No. 1. Properties of rational numbers.

 orderliness . For any rational numbers and there is a rule that allows you to uniquely identify between them one and only one of the three relations: "", "" or "". This rule is called ordering rule and is formulated as follows: two positive numbers are connected by the same relation as two integers; two non-positive numbers and are related by the same relation as two non-negative numbers and; if suddenly not negative, but negative, then.

summation of fractions

 Addition operation . summation rule, which puts them in correspondence with some rational number . In this case, the number itself is called sum numbers u is denoted, and the process of finding such a number is called summation. The summation rule has the following form: .

 multiplication operation . For any rational numbers and there is a so-called multiplication rule, which puts them in correspondence with some rational number . In this case, the number itself is called work numbers ii is denoted, and the process of finding such a number is also called multiplication. The multiplication rule is as follows: .

 Transitivity order relations. For any triple of rational numbers , and if less and less, then less, and if equal and equal, then equal.

 commutativity addition. From a change in the places of rational terms, the sum does not change.

 Associativity addition. The order in which three rational numbers are added does not affect the result.

 Availabilityzero . There is a rational number 0 that preserves every other rational number when summed.

 The presence of opposite numbers. Any rational number has an opposite rational number, which, when summed, gives 0.

 Commutativity of multiplication. By changing the places of rational factors, the product does not change.

 Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.

 Availabilityunits . There is a rational number 1 that preserves every other rational number when multiplied.

 Availabilityreciprocal numbers . Any non-zero rational number has an inverse rational number, multiplication by which gives 1.

 distributivity multiplication with respect to addition. The multiplication operation is consistent with the addition operation through the distribution law:

 Connection of the order relation with the operation of addition. The same rational number can be added to the left and right sides of a rational inequality.

 Connection of the order relation with the operation of multiplication. The left and right sides of a rational inequality can be multiplied by the same positive rational number.

 Axiom of Archimedes . Whatever the rational number , you can take so many units that their sum will exceed.

No. 2. Modulus of a real number.

Definition . The modulus of a non-negative real number x is the number itself: | x | = x; the modulus of a negative real number x is the opposite number: I x | = - x.

In short, it is written like this:

2. The geometric meaning of the modulus of a real number

Let us return to the set R of real numbers and its geometric models- number line. We mark two points a and b on the line (two real numbers a and b), denote by (a, b) the distance between the points a and b (- the letter of the Greek alphabet "ro"). This distance is equal to b - a, if b > a (Fig. 101), it is equal to a - b, if a > b (Fig. 102), finally, it is zero if a = b.

All three cases are covered by one formula:

b) Equation | x + 3.2 | = 2 rewrite in the form | x - (- 3.2) | \u003d 2 and further (x, - 3.2) \u003d 2. There are two points on the coordinate line that are removed from the point - 3.2 at a distance equal to 2. These are points - 5.2 and - 1.2 (Fig. .104). So the equation has two root: -5.2 and -1.2.

№4.SET OF REAL NUMBERS

The union of the set of rational numbers and the set of irrational numbers is called the set valid (or material ) numbers . The set of real numbers is denoted by the symbol R. Obviously, .

Real numbers are displayed on numerical axis Oh dots (Fig.). In this case, each real number corresponds to a certain point of the numerical axis, and each point of the axis corresponds to a certain real number.

Therefore, instead of the words "real number" you can say "point".

No. 5. number gaps.

Gap type	geometric images	Designation	Writing using inequalities
Interval
Half interval
Half interval
open beam
open beam

No. 6. Numeric function.

Let a number set be given If each number is assigned a single number y, then we say that on the set D numeric function :

y = f (x),

A bunch of D called function scope and denoted D (f (x)). The set of all elements f (x), where is called function range and denoted E (f (x)).

Number x often call function argument or an independent variable, and the number y- dependent variable or, in fact, function variable x. The number corresponding to the value is called function value at a point and denote or

To set a function f, you need to specify:

1) its domain of definition D (f (x));

2) specify the rule f, according to which each value is associated with some value y = f (x).

№7. inverse function,

Inverse function

If the roles of argument and function are reversed, then x becomes a function of y. In this case, one speaks of a new function called inverse function. Suppose we have a function:

v = u 2 ,

where u- argument, a v- function. If we reverse their roles, we get u as a function v :

If we denote the argument in both functions as x , and the function through y, then we have two functions:

each of which is the inverse of the other.

