Divide a circle into any number of equal parts. How to read a ruler

The theory of algebraic and transcendental numbers allowed mathematicians to solve three famous geometric problems that had remained unsolved since antiquity. We are referring to the “doubling the cube” problem, the “trisection of an angle” problem and the “squaring the circle” problem. These tasks relate to constructions using a compass and ruler and are as follows:

1) “Doubling the cube.” It is required to build a cube that has twice the volume compared to the given cube. Although the cube is a spatial figure, the problem is essentially planimetric. In fact, if we take the edge of a given cube as a unit of length (Fig. 16), then the task will be to construct a segment of length 1/2, since this will be the length of the edge of a cube that has twice the volume compared to the given one.

2) “Trisection of the angle.” Find a way by which, using only a compass and a ruler, you can divide any angle into three equal parts. There are some angles, such as 90° or 45°, that can be divided into three equal parts using a compass and ruler, but the so-called “common” angle cannot be divided into three equal parts using these tools.

3) “Squaring the circle.” Construct a square equal in area to a given circle, or, which is equivalent, construct a circle equal in area to a given square.

It is known that these three constructions are infeasible, that is, they cannot be performed using only a compass and a ruler. Many hobbyists continue to solve these problems without knowing that their efforts are wasted.

Although such amateurs are aware that no mathematician has yet been able to carry out these constructions, they are apparently unaware of the strictly proven impossibility of such constructions. From time to time amateur mathematicians find an approximate solution to one of these problems, but never, of course, find their exact solutions. It is clear what the difference is here: the problem of doubling a cube, for example, consists in constructing, using theoretically perfect drawing tools, a segment that would have a length not approximately but exactly equal to this number. The problem cannot be solved by constructing, for example, a segment of length, despite the fact that the numbers coincide to within six decimal places.

In the case of the angle trisection problem, there is a special source of misunderstanding.

Any angle can be divided into three equal parts if you use a ruler with divisions. Thus, the statement about the impossibility of dividing a common angle into three equal parts can be made only when it is assumed that the acceptable tools for construction are a compass and a ruler without divisions.

Since there is a lot of confusion regarding these three classical problems, we will now quickly explain how one can prove the impossibility of all three constructions. We cannot give complete proofs here, since the details are quite specialized. If the reader wants to get acquainted with them in detail, then he can refer to the book by R. Courant and G. Robbins, which contains a complete analysis of the problems of trisection of an angle and doubling of a cube (pp. 197-205). The proof of the impossibility of squaring a circle is much more complicated than the proof of the impossibility of the other two constructions.

How can we prove the impossibility of the constructions we are interested in? The first thing you need to understand to some extent is what length of segments can be constructed using a compass and ruler, if a segment of unit length is given. Without giving proof, we assert (and everyone familiar with geometric constructions will agree with us) that among the lengths that can be constructed are all the lengths obtained by successive extraction of square roots applied to rational numbers, for example.

All numbers obtained in this way are algebraic.

The four numbers (10), written out as an example, are respectively the roots of the following equations:

(11)

Let's take one of the equations, say (13), and check that the number

really is its root. Squaring both sides of the last equality, we get

Moving term 5 to the left and squaring it again, we find

Now squaring both sides again leads to equation (13).

Further, in addition to the fact that numbers (10) are respectively the roots of equations (11) - (14), none of these numbers are the roots of an equation with integer coefficients of a lower degree. Let's take, for example, the number . It satisfies equation (12) of degree 4, but does not satisfy any equation of degree 3, 2 or 1 with integer coefficients. (We do not prove this statement.) If an algebraic number is the root of an equation of degree with integer coefficients, but is not the root of any equation of lesser degree with integer coefficients, then it is called an algebraic number of degree. Thus, the numbers (10) are algebraic numbers of powers 2, 4, 8 and 16, respectively.

The above suggests the following main result about the lengths of segments that can be constructed using a compass and ruler:

Theorem on geometric constructions. The length of any segment that can be constructed starting from a given segment of unit length using a compass and ruler is an algebraic number of degree either 1, or 2, or 4, or 8,..., i.e., generally speaking, degrees , where is a non-negative integer.

We invite the reader to take this result on faith and, based on it, we will show that all three famous constructions are impossible.

Let's start with the doubling cube problem. As we saw above when formulating it, it is equivalent to the following: starting from a segment of unit length, construct a segment of length . But does the number satisfy the necessary conditions for this? It satisfies the equation

and this suggests that n is an algebraic number of degree 3. In fact, this is exactly the case, and to be convinced of this, you only need to show that the number does not satisfy any equation with integer coefficients of degree 1 or 2. Proof of this although not difficult, it requires some trickery and we will leave it until the next paragraph.

