Types of numerical intervals. Numerical segments, intervals, half-intervals and rays are called numerical intervals

Numeric interval

Interval, open span, interval- the set of points on the number line between two given numbers a And b, that is, a set of numbers x, satisfying the condition: a < x < b . The interval does not include ends and is denoted by ( a,b) (Sometimes ] a,b[ ), in contrast to the segment [ a,b] (closed interval), including the ends, that is, consisting of points.

In recording ( a,b), numbers a And b are called the ends of the interval. The interval includes all real numbers, the interval includes all numbers smaller a and the interval - all numbers are large a .

Term interval used in complex terms:

• upon integration - integration interval,
• when clarifying the roots of the equation - insulation span
• when determining the convergence of power series - interval of convergence of power series.

By the way, in English language in a word interval called a segment. And to denote the concept of interval, the term is used open interval.

Literature

• Vygodsky M. Ya. Handbook of higher mathematics. M.: “Astrel”, “AST”, 2002

see also

Links

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2010.

See what “Numerical interval” is in other dictionaries:

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Interval, open interval, interval is a set of points on a number line enclosed between two given numbers a and b, that is, a set of numbers x that satisfy the condition: a

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Answer - The set (-∞;+∞) is called a number line, and any number is a point on this line. Let a be an arbitrary point on the number line and δ

Positive number. The interval (a-δ; a+δ) is called the δ-neighborhood of point a.

A set X is bounded from above (from below) if there is a number c such that for any x ∈ X the inequality x≤с (x≥c) holds. The number c in this case is called the upper (lower) bound of the set X. A set that is bounded both above and below is called bounded. The smallest (largest) of the upper (lower) bounds of a set is called the exact upper (lower) bound of this set.

A numerical interval is a connected set of real numbers, that is, such that if 2 numbers belong to this set, then all the numbers between them also belong to this set. There are several somewhat different types of non-empty number intervals: Line, open ray, closed ray, segment, half-interval, interval

Number line

The set of all real numbers is also called the number line. They write.

In practice, there is no need to distinguish between the concept of a coordinate or number line in a geometric sense and the concept of a number line introduced by this definition. Therefore, these different concepts are denoted by the same term.

Open beam

The set of numbers such that is called an open number ray. They write or accordingly: .

Closed beam

The set of numbers such that is called a closed number line. They write or accordingly:.

A set of numbers is called a number segment.

Comment. The definition does not stipulate that . It is assumed that the case is possible. Then the numerical interval turns into a point.

Interval

A set of numbers such that is called a numerical interval.

Comment. The coincidence of the designations of an open beam, a straight line and an interval is not accidental. An open ray can be understood as an interval, one of whose ends is removed to infinity, and a number line - as an interval, both ends of which are removed to infinity.

Half-interval

A set of numbers such as this is called a numerical half-interval.

They write or, respectively,

3.Function.Graph of the function. Methods for specifying a function.

Answer - If two variables x and y are given, then the variable y is said to be a function of the variable x if such a relationship is given between these variables that allows for each value to uniquely determine the value of y.

The notation F = y(x) means that a function is being considered that allows for any value of the independent variable x (from among those that the argument x can generally take) to find the corresponding value of the dependent variable y.

Methods for specifying a function.

The function can be specified by a formula, for example:

y = 3x2 – 2.

The function can be specified by a graph. Using a graph, you can determine which function value corresponds to a specified argument value. This is usually an approximate value of the function.

4.Main characteristics of the function: monotonicity, parity, periodicity.

Answer - Periodicity Definition. A function f is called periodic if there is such a number
, that f(x+
)=f(x), for all x D(f). Naturally, there are countless numbers of such numbers. The smallest positive number ^ T is called the period of the function. Examples. A. y = cos x, T = 2 . V. y = tg x, T = . S. y = (x), T = 1. D. y = , this function is not periodic. Parity Definition. A function f is called even if the property f(-x) = f(x) holds for all x in D(f). If f(-x) = -f(x), then the function is called odd. If none of the indicated relations are satisfied, then the function is called a general function. Examples. A. y = cos (x) - even; V. y = tg (x) - odd; S. y = (x); y=sin(x+1) – functions of general form. Monotony Definition. A function f: X -> R is called increasing (decreasing) if for any
the condition is met:
Definition. A function X -> R is called monotonic on X if it is increasing or decreasing on X. If f is monotone on some subsets of X, then it is called piecewise monotone. Example. y = cos x - piecewise monotonic function.