EXAMPLES. These functions are inverse to each other:

1) sin x and arcsin x, since if y= sin x, then x= Arcsin y;

2) cos x and Arccos x, since if y= cos x, then x= Arccos y;

3) tan x and Arctan x, since if y= tan x, then x= Arctan y;

4) e x and ln x, since if y= e x, then x=ln y.

Inverse trigonometric functions- mathematical functions that are inverse to trigonometric functions. Inverse trigonometric functions usually include six functions:

arcsine(symbol: arcsin)

arc cosine(symbol: arccos)

arc tangent(designation: arctg; in foreign literature arctan)

arc tangent(designation: arcctg; in foreign literature arccotan)

arcsecant(symbol: arcsec)

arccosecant(designation: arccosec; in foreign literature arccsc)

№8. Basic elementary functions. Elementary Functions

It is worth noting that the inverse trigonometric functions are multi-valued (infinitely significant), when operating with them, the so-called principal values ​​are used.

№9. Complex numbers

are written as: a+ bi. Here a and b – real numbers, a i – imaginary unit, i.e. i 2 = –1. Number a called abscissa, a b – ordinate complex number a+ b.i. Two complex numbers a+ bi and a – bi called conjugate complex numbers.

Real numbers can be represented by points on a straight line, as shown in the figure, where point A represents the number 4, and point B represents the number -5. The same numbers can also be represented by segments OA, OB, taking into account not only their length, but also their direction.

Each point M of the number line depicts some real number (rational if the segment OM is commensurable with a unit of length, and irrational if it is incommensurable). Thus, there is no room on the number line for complex numbers.

But complex numbers can be represented on the number plane. To do this, we choose a rectangular coordinate system on the plane, with the same scale on both axes.

Complex number a + b i represented by the point M, in which the abscissa x is equal to the abscissa a complex number, and the ordinate of y is equal to the ordinate b complex number.

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 modulus of a complex number is determined and found.

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 . Thus, .

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 origin, so the distance from the origin to the point with coordinate 0 is zero (no single segment and no segment that makes up any fraction of the unit segment needs to be postponed in order to get from 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, i.e, . 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, i.e, . 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.

We already know that the set of real numbers $R$ is formed by rational and irrational numbers.

Rational numbers can always be represented as decimals (finite or infinite periodic).

Irrational numbers are written as infinite but non-recurring decimals.

The set of real numbers $R$ also includes the elements $-\infty $ and $+\infty $, for which the inequalities $-\infty

Consider ways to represent real numbers.

Common fractions

Ordinary fractions are written using two natural numbers and a horizontal fractional bar. The fractional bar actually replaces the division sign. The number below the line is the denominator (divisor), the number above the line is the numerator (divisible).

Definition

A fraction is called proper if its numerator is less than its denominator. Conversely, a fraction is called improper if its numerator is greater than or equal to its denominator.

For ordinary fractions, there are simple, practically obvious, comparison rules ($m$,$n$,$p$ are natural numbers):

1. of two fractions with the same denominators, the one with the larger numerator is larger, i.e. $\frac(m)(p) >\frac(n)(p) $ for $m>n$;
2. of two fractions with the same numerators, the one with the smaller denominator is larger, i.e. $\frac(p)(m) >\frac(p)(n) $ for $ m
3. a proper fraction is always less than one; improper fraction is always greater than one; a fraction whose numerator is equal to the denominator is equal to one;
4. Any improper fraction is greater than any proper fraction.

Decimal numbers

The notation of a decimal number (decimal fraction) has the form: integer part, decimal point, fractional part. The decimal notation of an ordinary fraction can be obtained by dividing the "angle" of the numerator by the denominator. This can result in either a finite decimal fraction or an infinite periodic decimal fraction.

Definition

The fractional digits are called decimal places. In this case, the first digit after the decimal point is called the tenths digit, the second - the hundredths digit, the third - the thousandths digit, etc.

Example 1

We determine the value of the decimal number 3.74. We get: $3.74=3+\frac(7)(10) +\frac(4)(100) $.

The decimal number can be rounded. In this case, you must specify the digit to which rounding is performed.

The rounding rule is as follows:

1. all digits to the right of this digit are replaced with zeros (if these digits are before the decimal point) or discarded (if these digits are after the decimal point);
2. if the first digit following the given digit is less than 5, then the digit of this digit is not changed;
3. if the first digit following the given digit is 5 or more, then the digit of this digit is increased by one.

Example 2

1. Let's round the number 17302 to the nearest thousand: 17000.
2. Let's round the number 17378 to the nearest hundred: 17400.
3. Let's round the number 17378.45 to tens: 17380.
4. Let's round the number 378.91434 to the nearest hundredth: 378.91.
5. Let's round the number 378.91534 to the nearest hundredth: 378.92.

