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 also $\left|x+y\right|\ge \left|x\right|-\left|y\right|$,$\ left|x-y\right|\ge \left|x\right|-\left|y\right|$;
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

Definition 1. Numerical axis A line is called a line with a reference point, scale and direction chosen on it.

Theorem 1. There is a one-to-one correspondence (bijection) between the points of the numerical axis and the real numbers.

Need. Let us show that each point of the numerical axis corresponds to a real number. To do this, set aside a scale segment of unit length

times so that point will lie to the left of the point , and the point
already to the right. Next segment
divide by
parts and set aside the segment and times so that point will lie to the left of the point , and the point
already to the right. Thus, at each stage, the number
,
… If this procedure ends at some stage, we will get the number
(point coordinate on the number line). If not, then we call the left boundary of any interval "number with a disadvantage", and the right one - "the number in excess", or "approximation of the number with a deficiency or excess, "and the number itself will be an infinite non-periodic (why?) decimal fraction. It can be shown that all operations with rational approximations of an irrational number are defined unambiguously.

Adequacy. Let us show that any real number corresponds to a single point on the number axis. 

Definition 2. If a
, then the number interval
calledsegment , if
, then the number interval calledinterval , if
, then the number interval
calledhalf-interval .

O
definition 3. If the segment
nested segments so that
, a
, then such a system is called SHS (nested segment system ).

Definition 4. They say that

(segment length
tends to zero , provided that
), if.

Definition 5. SVS, which
is called SSS (system of contracting segments).

Axiom of Cantor-Dedekind: In any SHS, there is at least one point that belongs to all of them at once.

Since rational approximations of the number can be represented by a system of contracting segments, then a rational number will correspond to a single point of the numerical axis if there is a single point in the system of contracting segments that belongs to all of them at once ( Cantor's theorem). Let's show this in reverse.

. Let be and two such points, and
,
. T
ak how,
, then
. But on the other side,
, and those. starting from some number
,
will be less than any constant. This contradiction proves what is required. ■

Thus, we have shown that the numerical axis is continuous (has no "holes") and no more numbers can be placed on it. However, we still do not know how to extract roots from any real numbers (in particular, from negative ones) and do not know how to solve equations like
. In Section 5 we will address this problem.

3. 4. The theory of faces

Definition 1. A bunch of
limited from above (from below ) if there is a number , such that
. Number calledtop (bottom ) edge .

Definition 2. A bunch oflimited if it is bounded both above and below.

Definition 3. Accurate top edge upper bounded set of real numbers
called :

(those. - one of the upper faces);

(those. - immovable).

Comment. Great upper bound (TSB) of a number set
denoted
(from lat. supremum- the smallest of the largest).

Comment. The corresponding definition for TNG ( exact bottom edge) give yourself. TNG number set
denoted
(from lat. infinum- the largest of the smallest).

Comment. may belong
, Or maybe not. Number is a TNG of the set of negative real numbers, and a TNG of the set of positive real numbers, but does not belong to either one or the other. Number is the TNG of the set of natural numbers and refers to them.

The question arises: does any bounded set have exact boundaries, and how many are there?

Theorem 1. Any non-empty set of real numbers bounded from above has a unique TVG. (similarly, formulate and prove the theorem for TNG on your own).

Design. A bunch of
non-empty set of real numbers bounded from above. Then
and
. Divide the segment

P
poles and call it a segment
one that has the following properties:

line segment
contains at least one point
. (for example, dot );

the whole set
lies to the left of the point , i.e.
.

Continuing this procedure, we get the CCC
. Thus, by Cantor's theorem, there is a unique point , belonging to all segments at once. Let us show that
.

Let us show that
(those. one of the edges). Assume the opposite that
. As
, then
once
,
, i.e.
, i.e.
. According to the point selection rule
, dot always to the left , i.e.
, therefore, and
. But is chosen so that all
, a
, i.e. and
. This contradiction proves this part of the theorem.

Let's show the immovability , i.e.
. Let's fix
and find a number. According
with rule 1 for choosing segments. We have just shown that
, i.e.
, or
. Thus
, or
. ■

An axis is a straight line on which one of the two possible directions is marked as positive (the opposite direction is considered negative). The positive direction is usually indicated by an arrow. The numerical (or coordinate) axis is the axis on which the starting point (or beginning) O and the scale unit or scale segment OE are selected (Fig. 1).

Thus, the numerical axis is given by indicating the direct direction, origin and scale.

Points on the number line represent real numbers. Integers are represented by points, which are obtained by laying off the scale segment the required number of times to the right of the beginning of O in the case of a positive integer and to the left in the case of a negative one. Zero is represented by the starting point O (the letter O itself is reminiscent of zero; it is the first letter of the word origo, meaning "beginning"). Fractional (rational) numbers are also simply represented by axis points; for example, to construct a point corresponding to the number , three scale segments and one third of the scale segment should be set aside to the left of O (point A in Fig. 1). In addition to point A in Fig. 1 shows more points B, C, D, representing the numbers -2, respectively; 3/2; 4.