Among numerical sets, that is sets, the objects of which are numbers, there are so-called numerical intervals. Their value is that it is very easy to imagine a set corresponding to a specified numerical interval, and vice versa. Therefore, with their help it is convenient to write down many solutions to an inequality.

In this article we will analyze all types of numerical intervals. Here we will give their names, introduce notations, depict numerical intervals on the coordinate line, and also show what simple inequalities correspond to them. In conclusion, let us visually present all the information in the form of a table of numerical intervals.

Page navigation.

Types of numerical intervals

Each numerical interval has four inextricably linked things:

• name of the number interval,
• corresponding inequality or double inequality,
• designation,
• and its geometric image in the form of an image on a coordinate line.

Any numerical interval can be specified by any of the last three methods in the list: either an inequality, or a notation, or its image on a coordinate line. Moreover, according to this method tasks, for example, on inequality, others can be easily restored (in our case, notation and geometric image).

Let's get down to specifics. Let us describe all numerical intervals from the four sides indicated above.

Let's start with a description of the numerical interval, called open number beam. Note that often the adjective “open” is omitted, leaving the name open beam.

This numerical interval corresponds to the simplest inequalities with one variable of the form x a , where a is some real number. That is, according to the meaning of the written inequalities, the open number ray consists of all that are less than the number a (in the case of the inequality x a).

The set of numbers satisfying the inequality x a as (a, +∞) .

It remains to show the geometric image of the open ray; from it it will become clear that the numerical interval in question did not receive such a name by chance. Let's turn to. It is known that there is a one-to-one correspondence between its points and real numbers, which allows the coordinate line to be called the number line. And when talking about comparing numbers we noted that the larger number is located on the coordinate line to the right of the smaller one, and the smaller number is located to the left of the larger one. Based on these considerations, the inequality x a – points lying to the right of point a. The number a itself does not satisfy these inequalities; to emphasize this, it is depicted in the drawing as a dot with an empty center. Above the points that correspond to numbers that satisfy the inequality, oblique shading is depicted:

From the given drawings it is clear that these numerical intervals correspond to parts of the number line, which are rays starting at point a, but excluding point a itself. In other words, these are rays without a beginning. Hence the name - open number beam.

Let us give several specific examples of open number rays. Thus, the strict inequality x>−3 defines an open number ray. It is also given by the notation (−3, ∞). And on the coordinate line, this numerical interval is a set of points lying to the right of the point with coordinate −3, not including this point itself. Another example: inequality x<2,3 , как и запись (−∞, 2,3) , задает открытый числовой луч, который следующим образом изображается на координатной прямой


Let's move on to numerical intervals of the following type - number rays. In geometric terms, their difference from open beams is that the beginning of the beam is not discarded. In other words, the geometric image of numerical intervals of this type is a full-fledged ray.

As for specifying numerical rays using inequalities, they are answered by the non-strict inequalities x≤a or x≥a. The following notations are accepted for them: (−∞, a] and . And the geometric image of a numerical segment is a segment along with its ends:


For example, a numerical segment that is given by a double inequality can be denoted as , on the coordinate line it corresponds to a segment with ends at points having coordinates root of two and root of three.

All that remains is to say about the numerical intervals called half-intervals. They represent, so to speak, an intermediate option between an interval and a segment, since they include one of the boundary points. Half-intervals are specified by double inequalities a

Table of numerical intervals

So, in the previous paragraph we defined and described the following numerical intervals:

• open number beam;
• number beam;
• interval;
• half-interval

For convenience, we summarize all the data on numerical intervals in a table. Let us enter into it the name of the numerical interval, the corresponding inequality, designation and image on the coordinate line. We get the following table of numerical intervals:

Bibliography.

• Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• Mordkovich A. G. Algebra. 9th grade. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich, P. V. Semenov. - 13th ed., erased. - M.: Mnemosyne, 2011. - 222 p.: ill. ISBN 978-5-346-01752-3.

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Slide captions:

7th grade Number intervals Math teacher: Bakhvalova G.S. Gymnasium No. 52

Lesson objectives: 1.Introduce the concept of a numerical interval; 2. Instill the skills of depicting numerical intervals on a number line and the ability to designate them. 3.Develop logical thinking: analyze, compare. Lesson plan: 1. Updating knowledge: “Coordinate axis.” 2. New topic: “Numerical intervals.” 3. Educational independent work. 4. Lesson summary.

Complete the task: 1. Mark on the number line the points with coordinates: A(-2); AT 5); O(0); C(5); D (-3).

Answer: 1. A(-2); AT 5); O(0); C(3); D(-3). 0 A B C 1 0 D

Complete the task: 2. Compare the numbers: -2 and 5; 5 and 0; -2 and –3; 5 and 3; 0 and –2.

Answer: -2 0; -2 > –3; 5 > 3; 0 > –2. check yourself

Complete the task orally: 3. Which of the given numbers on the number line is to the left: -2 or 5; 5 or 0; -2 or –3; 5 or 3; 0 or –2. CONCLUSION: of two numbers on the number line, the smaller number is located to the left, and the larger number is located to the right.

Let us mark points on the coordinate line with coordinates – 3 and 2. If the point is located between them, then it corresponds to a number that is greater than –3 and less than 2. The converse is also true: if the number x satisfies the condition - 3Slide 9

The set of all numbers satisfying the condition 3Slide 10

A number x that satisfies the condition -3 ≤x≤ 2 is represented by a point that either lies between the points with coordinates –3 and 2, or coincides with one of them. A set of such numbers is denoted [-3;2]. - 3 2 Write it down in your notebook Write it down in your notebook Write it down in your notebook

A number x that satisfies the condition x≤ 2 is represented by a point that either lies to the left of the point with coordinate 2 or coincides with it. The set of such numbers is denoted by (-∞;2]. 2 Write it down in your notebook Write it down in your notebook Write it down in your notebook

A number x that satisfies the condition x > -3 is represented by a point that either lies to the right of the point with coordinate -3. The set of such numbers denotes (-3; +∞). - 3 Write it down in your notebook Write it down in your notebook Write it down in your notebook

3 5 3 5 3 5 3 5 5 -7 3

Independent work OPTION 1 OPTION 4 OPTION 2 OPTION 3 CHOOSE AN OPTION Help me! And to me, and to me. Choose me! You'll help me, won't you?

OPTION 1 1.Draw numerical intervals on the coordinate line: a). ; b). (-2; + ∞); V). [ 3;5) ; g).(- ∞ ;5 ]. 2. Write down the numerical interval shown in the figure: 3. Which of the numbers -1.6; -1.5; -1; 0; 3; 5.1; 6.5 belong between: a). [-1.5;6.5]; b).(3; + ∞); V). (- ∞ ;1]. 3 7 -5 6 -7 c). A). b). 4. Indicate the largest integer belonging to the interval: a). [-12;-9]; b). (-1;17). THANK YOU!

OPTION 2 1.Draw numerical intervals on the coordinate line: a). [ - 3; 0) ; b). [ - 3 ; + ∞); V). (- thirty) ; g).(- ∞ ; 0) . 2. Write down the numerical interval shown in the figure: 3. Which of the numbers are 2, 2; - 2, 1; -1; 0; 0.5; 1; 8, 9 belong to the interval: a). (- 2 , 2 ; 8 , 9 ]; b).(- ∞ ;0 ] ; c). (1 ;+ ∞) . -5 6 3 7 c). A). b). 4. Indicate the largest integer belonging to the interval: a). [-12;-9) ; b). [ -1;17 ] . 2 Help me!

OPTION 3 1.Draw numerical intervals on the coordinate line: a). (-0.44;5) ; b). (10 ; + ∞); V). [ 0 ; 13) ; d).(- ∞ ; -0.44 ]. 2. Write down the numerical interval shown in the figure: 3. Name all the integers belonging to the interval: a). [- 3 ; 1 ]; b).(- 3; 1); at 3 ; 1) ; G). (- 3 ; 1 ]; . 7 20 -8 6 -7 c). A). b). 4. Indicate the smallest integer belonging to the interval: a). [-12;-9]; b). (-1;17 ] . Thank you, I’m very happy!

OPTION 4 1.Draw numerical intervals on the coordinate line: a). [ -4 ; -0.29 ]; b). (- ∞ ;+ ∞); V). [1.7;5.9); g).(0.01;+ ∞) . 2. Write down the numerical interval shown in the figure: 3. Name all the integers belonging to the interval: a). [- 4 ; 3 ]; b).(-4; 3); at 4 ; 3) ; G). (- 4 ; 3 ]; . -4 -1 -5 25 in). A). b). 4. Indicate the smallest integer belonging to the interval: a). [-12;-9) ; b). (-1;17]. -8 Well done!

Calling the test program If you still have free minutes, call the test program by clicking on the word “CALL” Homework You can solve another OPTION

Homework 1). Draw two number intervals on the same coordinate line such that they have common points (2 examples). 2). Draw two number intervals on the same coordinate line such that they do not have common points(2 examples). Shutdown

THANK YOU FOR YOUR WORK!!!