Converting a decimal number to a common fraction.

Case 1

A decimal number is a terminating decimal.

The conversion method is shown in the following example.

Example 2

We have: $3.74=3+\frac(7)(10) +\frac(4)(100) $.

Reduce to a common denominator and get:

The fraction can be reduced: $3.74=\frac(374)(100) =\frac(187)(50) $.

Case 2

A decimal number is an infinite recurring decimal.

The transformation method is based on the fact that the periodic part of a periodic decimal fraction can be considered as the sum of members of an infinite decreasing geometric progression.

Example 4

$0,\left(74\right)=\frac(74)(100) +\frac(74)(10000) +\frac(74)(1000000) +\ldots $. The first member of the progression is $a=0.74$, the denominator of the progression is $q=0.01$.

Example 5

$0.5\left(8\right)=\frac(5)(10) +\frac(8)(100) +\frac(8)(1000) +\frac(8)(10000) +\ldots $ . The first member of the progression is $a=0.08$, the denominator of the progression is $q=0.1$.

The sum of the terms of an infinite decreasing geometric progression is calculated by the formula $s=\frac(a)(1-q) $, where $a$ is the first term and $q$ is the denominator of the progression $ \left (0

Example 6

Let's convert the infinite periodic decimal fraction $0,\left(72\right)$ into a regular one.

The first member of the progression is $a=0.72$, the denominator of the progression is $q=0.01$. We get: $s=\frac(a)(1-q) =\frac(0.72)(1-0.01) =\frac(0.72)(0.99) =\frac(72)( 99) =\frac(8)(11) $. So $0,\left(72\right)=\frac(8)(11) $.

Example 7

Let's convert the infinite periodic decimal fraction $0.5\left(3\right)$ into a regular one.

The first member of the progression is $a=0.03$, the denominator of the progression is $q=0.1$. We get: $s=\frac(a)(1-q) =\frac(0.03)(1-0.1) =\frac(0.03)(0.9) =\frac(3)( 90) =\frac(1)(30)$.

So $0.5\left(3\right)=\frac(5)(10) +\frac(1)(30) =\frac(5\cdot 3)(10\cdot 3) +\frac( 1)(30) =\frac(15)(30) +\frac(1)(30) =\frac(16)(30) =\frac(8)(15) $.

Real numbers can be represented by points on the number line.

In this case, we call the numerical axis an infinite line on which the origin (point $O$), positive direction (indicated by an arrow) and scale (to display values) are selected.

Between all real numbers and all points of the numerical axis there is a one-to-one correspondence: each point corresponds to a single number and, conversely, each number corresponds to a single point. Therefore, the set of real numbers is continuous and infinite in the same way as the number axis is continuous and infinite.

Some subsets of the set of real numbers are called numerical intervals. The elements of a numerical interval are numbers $x\in R$ satisfying a certain inequality. Let $a\in R$, $b\in R$ and $a\le b$. In this case, the types of gaps can be as follows:

1. Interval $\left(a,\; b\right)$. At the same time $ a
2. Segment $\left$. Moreover, $a\le x\le b$.
3. Half-segments or half-intervals $\left$. At the same time $ a \le x
4. Infinite spans, e.g. $a

Of great importance is also a kind of interval, called the neighborhood of a point. The neighborhood of a given point $x_(0) \in R$ is an arbitrary interval $\left(a,\; b\right)$ containing this point inside itself, i.e. $a 0$ - 10th radius.

The absolute value of the number

The absolute value (or modulus) of a real number $x$ is a non-negative real number $\left|x\right|$, defined by the formula: $\left|x\right|=\left\(\begin(array)(c) (\; \; x\; \; (\rm on)\; \; x\ge 0) \\ (-x\; \; (\rm on)\; \; x

Geometrically, $\left|x\right|$ means the distance between the points $x$ and 0 on the real axis.

Properties of absolute values:

1. it follows from the definition that $\left|x\right|\ge 0$, $\left|x\right|=\left|-x\right|$;
2. for the modulus of the sum and for the modulus of the difference of two numbers, the inequalities $\left|x+y\right|\le \left|x\right|+\left|y\right|$, $\left|x-y\right|\le \left|x\right|+\left|y\right|$ and $\left|x+y\right|\ge \left|x\right|-\left|y\right|$,$\ left|x-y\right|\ge \left|x\right|-\left|y\right|$;
3. the modulus of the product and the modulus of the quotient of two numbers satisfy the equalities $\left|x\cdot y\right|=\left|x\right|\cdot \left|y\right|$ and $\left|\frac(x)( y) \right|=\frac(\left|x\right|)(\left|y\right|) $.

