Image of real numbers on the coordinate line. Definition of the modulus of a number. The geometric meaning of the module. Determining the modulus of a number through the arithmetic square root

In this article, we will analyze in detail the absolute value of a number. We will give various definitions of the modulus of a number, introduce notation and give graphic illustrations. In this case, we consider various examples of finding the modulus of a number by definition. After that, we list and justify the main properties of the module. At the end of the article, we will talk about how the module is defined and located. complex number.

Page navigation.

Modulus of number - definition, notation and examples

First we introduce modulus designation. The module of the number a will be written as , that is, to the left and to the right of the number we will put vertical lines that form the sign of the module. Let's give a couple of examples. For example, modulo -7 can be written as ; module 4,125 is written as , and module is written as .

The following definition of the module refers to, and therefore, to, and to integers, and to rational and irrational numbers, as to the constituent parts of the set of real numbers. We will talk about the modulus of a complex number in.

Definition.

Modulus of a is either the number a itself, if a is a positive number, or the number −a, the opposite of the number a, if a is a negative number, or 0, if a=0 .

The voiced definition of the modulus of a number is often written in the following form , this notation means that if a>0 , if a=0 , and if a<0 .

The record can be represented in a more compact form . This notation means that if (a is greater than or equal to 0 ), and if a<0 .

There is also a record . Here, the case when a=0 should be explained separately. In this case, we have , but −0=0 , since zero is considered a number that is opposite to itself.

Let's bring examples of finding the modulus of a number with a given definition. For example, let's find modules of numbers 15 and . Let's start with finding . Since the number 15 is positive, its modulus is, by definition, equal to this number itself, that is, . What is the modulus of a number? Since is a negative number, then its modulus is equal to the number opposite to the number, that is, the number . In this way, .

In conclusion of this paragraph, we give one conclusion, which is very convenient to apply in practice when finding the modulus of a number. From the definition of the modulus of a number it follows that the modulus of a number is equal to the number under the sign of the modulus, regardless of its sign, and from the examples discussed above, this is very clearly visible. The voiced statement explains why the modulus of a number is also called the absolute value of the number. So the modulus of a number and the absolute value of a number are one and the same.

Modulus of a number as a distance

Geometrically, the modulus of a number can be interpreted as distance. Let's bring determination of the modulus of a number in terms of distance.

Definition.

Modulus of a is the distance from the origin on the coordinate line to the point corresponding to the number a.

This definition is consistent with the definition of the modulus of a number given in the first paragraph. Let's explain this point. The distance from the origin to the point corresponding to a positive number is equal to this number. Zero corresponds to the reference point, therefore the distance from the reference point to the point with coordinate 0 is equal to zero (no single segment and no segment constituting any fraction of a single segment is needed to get from the point O to the point with coordinate 0). The distance from the origin to a point with a negative coordinate is equal to the number opposite to the coordinate of the given point, since it is equal to the distance from the origin to the point whose coordinate is the opposite number.

For example, the modulus of the number 9 is 9, since the distance from the origin to the point with coordinate 9 is nine. Let's take another example. The point with coordinate −3.25 is at a distance of 3.25 from point O, so .

The sounded definition of the modulus of a number is a special case of defining the modulus of the difference of two numbers.

Definition.

Difference modulus of two numbers a and b is equal to the distance between the points of the coordinate line with coordinates a and b .

That is, if points on the coordinate line A(a) and B(b) are given, then the distance from point A to point B is equal to the modulus of the difference between the numbers a and b. If we take point O (reference point) as point B, then we will get the definition of the modulus of the number given at the beginning of this paragraph.

Determining the modulus of a number through the arithmetic square root

Sometimes found determination of the modulus through the arithmetic square root.

For example, let's calculate the modules of the numbers −30 and based on this definition. We have . Similarly, we calculate the modulus of two-thirds: .

The definition of the modulus of a number in terms of the arithmetic square root is also consistent with the definition given in the first paragraph of this article. Let's show it. Let a be a positive number, and let −a be negative. Then and , if a=0 , then .

Module Properties

The module has a number of characteristic results - module properties. Now we will give the main and most commonly used of them. When substantiating these properties, we will rely on the definition of the modulus of a number in terms of distance.

Let's start with the most obvious module property − modulus of a number cannot be a negative number. In literal form, this property has the form for any number a . This property is very easy to justify: the modulus of a number is the distance, and the distance cannot be expressed as a negative number.

Let's move on to the next property of the module. The modulus of a number is equal to zero if and only if this number is zero. The modulus of zero is zero by definition. Zero corresponds to the origin, no other point on the coordinate line corresponds to zero, since each real number is associated with a single point on the coordinate line. For the same reason, any number other than zero corresponds to a point other than the origin. And the distance from the origin to any point other than the point O is not equal to zero, since the distance between two points is equal to zero if and only if these points coincide. The above reasoning proves that only the modulus of zero is equal to zero.

Move on. Opposite numbers have equal modules, that is, for any number a . Indeed, two points on the coordinate line, whose coordinates are opposite numbers, are at the same distance from the origin, which means that the modules of opposite numbers are equal.

