Real numbers, image on the number axis. Real numbers, image on the number axis Properties of absolute quantities

We already know that the set of real numbers $R$ consists of rational and irrational numbers.

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

Irrational numbers are written as infinite but non-periodic decimal fractions.

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

Let's look at ways to represent real numbers.

Common fractions

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

Definition

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

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

1. of two fractions with the same denominators, the one with the larger numerator is greater, that is, $\frac(m)(p) >\frac(n)(p) $ for $m>n$;
2. of two fractions with the same numerators, the one with the smaller denominator is greater, that is, $\frac(p)(m) >\frac(p)(n) $ for $ m
3. a proper fraction is always less than one; an improper fraction is always greater than one; a fraction in which the numerator is equal to the denominator is equal to one;
4. Every improper fraction is greater than every 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 a common fraction can be obtained by dividing the numerator by the denominator with the “angle”. This can result in either a finite decimal fraction or an infinite periodic decimal fraction.

Definition

The digits of the fractional part are called decimals. 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

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 indicate 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 a given digit is less than 5, then the digit of this digit is not changed;
3. if the first digit following a 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 thousands: 17000.
2. Let's round the number 17378 to hundreds: 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.

Convert a decimal number to a fraction.

Case 1

A decimal number represents a terminating decimal fraction.

The following example demonstrates the conversion method.

Example 2

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

We reduce it to a common denominator and get:

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

Case 2

A decimal represents an infinite periodic decimal fraction.

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

Example 4

$0,\left(74\right)=\frac(74)(100) +\frac(74)(10000) +\frac(74)(1000000) +\ldots $. The first term 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 term 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 term 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) $. Thus, $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 term 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) $.

Thus, $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 axis.

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

There is a one-to-one correspondence between all real numbers and all points on the number axis: each point corresponds to a single number and, conversely, each number corresponds to a single point. Consequently, the set of real numbers is continuous and infinite, just as the number line is continuous and infinite.

Some subsets of the set of real numbers are called numerical intervals. The elements of a numerical interval are numbers $x\in R$ that satisfy a certain inequality. Let $a\in R$, $b\in R$ and $a\le b$. In this case, the types of intervals 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 $\left$. Moreover $ a \le x
4. Infinite intervals, for example $a

A type of interval called a neighborhood of a point is also important. The neighborhood of a given point $x_(0) \in R$ is an arbitrary interval $\left(a,\; b\right)$ containing this point inside itself, that is, $a 0$ is its radius.

Absolute value of a number

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

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

Properties of absolute values:

1. from the definition it follows 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 following inequalities are valid: $\left|x+y\right|\le \left|x\right|+\left|y\right|$, $\left|x-y\right|\le \left|x\right|+\left|y\right|$, as well as $\left|x+y\right|\ge \left|x\right|-\left|y\right|$,$\ left|x-y\right|\ge \left|x\right|-\left|y\right|$;
3. for the modulus of the product and the modulus of the quotient of two numbers, the following equalities are true: $\left|x\cdot y\right|=\left|x\right|\cdot \left|y\right|$ 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 $a>0$, we 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. Number axis is called a straight line with the origin, scale and direction selected on it.

Theorem 1. There is a one-to-one correspondence (bijection) between points on the number line and real numbers.

Necessity. Let us show that each point on the number line corresponds to a real number. To do this, let us set aside a scale segment of unit length

if so, that's the point will lie to the left of the point , and point
already more to the right. Next segment
divide by
parts and set aside the segment and if so, that's the point will lie to the left of the point , and point
already more to the right. Thus, at each stage the number
,
... If this procedure ends at some point, we will get the number
(point coordinate on the number axis). If not, then let’s call the left boundary of any interval a “number” with a disadvantage”, and the right one – “with a number with excess", or "approximation of the number with 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 determined unambiguously.

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

Definition 2. If
, then the numerical interval
calledsegment , If
, then the numerical interval calledinterval , If
, then the numerical interval
calledhalf-interval .

ABOUT
definition 3. If in a segment
segments are nested so that
, A
, then such a system is called SHS (system of nested segments ).

Definition 4. They say that

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

Definition 5. SBC, which has
called CSS (contracting segment system).

Cantor-Dedekind axiom: 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 the rational number will correspond to a single point on the numerical axis if in the system of contracting segments there is a single point that belongs to all of them at once ( Cantor's theorem). Let's show this by contradiction.

. Let And two such points, and
,
. T
how, how,
, That
. But in other way,
, and those. starting from some number
,
will be less than any constant. This contradiction proves what is required.

Thus, we have shown that the number 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 paragraph 5 we will solve this problem.

3. 4. Theory of edges

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. Exact upper edge bounded above set of real numbers
called :

(those. – one of the upper faces);

(those. – non-movable).

Comment. Exact supremum bound (SUB) of a number set
denoted by
(from lat. supremum- the smallest of the big).

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

Comment. may belong
, Or maybe not. Number is a TVG of the set of negative real numbers, and a TVG of the set of positive real numbers, but does not belong to either one or the other. Number is a 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 above has a unique TVG. (similarly, formulate and prove the theorem for TNG yourself).

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

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

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

all the multitude
lies to the left of the point , i.e.
.

Continuing this procedure, we obtain SSS
. Thus, according to Cantor’s theorem, there is a unique point , belonging to all segments at once. Let's show that
.

Let's show that
(those. - one of the faces). Let's assume the opposite, that
. Because
, That
as soon as
,
, i.e.
, i.e.
. According to the point selection rule
, dot always to the left , i.e.
, therefore, and
. But is chosen so that everything
, A
, i.e. And
. This contradiction proves this part of the theorem.

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

2 EQUATIONS AND INEQUALITIES OF THE FIRST DEGREE
Start studying the topic by solving 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) - 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| - the number is non-negative;
4) the 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 no more than the sum of the absolute values ​​of the terms.

178. What can you say 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 down in ascending order all three-digit numbers, each of which contains the digits 2; 0; 5, and connect them with an inequality sign.

181. 1) For a single measurement of a certain length l found that it is more than 217 cm, but less than 218 cm. Write down the measurement result, taking these numbers as boundaries for the length value l.

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

182. When weighing a certain object with an accuracy of 0.05 kg, the weight was obtained
P ≈ 26.4 kg. Indicate the weight limits for this item.

183. Where on the number axis 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 multiple of 9 lies between 141 and 152? Give an illustration of a number line.

186. Determine which of 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 that contain proper fractions? Is it possible to specify two integers with all the improper fractions between them?

188. Purchased 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 less in physics than in history?

189. During an algebra lesson, the knowledge of three students was tested. What grade did each student receive if it is known that the first received a score higher than the second, and the second received a higher score than the third, and the number of points received by each student is more than two?

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

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

192. Given inequality A< b. Is the inequality true? A > - b?

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

194. Multiply both sides of the inequality by the factor indicated 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. Lead to a whole type 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 the 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 indicated 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 the inequalities term by term:

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 half-perimeter.

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

203. Subtract the second inequality term by term 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 , That - A< х < а .

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

206. Indicate on a 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 shorthand 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. Measuring the same distance five times 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 limits of absolute and relative errors.

211. When weighing the load, 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 the absolute error
Δ a and establish 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 using a map, they found that it was more than 24.4 cm, but less than 24.8 cm. Find the actual distance between the 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 inequality A > b. Is it always A 2 > b 2? Give examples.

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

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

An axis is a straight line on which one of two possible directions is selected 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 origin) O and the scale unit or scale segment OE are selected (Fig. 1).

Thus, the number axis is specified by indicating the direction, origin and scale on the line.

Real numbers are represented using points on the number axis. Integers are represented by points, which are obtained by putting the scale segment the required number of times to the right from the beginning 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 another third part 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 also shows points B, C, D, respectively representing the numbers -2; 3/2; 4.

There is an infinite number of integers, but on the number axis integers are depicted by points located “sparsely”; the integer points of the axis are spaced from their neighbors by one scale unit. Rational points are located very “densely” on the axis - it is not difficult to show that on any however small portion 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 on the number axis we construct a segment OA equal to the hypotenuse OS of the right triangle OEC with legs, then the length of this segment (according to the Pythagorean theorem, paragraph 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 in numbers (rational numbers!) 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 on 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 very specific point on the number axis. A one-to-one correspondence is established between real numbers and points on the number axis.

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

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

The number whose image is this 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 OA/OE of the segment OA to the scale segment OE, which is assigned a minus sign for points lying from the origin O in the negative direction.

Let us now introduce rectangular Cartesian coordinates on the plane. Let's 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. Point O is called the origin of coordinates, the Ox and Oy axes are called the coordinate axes (the Ox axis is called the abscissa axis, the Oy axis is the ordinate axis). In Fig. 3, as usual, the abscissa axis is horizontal, the ordinate axis is vertical. The plane on which the coordinate system is specified is called the coordinate plane.

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

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

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

So, each point on the plane is associated with an ordered pair of real numbers (x, y) - the 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 serves as the abscissa and y as the ordinate. Defining a rectangular Cartesian coordinate system in a plane establishes a one-to-one correspondence between points on 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 a y ordinate equal to zero, points on the Oy axis have an abscissa equal to zero. Both coordinates of the origin O are equal to zero: .

Example 1. Construct points on a 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 symmetrical with it relative to the Ox, Oy axes and the origin of coordinates: a point symmetrical with M relative to the Ox axis will have the coordinates of a point symmetrical with M relative to the coordinate, and finally, at a point symmetrical with M relative to the origin, the coordinates will be (-x, -y).

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

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

The Cartesian rectangular coordinate system 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. In Fig. Figure 6 shows two coordinate systems simultaneously: 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 origin in the old system are denoted by , then the connection between the old coordinates of point M and its new coordinates (x, y) will be expressed by the formulas

These formulas are called coordinate system transfer formulas; when drawing them according to Fig. 6, the most convenient position of point M was selected, lying in the first quadrant of both the old and new systems.

You can make sure that formulas (8.1) remain true for any location of the point M.

The position of 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, point M with the beginning of 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. The angle is defined as the angle by which the axis must be rotated before it aligns with OM, and is considered positive , if the rotation is made 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, the pair of numbers is the polar coordinates of the point M. As you can see, to determine the polar coordinates of a point, you only need to specify one coordinate axis Ox (called in this case the polar axis). It is convenient, however, to consider both polar and Cartesian rectangular coordinates simultaneously, as is done in Fig. 7.

The polar angle of a point is determined by specifying the point ambiguously: if is one of the polar angles of the point, then every 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 specific 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 (clause 97). These relations allow you to find Cartesian coordinates from given polar ones. The following formulas:

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

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

A point specified by its polar coordinates is constructed (without calculating Cartesian coordinates) according to its polar angle and radius.

Example 2. Find the Cartesian coordinates of the points.