Based on the definition of the absolute value for an arbitrary number $a>0$, one can also establish the equivalence of the following pairs of inequalities:

1. if $ \left|x\right|
2. if $\left|x\right|\le a$ then $-a\le x\le a$;
3. if $\left|x\right|>a$ then either $xa$;
4. if $\left|x\right|\ge a$, then either $x\le -a$ or $x\ge a$.

Example 8

Solve the inequality $\left|2\cdot x+1\right|

This inequality is equivalent to the inequalities $-7

From here we get: $-8

The video lesson "The geometric meaning of the modulus of a real number" is a visual aid for a mathematics lesson on the relevant topic. In the video tutorial, the geometric meaning of the module is examined in detail and clearly, after which it is shown with examples how the module of a real number is found, and the solution is accompanied by a picture. The material can be used at the stage of explaining a new topic as a separate part of the lesson or providing clarity to the teacher's explanation. Both options help to increase the effectiveness of the mathematics lesson, help the teacher achieve the goals of the lesson.

This video tutorial contains constructions that clearly demonstrate the geometric meaning of the module. To make the demonstration more visual, these constructions are performed using animation effects. To make the learning material easier to remember, important theses are highlighted in color. The solution of examples is considered in detail, which, due to animation effects, is presented in a structured, consistent, understandable way. When compiling the video, tools were used that help make the video lesson an effective modern learning tool.

The video starts by introducing the topic of the lesson. A construction is being performed on the screen - a ray is shown on which points a and b are marked, the distance between which is marked as ρ(a;b). It is reminded that the distance is measured on the coordinate ray by subtracting a smaller number from a larger number, that is, for this construction, the distance is equal to b-a for b>a and equal to a-b for a>b. The construction is shown below, on which the marked point a lies to the right of b, that is, the corresponding numerical value is greater than b. Below we note one more case when the positions of points a and b coincide. In this case, the distance between the points is equal to zero ρ(a;b)=0. All together these cases are described by one formula ρ(a;b)=|a-b|.

Next, we consider the solution of problems in which knowledge about the geometric meaning of the module is applied. In the first example, it is necessary to solve the equation |x-2|=3. It is noted that this is an analytical form of writing this equation, which we translate into geometric language to find a solution. Geometrically, this problem means that it is necessary to find points x for which the equality ρ(x;2)=3 will be true. On the coordinate line, this will mean that the points x are equidistant from the point x \u003d 2 at a distance of 3. To demonstrate the solution on the coordinate line, a ray is drawn on which point 2 is marked. At a distance of 3 from the point x \u003d 2, points -1 and 5 are marked. Obviously that these marked points will be the solution of the equation.

To solve the equation |x+3,2|=2, it is proposed to bring it first to the form |a-b| in order to solve the task on the coordinate line. After transformation, the equation takes the form |x-(-3,2)|=2. This means that the distance between the point -3.2 and the desired points will be equal to 2, that is, ρ (x; -3.2) = 2. Point -3,2 is marked on the coordinate line. Points -1.2 and -5.2 are located at a distance of 2 from it. These points are marked on the coordinate line and are indicated as a solution to the equation.

The solution of another equation |x|=2.7 considers the case when the desired points are located at a distance of 2.7 from point 0. The equation is rewritten as |x-0|=2.7. At the same time, it is indicated that the distance to the desired points is determined as ρ(x;0)=2.7. Point 0 is marked on the coordinate line. At a distance of 2.7 from point 0, points -2.7 and 2.7 are placed. These points are marked on the constructed line, they are the solutions of the equation.

To solve the following equation |x-√2|=0, no geometric interpretation is required, since if the modulus of the expression is zero, this means that this expression is equal to zero, that is, x-√2=0. It follows from the equation that x=√2.

The following example deals with solving equations that require a transformation before being solved. In the first equation |2x-6|=8 there is a numerical coefficient 2 in front of x. |=2|x-3|. After that, the right and left parts of the equation are reduced by 2. We get an equation of the form |x-3|=4. This equation of an analytical form is translated into the geometric language ρ(х;3)=4. We mark point 3 on the coordinate line. From this point we set aside points located at a distance of 4. The solution to the equation will be points -1 and 7, which are marked on the coordinate line. The second considered equation |5-3x|=6 also contains a numerical coefficient in front of the variable x. To solve the equation, the coefficient 3 is taken out of brackets. The equation becomes |-3(x-5/3)|=3|x-5/3|. The right and left sides of the equation can be reduced by 3. After that, an equation of the form |x-5/3|=2 is obtained. We pass from the analytical form to the geometric interpretation ρ(х;5/3)=2. A drawing is constructed for the solution, on which a coordinate line is depicted. The point 5/3 is marked on this line. At a distance of 2 from the point 5/3 are the points -1/3 and 11/3. These points are the solutions of the equation.

The last considered equation |4x+1|=-2. To solve this equation, transformations and geometric representation are not required. The left side of the equation obviously produces a non-negative number, while the right side contains the number -2. Therefore, this equation has no solutions.

The video lesson "The geometric meaning of the modulus of a real number" can be used in a traditional mathematics lesson at school. The material can be useful for a teacher who provides distance education. A detailed, understandable explanation of the solution of tasks that use the function of the module will help a student who masters the topic on his own to master the material.

A number line, a number axis, is a line on which real numbers are depicted. On the straight line, the origin is chosen - the point O (point O represents 0) and the point L, representing the unit. The point L usually stands to the right of the point O. The segment OL is called the unit segment.

The points to the right of point O represent positive numbers. Dots to the left of the dot. Oh, depict negative numbers. If the point X represents a positive number x, then the distance OX = x. If the point X represents a negative number x, then the distance OX = - x.

The number showing the position of a point on a straight line is called the coordinate of this point.

Point V shown in the figure has a coordinate of 2, and point H has a coordinate of -2.6.

The modulus of a real number is the distance from the origin to the point corresponding to this number. Designate the modulus of the number x, so: | x |. Obviously, | 0 | = 0.

If the number x is greater than 0, then | x | = x, and if x is less than 0, then | x | = - x. On these properties of the module, the solution of many equations and inequalities with the module is based.

Example: Solve Equation | x - 3 | = 1.

Solution: Consider two cases - the first case, when x -3 > 0, and the second case, when x - 3 0.

1. x - 3 > 0, x > 3.

In this case | x - 3 | = x - 3.

The equation takes the form x - 3 \u003d 1, x \u003d 4. 4\u003e 3 - satisfy the first condition.

2. x -3 0, x 3.

In this case | x - 3 | = - x + 3

The equation takes the form x + 3 \u003d 1, x \u003d - 2. -2 3 - satisfy the second condition.

Answer: x = 4, x = -2.

Numeric expressions.

A numeric expression is a collection of one or more numbers and functions connected by arithmetic operators and brackets.
Examples of numeric expressions:

The value of a numeric expression is a number.
Operations in numerical expression are performed in the following sequence:

1. Actions in brackets.

2. Calculation of functions.

3. Exponentiation

4. Multiplication and division.

5. Addition and subtraction.

6. Operations of the same type are performed from left to right.

So the value of the first expression will be the number itself 12.3
In order to calculate the value of the second expression, we will perform the actions in the following sequence:

1. Perform the actions in brackets in the following sequence - first we raise 2 to the third power, then subtract 11 from the resulting number:

3 4 + (23 - 11) = 3 4 + (8 - 11) = 3 4 + (-3)

2. Multiply 3 by 4:

3 4 + (-3) = 12 + (-3)

3. Perform the operations sequentially from left to right:

12 + (-3) = 9.
An expression with variables is a collection of one or more numbers, variables and functions connected by arithmetic operators and brackets. The values ​​of expressions with variables depend on the values ​​of the variables included in it. The sequence of operations here is the same as for numerical expressions. It is sometimes useful to simplify expressions with variables by performing various actions - parentheses, parenthesis expansion, grouping, reduction of fractions, reduction of similar ones, etc. Also, to simplify expressions, various formulas are often used, for example, abbreviated multiplication formulas, properties of various functions, etc.

Algebraic expressions.

An algebraic expression is one or more algebraic quantities (numbers and letters) interconnected by signs of algebraic operations: addition, subtraction, multiplication and division, as well as extracting the root and raising to an integer power (moreover, the root and exponent must necessarily be integers) and signs of the sequence of these actions (usually brackets of various kinds). The number of values ​​included in the algebraic expression must be finite.

An example of an algebraic expression:

"Algebraic expression" is a syntactic concept, that is, something is an algebraic expression if and only if it obeys certain grammatical rules (see Formal grammar). If the letters in an algebraic expression are considered variables, then the algebraic expression acquires the meaning of an algebraic function.