There are an infinite number of integers, but on the numerical axis, integers are represented by points located “rarely”, integer points of the axis are separated from neighboring ones by a scale unit. Rational points are located on the axis very "densely" - it is easy to show that on any arbitrarily small section of the axis there are infinitely many points representing rational numbers. However, there are points on the number line that are not images of rational numbers. So, if you construct a segment OA on the real axis, equal to the hypotenuse OS of a right triangle OEC with legs, then the length of this segment (according to the Pythagorean theorem, p. 216) will be equal and point A will not be an image of a rational number.

Historically, it was the fact of the existence of segments whose lengths cannot be expressed by a number (a rational number!), that led to the introduction of irrational numbers.

The introduction of irrational numbers, which together with rational numbers form the set of all real numbers, leads to the fact that each point of the number axis corresponds to a single real number, the image of which it serves. On the contrary, each real number is represented by a well-defined point on the numerical axis. A one-to-one correspondence is established between real numbers and points of the numerical axis.

Since we think of the number axis as a continuous line, and its points are in one-to-one correspondence with the real numbers, we are talking about the continuity property of the set of real numbers (item 6).

We also note that in a certain sense (we do not specify it) there are incomparably more irrational numbers than rational ones.

The number represented by a given point A of the numerical axis is called the coordinate of this point; the fact that a is the coordinate of point A is written as follows: A (a). The coordinate of any point A is expressed as the ratio of OA / OE of the OA segment to the scale segment OE, to which, for points lying from the beginning of O in the negative direction, a minus sign is assigned.

We now introduce rectangular Cartesian coordinates on the plane. Let us take two mutually perpendicular numerical axes Ox and Oy, having a common origin O and equal scale segments (in practice, coordinate axes with different scale units are often used). Let's say that these axes (Fig. 3) form a Cartesian rectangular coordinate system on the plane. The point O is called the origin of coordinates, the Ox and Oy axes are the coordinate axes (the Ox axis is called the abscissa axis, the Oy axis is the ordinate axis). On fig. 3, as usual, the abscissa is horizontal, the y-axis is vertical. The plane on which the coordinate system is given is called the coordinate plane.

Each point of the plane is assigned a pair of numbers - the coordinates of this point relative to the given coordinate system. Namely, we take rectangular projections of the point M on the axes Ox and Oy, the corresponding points on the axes Ox, Oy are indicated in Fig. 3 through

The point has, as a point of the numerical axis, the coordinate (abscissa) x, the point, as a point of the numerical axis, the coordinate (ordinate) y. These two numbers y (written in the indicated order) are called the coordinates of the point M.

At the same time, they write: (x, y).

So, each point of the plane is assigned an ordered pair of real numbers (x, y) - Cartesian rectangular coordinates of this point. The term "ordered pair" indicates that one should distinguish between the first number of the pair - the abscissa and the second - the ordinate. On the contrary, each pair of numbers (x, y) defines a single point M for which x is the abscissa and y is the ordinate. Setting a rectangular Cartesian coordinate system in the plane establishes a one-to-one correspondence between the points of the plane and ordered pairs of real numbers.

The coordinate axes divide the coordinate plane into four parts, four quadrants. The quadrants are numbered as shown in Fig. 3, in Roman numerals.

The signs of a point's coordinates depend on which quadrant it lies in, as shown in the following table:

Points lying on the axis have an ordinate y equal to zero, points on the axis Oy have an abscissa equal to zero. Both coordinates of the origin O are equal to zero: .

Example 1. Construct points on the plane

The solution is given in Fig. 4.

If the coordinates of a certain point are known, then it is easy to indicate the coordinates of points that are symmetrical with it about the axes Ox, Oy and the origin: a point symmetrical with M about the Ox axis will have the coordinates of a point symmetrical with M about the coordinate, finally, at a point symmetrical with M relative to the origin, the coordinates will be (-x, -y).

You can also specify a relationship between the coordinates of a pair of points that are symmetric with respect to the bisector of the coordinate angles (Fig. 5); if one of these points M has coordinates x and y, then the y of the second abscissa is equal to the ordinate of the first point, and the ordinate is the abscissa of the first point.

In other words, the coordinates of the point N, symmetrical with M with respect to the bisector of the coordinate angles, will be To prove this position, consider right-angled triangles O AM and OBN. They are located symmetrically with respect to the bisector of the coordinate angle and therefore are equal. Comparing their respective legs, we will verify the correctness of our statement.

The system of Cartesian rectangular coordinates can be transformed by moving its origin O to a new point O without changing the direction of the axes and the size of the scale segment. On fig. Figure 6 shows two coordinate systems at the same time: the “old” one with the origin O and the “new” one with the origin O. An arbitrary point M now has two pairs of coordinates, one relative to the old coordinate system, the other relative to the new one. If the coordinates of the new beginning in the old system are denoted by , then the relationship between the old coordinates of the point M and its new coordinates (x, y) is expressed by the formulas

These formulas are called coordinate system transfer formulas; when they are displayed in Fig. 6, the most convenient position of the point M, which lies in the first quadrant of both the old and the new systems, is chosen.

It can be seen that formulas (8.1) remain valid for any location of the point M.

The position of the point M on the plane can be specified not only by its Cartesian rectangular coordinates y, but also in other ways. Let us connect, for example, the point M with the origin O (Fig. 7) and consider the following two numbers: the length of the segment and the angle of inclination of this segment to the positive direction of the axis; , if the rotation is counterclockwise, and negative otherwise, as is customary in trigonometry. The segment is called the polar radius of the point M, the angle is the polar angle, a pair of numbers is the polar coordinates of the point M. As you can see, to determine the polar coordinates of the point, you need to specify only one coordinate axis Ox (called in this case the polar axis). It is convenient, however, to consider simultaneously both polar and Cartesian rectangular coordinates, as is done in Fig. 7.

The polar angle of a point is defined ambiguously by specifying a point: if is one of the polar angles of a point, then any angle

will be its polar angle. Specifying the polar radius and angle determines the position of the point in a unique way. The origin O (called the pole of the polar coordinate system) has a radius equal to zero, no definite polar angle is assigned to the point O.

There are the following relationships between the Cartesian and polar coordinates of a point:

directly following from the definition of trigonometric functions (Sec. 97). These relations make it possible to find Cartesian coordinates from given polar coordinates. The following formulas:

allow solving the inverse problem: using the given Cartesian coordinates of a point, find its polar coordinates.

In this case, by the value (or ), you can find two possible values ​​​​of the angle within the first circle; one of them is chosen by the sign coef. You can also determine the angle by its tangent: , but in this case, the quarter in which lies is specified by the sign coef or .

A point given by its polar coordinates is constructed (without calculation of Cartesian coordinates) by its polar angle and radius.

Example 2. Find the Cartesian coordinates of points.

2 EQUATIONS AND INEQUALITIES OF THE FIRST DEGREE
Start studying the topic by solving the repetition problems from Chapter 1

§ 4. INEQUALITIES

Numerical inequalities and their properties

175. Put an inequality sign between numbers a and b if it is known that:
1) (a - b) is a positive number;
2) (a - b) - a negative number;
3) (a - b) is a non-negative number.

176. X, if:
1) X> 0; 2) X < 0; 3) 1 < X; 4) X > -3,2?

177. Write using inequality signs that:
1) X- positive number;
2) at-a negative number;
3) | a| - non-negative number;
4) arithmetic mean of two positive numbers a and b not less than their geometric mean;
5) the absolute value of the sum of two rational numbers a and b not more than the sum of the absolute values ​​of the terms.

178. What can be said about the signs of numbers a and b, if:

1) a b> 0; 2) a / b > 0; 3) a b< 0; 4) a / b < 0?

179. 1) Arrange the following numbers in ascending order, connecting them with an inequality sign: 0; -5; 2. How to read this entry?

2) Arrange the following numbers in descending order, connecting them with an inequality sign: -10; 0.1;-2/3. How to read this entry?

180. Write out in ascending order all three-digit numbers, each of which contains the numbers 2; 0; 5, and connect them with an inequality sign.

181. 1) When measuring a certain length once l found that it is more than 217 cm, but less than 218 cm. Record the result of the measurement, taking these numbers as the boundaries of the length value l.

2) When weighing an object, it turned out that it is heavier than 19.5 G, but lighter than 20.0 G. Write down the weighing result indicating the boundaries.

182. When weighing an object with an accuracy of 0.05 kg, we received the weight
Р ≈ 26.4 kg. Specify the limits of the weight of this item.

183. Where on the number line lies the point representing the number X, if:
1) 3 < X < 10; 2) - 2 < X < 7; 3) - 1 > X > - 6?

184. Find and indicate integer values ​​on the number axis X, satisfying the inequalities.

1) 0,2 < X <4;
2)-3 < X <2;
3) 1 / 2 < X< 5;
4) -1< X<;3.

185. What is a multiple of 9 between 141 and 152? Give an illustration on the number line.

186. Determine which of the two numbers is greater, if it is known that each of them is greater than 103 and less than 115, and the first number is a multiple of 13, and the second is a multiple of 3. Give a geometric illustration.

187. What are the nearest whole numbers between proper fractions? Is it possible to specify two integers between which all improper fractions are enclosed?

188. Bought 6 books on mathematics, physics and history. How many books were bought in each subject if more books were bought in mathematics than in history, and fewer in physics than in history?

189. At the algebra lesson, the knowledge of three students was tested. What grade did each student get if it is known that the first one got more than the second, and the second one more than the third, and the number of points received by each student is more than two?

190. In a chess tournament, chess players A, B, C and D achieved the best results. Is it possible to find out what place each of the participants in the tournament took if it is known that A scored more points than D, and B less than C?

191. Given the inequality a > b. Is it always a c > b c? Give examples.

192. Given the inequality a< b. Is the inequality correct? a > - b?

193. Is it possible, without changing the sign of inequality, to multiply both parts of it by the expression X 2 + 1, where X- any rational number?

194. Multiply both sides of the inequality by the factor given in brackets.

1)-3 < 1 (5); 2) 2 < 5 (-1); 3) X > 2 (X);
4) a < - 1 (a); 5) b < - 3 (-b); 6)X -2 > 1 (X).

195. Bring to the whole form of inequality:

196. Given a function y = kx, where k at with increasing argument X if: 1) k> 0; 2) k < 0? Обосновать ответы.

197. Given a function y = kx + b, where k =/= 0, b=/= 0. How function values ​​change at with decreasing argument values X if: 1) k > 0; 2) k < 0? Обосновать ответы.

198. Prove that if a > b and with> 0, then a / c > b / c; if a > b and with< 0, то a / c < b / c .

199. Divide both sides of the inequality by the numbers in brackets:

1) - 6 < 3 (1 / 3); 2) 4 > -1,5 (-1); 3) a < - 2a 2 (a);
4) a > a 2 (a); 5) a 3 > a 2 (-a).

200. Add term by term inequalities:

1) 12 > 11 and 1 > -3;
2) -5 < 2 и 4 < 8,2;
3) a - 2 < 8 + b and 5 - 2 a < 2 - b;
4) X 2 + 1 > 2X and X - 3 < 9 - X 2 .

201. Prove that each diagonal of a convex quadrilateral is less than its semiperimeter.

202. Prove that the sum of two opposite sides of a convex quadrilateral is less than the sum of its diagonals.

203. Subtract term by term the second inequality from the first:

1)5 > 2; -3 < 1;
2) 0,2 < 3; 0,3 > -2;
3) 7 < 11; -4 < -3;
4) 2a- 1 > 3b; 2b > 3.

204. Prove that if | x |< а , then - a< х < а .

205. Write the following inequalities as double inequalities:
1) | t |< 1; 2) | X - 2 | < 2.

206. Specify on the number axis the set of all values X satisfying the inequalities: 1) | X |< 2; 2) | X | < 1; 3) | X | > 3; 4) | X - 1 | < 1.

207. Prove that if - a< х < а , then | x |< a.

208. Replace double inequalities with an abbreviated notation:
1) -2 < a < 2; 2) -1 < 2P < 1; 3) 1 < x < 3.

209. Approximate Length l= 24.08(±0.01) mm. Set length limits l.

210. Five times measurement of the same distance using a meter ruler gave the following results: 21.56; 21.60; 21.59; 21.55; 21.61 (m). Find the arithmetic mean of the measurement results, indicating the boundaries of the absolute and relative errors.

211. When weighing the cargo, P = 16.7 (± 0.4%) kg was obtained. Find the limits of the weight R.

212. a≈ 16.4, relative error ε = 0.5%. Find Absolute Error
Δ a and set the boundaries between which the approximate number lies.

213. Determine the limit of the relative error of the approximate value of each of the following numbers, if the approximate value is taken with the specified number of correct digits: 1) 11 / 6 with three correct digits; 2) √5 with four correct digits.

214. When measuring the distance between two cities on a map, they found that it is more than 24.4 cm, but less than 24.8 cm. Find the actual distance between cities and the absolute calculation error if the map scale is 1: 2,500,000.

215. Perform calculations and determine the absolute and relative errors of the result: x = a + b - c, if a= 7.22 (±0.01); 3.14< b < 3,17; with= 5.4(±0.05).

216. Multiply the inequalities term by term:

1) 7 > 5 and 3 > 2; 2) 3< 5 и 2 / 3 <2;

3) - 6 < - 2 и - 3 < - 1; 4)a> 2 and b < -2.

217. Given the inequality a > b. Is it always a 2 > b 2? Give examples.

218. If a a > b > 0 and P is a natural number, then up > b. Prove.

219. Which is greater: (0.3) 20 or (0.1) 10 ?

220. If a a > b > 0 or b< а < 0 then 1 / a < 1 / b. Prove.

221. Calculate the area of ​​a rectangular land plot with a length of 437 m and a width of 162 m, if an error of ±2 m is possible when measuring the length of the plot, and an error of ±1 m when measuring the width.