:
In particular,
Represents the set of positive real numbers, also recorded as
。
Represents a set of nonnegative real numbers.
If the interval is unilateral and unbounded, it is also called ray orSemistraight line。If it contains finite endpoints, it is called a closed ray or a closed half line.If it does not contain finite endpoints, it is called an open ray or an open half line.[4]
The above five marks are generally used, and
The writing principle of "etc." is quite rare.Some authors assume that the interval is a subset of the set of real numbers. For them, these expressions are either meaningless or just like parentheses.In the latter case, we can write
。So the set of real numbers can be regarded as an open and closed interval.
If we considerExtended Real AxisThen these four writing methods are numbered intervals.
Generally speaking, for integers a and b, the specific writing is:
。
In addition to [a.. B], there are also {a.. B} and aB is written in the same way.
The notation of [a.. B] is used in some programming languages, such asPascalandHaskell。
If an integer interval is bounded, it must contain the minimum a and the maximum b.Therefore, if you want to define an interval with the minimum or maximum number removed, you can simply use [a.. B-1], [a+1.. B] or [a+1.. B-1].It is not necessary to introduce the notation of [a.b) or (a.b) like the real number interval.
classification
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The real number interval can be divided into 11 types, as listed below.Where a and b are real numbers, a is the left end of the interval, and b is the right end of the intervalPoint.[8]
#1. # 4, # 8, # 10, and # 11 can be called "open intervals" (under standard topologyOpen set）, # 1, # 2, # 3, # 7, # 9, and # 11 can be called "closed intervals" (under standard topologyClosed set）。#3 and # 4 are sometimes referred to as "semi open intervals" or "semi closed intervals"#1 and # 11 are both "open" and "closed", not "semi open" and "semi closed".
Interval representation
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Interval representation means to visually represent the range of an interval on the real number line.It also means that it is given in the form of interval (including aunknown numberX)InequalityThe solution set of.
The interval also covers all convex subsets of the set of real numbers.In addition, let X be
If Y is the smallest closed interval containing X (that is, if Z is another closed interval containing X, Y is also contained in Z), then Y isconvex hull。actually,
。
Of any group of intervalsintersectionIt is still an interval.Two intervalUnionIs an interval, if and only if their intersection is not empty, or if the endpoint not included in one interval is exactly the endpoint included in another interval.For example:
When n=2, generally speaking, it defines a rectangle whose length and width are parallel to two coordinate axes.When n=3, a box is generally defined, and its sides are also parallel to the coordinate axis.
Complex interval
The interval of complex number can be defined asComplex planeTwo reasonable choices are rectangle or disc.[6]
interval arithmetic
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Interval arithmetic is also called interval mathematicsinterval analysis Interval calculation was introduced in the 1950s and 1960s as a tool to calculate rounding errors in numerical analysis.
The basic operation of interval arithmetic is for subsets on the real number line
and
：
[a,b]-[c,d]=[a-d,b-c]
Divided by an interval containing zero, there is no arithmetic definition of the basic interval.