The next module property is: the modulus of the product of two numbers is equal to the product of the modules of these numbers, that is, . By definition, the modulus of the product of numbers a and b is either a b if , or −(a b) if . It follows from the rules of multiplication of real numbers that the product of moduli of numbers a and b is equal to either a b , , or −(a b) , if , which proves the considered property.

The modulus of the quotient of dividing a by b is equal to the quotient of dividing the modulus of a by the modulus of b, that is, . Let us justify this property of the module. Since the quotient is equal to the product, then . By virtue of the previous property, we have . It remains only to use the equality , which is valid due to the definition of the modulus of the number.

The following module property is written as an inequality: , a , b and c are arbitrary real numbers. The written inequality is nothing more than triangle inequality. To make this clear, let's take the points A(a) , B(b) , C(c) on the coordinate line, and consider the degenerate triangle ABC, whose vertices lie on the same line. By definition, the modulus of the difference is equal to the length of the segment AB, - the length of the segment AC, and - the length of the segment CB. Since the length of any side of a triangle does not exceed the sum of the lengths of the other two sides, the inequality , therefore, the inequality also holds.

The inequality just proved is much more common in the form . The written inequality is usually considered as a separate property of the module with the formulation: “ The modulus of the sum of two numbers does not exceed the sum of the moduli of these numbers". But the inequality directly follows from the inequality , if we put −b instead of b in it, and take c=0 .

Complex number modulus

Let's give determination of the modulus of a complex number. Let us be given complex number, written in algebraic form , where x and y are some real numbers, representing, respectively, the real and imaginary parts of a given complex number z, and is an imaginary unit.

Definition.

The modulus of a complex number z=x+i y is called the arithmetic square root of the sum of the squares of the real and imaginary parts of a given complex number.

The modulus of a complex number z is denoted as , then the sounded definition of the modulus of a complex number can be written as .

This definition allows you to calculate the modulus of any complex number in algebraic notation. For example, let's calculate the modulus of a complex number. In this example, the real part of the complex number is , and the imaginary part is minus four. Then, by the definition of the modulus of a complex number, we have .

The geometric interpretation of the modulus of a complex number can be given in terms of distance, by analogy with the geometric interpretation of the modulus of a real number.

Definition.

Complex number modulus z is the distance from the beginning of the complex plane to the point corresponding to the number z in this plane.

According to the Pythagorean theorem, the distance from the point O to the point with coordinates (x, y) is found as , therefore, , where . Therefore, the last definition of the modulus of a complex number agrees with the first.

This definition also allows you to immediately indicate what the modulus of a complex number z is, if it is written in trigonometric form as or in exponential form. Here . For example, the modulus of a complex number is 5 , and the modulus of the complex number is .

It can also be seen that the product of a complex number and its complex conjugate gives the sum of the squares of the real and imaginary parts. Really, . The resulting equality allows us to give one more definition of the modulus of a complex number.

Definition.

Complex number modulus z is the arithmetic square root of the product of this number and its complex conjugate, that is, .

In conclusion, we note that all the properties of the module formulated in the corresponding subsection are also valid for complex numbers.

Bibliography.

• Vilenkin N.Ya. etc. Mathematics. Grade 6: textbook for educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
• Lunts G.L., Elsgolts L.E. Functions of a complex variable: a textbook for universities.
• Privalov I.I. Introduction to the theory of functions of a complex variable.

Back forward

Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested in this work, please download the full version.

Goals:

Equipment: projector, screen, personal computer, multimedia presentation

During the classes

1. Organizational moment.

2. Actualization of students' knowledge.

2.1. Answer student questions for homework.

2.2. Solve the crossword puzzle (repetition of theoretical material) (Slide 2):

- Having solved the crossword puzzle, read the title of the topic of today's lesson in the highlighted vertical column. (Slides 3, 4)

3. Explanation new topic.

3.1. - Guys, you have already met with the concept of a module, used the designation | a| . Previously, it was only about rational numbers. Now we need to introduce the concept of modulus for any real number.

Each real number corresponds to a single point on the number line, and, conversely, to each point on the number line, there corresponds a single real number. All basic properties of actions on rational numbers are also preserved for real numbers.

The concept of the modulus of a real number is introduced. (Slide 5).

Definition. The modulus of a non-negative real number x call this number itself: | x| = x; modulo a negative real number X call the opposite number: | x| = – x .

– Write in your notebooks the topic of the lesson, the definition of the module:

In practice, various module properties, for example. (Slide 6) :

Perform orally No. 16.3 (a, b) - 16.5 (a, b) on the application of the definition, properties of the module. (Slide 7) .

3.4. For any real number X can be calculated | x| , i.e. we can talk about the function y = |x| .

Task 1. Draw a graph and list the properties of a function y = |x| (Slides 8, 9).

One student on the board builds a graph of a function

Fig 1.

Properties are listed by students. (Slide 10)

1) Domain of definition - (- ∞; + ∞) .

2) y = 0 at x = 0; y > 0 for x< 0 и x > 0.

3) The function is continuous.

4) y max = 0 for x = 0, y max does not exist.

5) The function is limited from below, not limited from above.

6) The function decreases on the ray (– ∞; 0) and increases on the ray )