Since there is an algebraic number of degree 3, then, by virtue of the theorem formulated above on geometric constructions, it is impossible to construct a segment of length , based on a segment of unit length. Thus, it is impossible to double the cube.

Let us now consider the problem of trisection of an angle. To establish the impossibility of trisection in the general case, it is enough to show that a certain fixed angle cannot be divided into three identical parts by a compass and a ruler. Let's take an angle of 60°. Trisection of an angle of 60° means constructing an angle of 20°. This comes down to constructing, based on a given segment of unit length, a segment having length . To verify this, consider a triangle with a base of length 1 and with angles at the base of 60° and 90°, i.e. triangle ABC with a base and angles BAC - 60° and (Fig. 17). On side BC, take point D so that angle BAD is 20°. From elementary trigonometry we know that

Thus, trisection of an angle of 60° is reduced to constructing a segment of length . But this, in turn, comes down to constructing a segment of length , since they are numbers that are inverse to each other, and it is well known that if you can construct a segment of a certain given length, then you can also construct a segment of the inverse length.

AB = 6 cm. = 60 mm. IIIIIIIII IIIIIIIII IIIIIIIII IIIIIIIII IIIIIIIII IIIIIIIII IIIIIIIII IIIIIIIII IIIIIIIII IIIIIIIII III. The lengths of the segments are measured with a ruler. There are strokes on the ruler. They break the ruler into equal parts. These parts are called divisions. All divisions of the ruler form a scale. The division value is 1 cm. Mm.

Slide 5 from the presentation “Scales and coordinates grade 5”. The size of the archive with the presentation is 482 KB.

Mathematics 5th grade

summary of other presentations

“Math quiz with answers” ​​- Subtotals. Who calculates better? Team awards. The numbers are in order. Team presentation. Math quiz. Jury. It's time to rest. Look at the picture. Quatrain. Rebus. Who will write the required numbers in the squares faster? Crossword. Decipher the mathematical terms. Repetition of educational material. Anagrams.

“Constructing angles” - Vertex. Sharp corner. Measuring angles. ?Aov, ?voa, ?o. Construct an acute angle. Construct an angle of 78°. Swap notebooks with your desk neighbor. Construction and measurement of angles. Unfolded angle. Protractor. Check each other's work. Construction of angles. Side. Work in pairs. Obtuse angle. Degree. Do the same task, constructing angles of 145o and 90o. Ask your seatmate to check your formation. Do the same task by constructing an obtuse angle.

“Arithmetic mean” - Checking tasks on cards. The arithmetic mean of four numbers. Sum of numbers. Find the arithmetic mean. Task. Verbal counting. Using the answers you found and the data in the table, fill in the blanks. Average. Sum of eight numbers. Individual work. Let the smaller number be x, then the larger number is 3.2x. Intelligence challenge.

“Mathematics “Mixed Numbers”” - One point two thirds. Mixed number. Separate the whole part from an improper fraction. Numerator of the fractional part. Mathematical dictation. Adding and subtracting ordinary fractions. In class. Denominator of the fractional part. A number consisting of an integer part and a fractional part is called a mixed number. Divide each apple into three equal parts. Express a mixed number as an improper fraction. Mixed numbers.

“Laws of addition and subtraction” - Laws of subtraction. Integers. Subtracting zero does not change the number. Add all natural numbers. Commutative (commutative) property. Combinative (associative) property. Laws of addition and subtraction. Letter entry. Law of absorption of zero. The property of subtracting a sum from a number. Zero. Find the meaning of the expression. Examples of application of laws.

“Writing natural numbers” - The number 1 is not the smallest natural number. Notation of natural numbers. Compare the numbers. What numbers represent the entries? What categories do you know? Formulation of the problem. Arabic numerals. Notation of numbers using Roman numerals. Calculate. Graphic dictation. Answer the questions. A rebus is a riddle in which the searched word is represented in letters. 0 is not a natural number. Lesson objectives. How big is a million?

A circle is a closed curved line, each point of which is located at the same distance from one point O, called the center.

Straight lines connecting any point on a circle to its center are called radii R.

The straight line AB connecting two points of a circle and passing through its center O is called diameter D.

The parts of circles are called arcs.

The straight line CD connecting two points on a circle is called chord.

A straight line MN that has only one common point with a circle is called tangent.

The part of the circle bounded by the chord CD and the arc is called segment.

