Image of numbers on the number line. Image of real numbers on the number line. intervals. Determining the modulus of a number through the arithmetic square root

Equations with modules, methods of solutions. Part 1.

Before proceeding directly to the study of techniques for solving such equations, it is important to understand the essence of the module, its geometric value. It is in understanding the definition of the module and its geometric meaning that the main methods for solving such equations are laid. The so-called method of intervals when opening modular brackets is so effective that using it it is possible to solve absolutely any equation or inequality with modules. In this part, we will study in detail two standard methods: the method of intervals and the method of replacing an equation by a population.

However, as we will see, these methods are always effective, but not always convenient and can lead to long and even not very convenient calculations, which naturally require more time to solve them. Therefore, it is important to know those methods that greatly simplify the solution of certain structures of equations. Squaring both parts of an equation, the method of introducing a new variable, the graphical method, solving equations containing the modulus under the modulus sign. We will cover these methods in the next section.

Definition of the modulus of a number. The geometric meaning of the module.

First of all, let's get acquainted with geometric sense module:

modulo number a (|a|) call the distance on the number line from the origin (point 0) to the point A(a).

Based on this definition, consider some examples:

|7| is the distance from 0 to point 7, of course it is 7. → | 7 |=7

|-5| is distance from 0 to point -5 and it is equal to: 5. → |-5| = 5

We all understand distance cannot be negative! Therefore |x| ≥ 0 always!

Solve the equation: |x |=4

This equation can be read like this: the distance from point 0 to point x is 4. Yeah, it turns out that from 0 we can move both to the left and to the right, which means moving to the left by a distance equal to 4 we will end up at the point: -4, and moving to the right we will end up at the point: 4. Indeed, |-4 |=4 and |4 |=4.

Hence the answer is x=±4.

If you carefully study the previous equation, you will notice that: the distance to the right along the number line from 0 to the point is equal to the point itself, and the distance to the left from 0 to the number is equal to the opposite number! Realizing that to the right of 0 positive numbers, and negative to the left of 0, we formulate definitions of the modulus of a number: modulus (absolute value) of a number X(|x|) is called the number itself X, if x ≥0, and the number is X if x<0.

Here we need to find a set of points on the number line, the distance from 0 to which will be less than 3, let's imagine a number line, point 0 on it, go left and count one (-1), two (-2) and three (-3), stop. Further points will go that lie further than 3 or the distance to which from 0 is greater than 3, now we go to the right: one, two, three, stop again. Now we select all our points and get the interval x: (-3; 3).

It is important that you see this clearly, if it still doesn’t work out, draw on paper and see that this illustration is completely clear to you, do not be lazy and try to see in your mind the solutions to the following tasks:

|x |=11, x=? |x|=-5, x=?

| x |<8, х-? |х| <-6, х-?

|x|>2, x-? |x|> -3, x-?

|π-3|=? |-x²-10|=?

|√5-2|=? |2x-x²-3|=?

|x²+2|=? |х²+4|=0

|x²+3x+4|=? |-x²+9| ≤0

Pay attention to the strange tasks in the second column? Indeed, the distance cannot be negative, therefore: |x|=-5- has no solutions, of course, it cannot be less than 0, therefore: |x|<-6 тоже не имеет решений, ну и естественно, что любое расстояние будет больше отрицательного числа, значит решением |x|>-3 are all numbers.

After you learn how to quickly see drawings with solutions, read on.

REAL NUMBERS II

§ 44 Geometric representation of real numbers

Geometrically real numbers, like rational numbers, are represented by points on a straight line.

Let l - an arbitrary straight line, and O - some of its points (Fig. 58). Every positive real number α put in correspondence the point A, lying to the right of O at a distance of α units of length.

If, for example, α = 2.1356..., then

2 < α < 3
2,1 < α < 2,2
2,13 < α < 2,14

etc. It is obvious that the point A in this case must be on the line l to the right of the points corresponding to the numbers

2; 2,1; 2,13; ... ,

but to the left of the points corresponding to the numbers

3; 2,2; 2,14; ... .

It can be shown that these conditions define on the line l the only point A, which we consider as the geometric image of a real number α = 2,1356... .

Likewise, every negative real number β put in correspondence the point B lying to the left of O at a distance of | β | units of length. Finally, we assign the point O to the number "zero".

So, the number 1 will be displayed on a straight line l point A, located to the right of O at a distance of one unit of length (Fig. 59), the number - √2 - point B, lying to the left of O at a distance of √2 units of length, etc.

Let's show how on a straight line l using a compass and a ruler, you can find points corresponding to the real numbers √2, √3, √4, √5, etc. To do this, first of all, we will show how to construct segments whose lengths are expressed by these numbers. Let AB be a segment taken as a unit of length (Fig. 60).

At point A, we restore a perpendicular to this segment and set aside on it the segment AC, equal to the segment AB. Then, applying the Pythagorean theorem to the right triangle ABC, we get; BC \u003d √AB 2 + AC 2 \u003d √1 + 1 \u003d √2

Therefore, the segment BC has length √2. Now let us restore the perpendicular to the segment BC at the point C and choose the point D on it so that the segment CD is equal to unit length AB. Then from the right triangle BCD we find:

ВD \u003d √BC 2 + CD 2 \u003d √2 + 1 \u003d √3

Therefore, the segment BD has length √3. Continuing the described process further, we could get segments BE, BF, ..., whose lengths are expressed by the numbers √4, √5, etc.