The part of a circle bounded by two radii and an arc is called sector.

Two mutually perpendicular horizontal and vertical lines intersecting at the center of a circle are called axes of the circle.

The angle formed by two radii KOA is called central angle.

Two mutually perpendicular radius make an angle of 90 0 and limit 1/4 of the circle.

We draw a circle with horizontal and vertical axes, which divide it into 4 equal parts. Drawing with a compass or square at 45 0, two mutually perpendicular lines divide the circle into 8 equal parts.

Dividing a circle into 3 and 6 equal parts (multiples of 3 to three)

To divide a circle into 3, 6 and a multiple of them, draw a circle of a given radius and the corresponding axes. Division can begin from the point of intersection of the horizontal or vertical axis with the circle. The specified radius of the circle is plotted 6 times successively. Then the resulting points on the circle are sequentially connected by straight lines and form a regular inscribed hexagon. Connecting points through one gives an equilateral triangle, and dividing the circle into three equal parts.

The construction of a regular pentagon is carried out as follows. We draw two mutually perpendicular circle axis equal to the diameter of the circle. Divide the right half of the horizontal diameter in half using arc R1. From the resulting point “a” in the middle of this segment with radius R2, draw a circular arc until it intersects with the horizontal diameter at point “b”. With radius R3, from point “1”, draw a circular arc until it intersects with a given circle (point 5) and obtain the side of a regular pentagon. The distance "b-O" gives the side of a regular decagon.

Dividing a circle into N number of identical parts (constructing a regular polygon with N sides)

This is done as follows. We draw horizontal and vertical mutually perpendicular axis of the circle. From the top point “1” of the circle, draw a straight line at an arbitrary angle to the vertical axis. We lay out equal segments of arbitrary length on it, the number of which is equal to the number of parts into which we divide the given circle, for example 9. We connect the end of the last segment to the lower point of the vertical diameter. We draw lines parallel to the resulting one from the ends of the set aside segments until they intersect with the vertical diameter, thus dividing the vertical diameter of a given circle into a given number of parts. With a radius equal to the diameter of the circle, from the bottom point of the vertical axis we draw an arc MN until it intersects with the continuation of the horizontal axis of the circle. From points M and N we draw rays through even (or odd) division points of the vertical diameter until they intersect with the circle. The resulting segments of the circle will be the required ones, because points 1, 2, …. 9 divide the circle into 9 (N) equal parts.

I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I ? AB = 3 cm 8 mm Write down the length of the segment. AB = 38 mm

1 One division corresponds to 1 hour. In addition, the watch dial is divided into 60 small divisions. One small division corresponds to 1 minute. In some instruments, the scales are located on circles or arcs of circles. On the watch dial, the entire circumference is divided into 12 large divisions.

The figure shows a scale of a device indicating how many liters of gasoline are left in the car's tank. How many liters of gasoline are in the tank now? l b) will 30 l be consumed when moving? How many divisions and in which direction will the arrow of the device move if: a) another 20 liters of gasoline are poured into the gas tank;

Pick up a weight to find out the weight of a melon. CHECK 1kg 100g 1kg 3kg 3kg 2kg

CHECK 3kg 50g Pick up the weight to find out the weight of the watermelon. 2kg 1kg 3kg 3kg

CHECK 5kg 450g Pick up the weights to find out the weight of the pumpkins. 3kg 3kg 1kg 2kg 2kg

CHECK 20 kg 800g 20kg Pick up the weight to find out the weight of the snowman. 5kg 2kg

I IIII I IIII I IIII I IIII I The figure shows a scale. What numbers correspond to points A, B, C and D on this scale? 30 CBD

On the time scale, divisions represent one century. Show on the scale: a) a) the beginning and end of the second century; I I I I I I I I I I I I I I I I I I I II II III VI V VII VI VIII IX X XI XII XIII XIV XV XVII XVI XVIII XIX XX b) b) the end of the sixth century; c) c) seventh century; d) d) mid-twelfth century; e) e) the first half of the seventeenth century. a c b d e

They write: O(0), E(1), A(2), B(3), etc. Step by step we get an infinite scale. coordinate ray It is called a coordinate ray. coordinates The numbers 0, 1, 2, 3, ..., corresponding to points O, E, A, B ... are called the coordinates of these points. Let's draw the ray OX so that it goes from left to right. Let's mark some point E on this ray with a unit segment. Above the beginning of the ray we write the number 0, and above the point E - the number 1. The segment OE is called a unit segment. 01E OX 2A3B456