Now on the line l it is easy to find those points that serve as a geometric representation of the numbers √2, √3, √4, √5, etc.

Putting, for example, to the right of the point O the segment BC (Fig. 61), we get the point C, which serves as a geometric representation of the number √2. In the same way, putting off the segment BD to the right of the point O, we get the point D", which is the geometric image of the number √3, etc.

However, one should not think that with the help of a compass and a ruler on a number line l one can find a point corresponding to any given real number. It has been proven, for example, that, having only a compass and a ruler at your disposal, it is impossible to construct a segment whose length is expressed by the number π = 3.14 ... . So on the number line l using such constructions, it is impossible to indicate a point corresponding to this number. Nevertheless, such a point exists.

So for every real number α it is possible to associate some well-defined point of the line l . This point will be separated from the starting point O at a distance of | α | units of length and be to the right of O if α > 0, and to the left of O if α < 0. Очевидно, что при этом двум неравным действительным числам будут соответствовать две различные точки прямой l . Indeed, let the number α corresponds to point A, and the number β - point B. Then, if α > β , then A will be to the right of B (Fig. 62, a); if α < β , then A will lie to the left of B (Fig. 62, b).

Speaking in § 37 about the geometric representation of rational numbers, we posed the question: can any point of a straight line be considered as a geometric image of some rational numbers? At that time we could not give an answer to this question; now we can answer it quite definitely. There are points on the line that serve as a geometric representation of irrational numbers (for example, √2). Therefore, not every point on a straight line represents a rational number. But in this case, another question arises: can any point of the real line be considered as a geometric image of some valid numbers? This issue has already been resolved positively.

Indeed, let A be an arbitrary point on the line l , lying to the right of O (Fig. 63).

The length of the segment OA is expressed by some positive real number α (see § 41). Therefore point A is the geometric image of the number α . Similarly, it is established that each point B, lying to the left of O, can be considered as a geometric image of a negative real number - β , Where β - the length of the segment VO. Finally, the point O serves as a geometric representation of the number zero. It is clear that two distinct points of the line l cannot be the geometric image of the same real number.

For the reasons stated above, a straight line on which some point O is indicated as the "initial" point (for a given unit of length) is called number line.

Conclusion. The set of all real numbers and the set of all points of the real line are in a one-to-one correspondence.

This means that each real number corresponds to one, well-defined point of the number line, and, conversely, to each point of the number line, with such a correspondence, there corresponds one, well-defined real number.

Exercises

320. Find out which of the two points is on the number line to the left and which to the right, if these points correspond to numbers:

a) 1.454545... and 1.455454...; c) 0 and - 1.56673...;

b) - 12.0003... and - 12.0002...; d) 13.24... and 13.00....

321. Find out which of the two points is further from the starting point O on the number line, if these points correspond to numbers:

a) 5.2397... and 4.4996...; .. c) -0.3567... and 0.3557... .

d) - 15.0001 and - 15.1000...;

322. In this section it was shown that to construct a segment of length √ n using a compass and straightedge, you can do the following: first construct a segment with a length of √2, then a segment with a length of √3, etc., until we reach a segment with a length of √ n . But for every fixed P > 3 this process can be accelerated. How, for example, would you begin to build a segment of length √10?

323*. How to use a compass and ruler to find a point on the number line corresponding to the number 1 / α , if the position of the point corresponding to the number α , known?

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):

of two fractions with the same denominators, the one with the larger numerator is larger, i.e. $\frac(m)(p) >\frac(n)(p) $ for $m>n$;
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
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;
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:

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);
if the first digit following the given digit is less than 5, then the digit of this digit is not changed;
if the first digit following the given digit is 5 or more, then the digit of this digit is increased by one.

Example 2

Let's round the number 17302 to the nearest thousand: 17000.
Let's round the number 17378 to the nearest hundred: 17400.
Let's round the number 17378.45 to tens: 17380.
Let's round the number 378.91434 to the nearest hundredth: 378.91.
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 straight line, on which the origin (point $O$), positive direction (indicated by an arrow) and scale (to display values) are selected.

There is a one-to-one correspondence between all real numbers and all points of the numerical axis: 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:

Interval $\left(a,\; b\right)$. At the same time $ a
Segment $\left$. Moreover, $a\le x\le b$.
Half-segments or half-intervals $\left$. At the same time $ a \le x
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:

it follows from the definition that $\left|x\right|\ge 0$, $\left|x\right|=\left|-x\right|$;
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 also $\left|x+y\right|\ge \left|x\right|-\left|y\right|$,$\ left|x-y\right|\ge \left|x\right|-\left|y\right|$;
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:

if $ \left|x\right|
if $\left|x\right|\le a$ then $-a\le x\le a$;
if $\left|x\right|>a$ then either $xa$;
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

Back forward

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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):

A combination of mathematical symbols expressing some

statement. ( Formula.)
Infinite decimal non-periodic fractions. ( Irrational numbers)

A digit or group of digits repeated in an infinite decimal. ( Period.)

Numbers used to count things. ( natural numbers.)

Infinite decimal periodic fractions. (Rational numbers .)

Rational numbers + irrational numbers = ? (Valid numbers .)

- 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 of the 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: