Presentation on the topic of the set of real numbers. Presentation "set of real numbers". Set of irrational numbers

“The set of real numbers” is an interesting and extensive topic from school algebra. Since schoolchildren have already become familiar with the sets of rational and irrational numbers, they can move on to studying real numbers, because they include both the first and second sets.

slides 1-2 (Presentation topic “The set of real numbers”, definition of the set of real numbers)

Like any other set, the set of real numbers has a letter designation - R. This concept covers all infinite and all finite decimal fractions. Thus, the set of all real numbers can be written as an interval from minus infinity to plus infinity, or vice versa, the essence of which does not change. The first slide demonstrates this information.

slides 3-4 (examples)

Further, on the next page of the presentation “The Set of Real Numbers” text information is provided. It talks about what a coordinate line is as a geometric model, and what a number line is. Before giving the definition, the slide contains some preface, that is, text from which you can better understand the essence of the definition. As you can see, the definitions are highlighted yellow, and the concept itself is in red. This will help students better concentrate on this concept and remember it better visually.

Next, next page, contains a geometric notation of the number line, that is, a drawing. Below are basic formulas that will be very useful in transforming or simplifying cumbersome and simple expressions. These include the formula for the difference of squares, the rule of displacement for sums and products, the associative rule, etc. Schoolchildren have already been familiar with some of these rules in previous algebra lessons. It will be useful to remember this material.

The next slide provides a definition in which case the number “a” will be called less (or greater) than some other number. We are talking about real numbers.

slides 7-8 (examples)

Below we demonstrate through comparison signs the cases in which some real number “a” (or expression) is positive or negative.

On the next slide, a certain number “a”, belonging to the set of real numbers, is compared with zero using the “greater than or equal to” or “less than or equal” signs. The inequalities themselves are written on the left, and the conclusions on the right.

Let's move on to the next slide. It is dedicated to practical examples. The first example asks you to compare a fractional number with a positive integer. At first, students can try to cope with the example on their own. Below is the solution.

The second example is to compare the sum of a rational and irrational number of numbers with an integer positive number. As can be seen from the solution, during transformations the irrational number is in the form square root is written through an infinite non-periodic fraction.

The third example is the simplest. After all, it is proposed to compare negative number with positive. And it doesn’t matter at all which sets these numbers belong to. Just look at their signs.

slide 9 (example)

The last slide also includes examples with solutions. If schoolchildren manage to understand practical examples, they will be able to independently cope with similar tasks from homework or independent and control work.

Explanatory note to the resource

“Real numbers and operations on them” (first lesson in 10th grade)

Author - mathematics teacher Bystrykh Valentina Nikolaevna
Educational Institution – Municipal educational institution"Average secondary school No. 8" Krasnovishersk, Perm region.

Item algebra and beginnings of analysis

Class – 10

Topic – “ Real numbers and operations on them" (first lesson in 10th grade)

Educational and methodological support:

Algebra named after the beginning of analysis: A.P. Ivanov “Tests and tests in mathematics", Moscow, MIPT, 2002

Lesson time– 90 minutes

Equipment and materials for the lesson: projector, screen (, presentation to accompany the lesson.

Lesson structure:

A lesson in repetition and generalization of knowledge acquired in basic school.

1. Phys. Just a minute
2. Determining the topic of the lesson
3. Formulation of goals and objectives by students using keywords
4. Lecture
5. Practical tasks
6. Lesson summary

1. Wednesday - Microsoft Office PowerPoint 2007
2. Type of media product - visual presentation of what is being studied educational material, which can be used by the teacher in the classroom and for self-study material by students.
3. Target groups – teachers, students.
4. Function in educational process – educational, illustrative and training.

The presentation consists of 15 slides.

• Slides can be changed by clicking the mouse.
• The first slide has audio.
• A hyperlink is inserted on slides 9 – 14.
• All other slides are animated at the click of a mouse.

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Preview:

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Slide captions:

Real numbers and transformations of algebraic expressions

Purpose of the lesson: Repeat, Distinguish, Develop, Evaluate

At home: theory (10) (3)

Natural numbers(N) – a unit or a collection of several units (1; 2;…9 – a series of natural numbers) Integers (Z) – natural numbers, the opposite of natural numbers and zero Rational numbers(Q) - integers, positive and negative fractions Real numbers (R) - rational and irrational numbers Irrational numbers (||) - infinite non-periodic fractions

Natural numbers (N) Primes - divisible by themselves and by one Even - divisible by 2 and the number 0. (2p) Odd - the rest (2p+1; 2p-1). Signs of divisibility: By 2 - By 3 - By 5 - By 9 - By 10 - Any composite number can be decomposed into prime factors Assignment: factor numbers into prime factors; 1260; 248; 4725 Find the LCM and GCD of the numbers (54; 72;) ;(96; 124)(125; 325); (34; 68) Compounds – the rest.

Rational numbers (Q) A fraction (part) of a unit or a collection of several identical fractions of a unit is called an ordinary fraction. A fraction whose denominator is one with one or more zeros is called a decimal fraction 2/3 = 0.666... ​​- infinite periodic fraction, 0.666... ​​= 0 ,(6) 0,(68) – pure periodic fraction 1, 4(35) – mixed periodic fraction

The rule for converting a mixed periodic fraction into an ordinary fraction To convert a pure periodic fraction into an ordinary fraction, you need to make its period the numerator, and write the number 9 in the denominator as many times as there are digits in the period. To convert a mixed periodic fraction into an ordinary fraction, you need to subtract the number after the decimal point before the second period from the number after the decimal point before the first period, and make this difference the numerator, and write the number 9 in the denominator as many times as there are digits in the period, with with as many zeros on the right as there are digits between the decimal point and the first period.

1 2 4 3 9 10 11 12 13 14

5 6 7 9 10 11 12 13 14

11 10 9 8 9 10 11 12 13 14

9 10 11 12 13 14

9 10 11 12 13 14

15 14 13 12 9 10 11 12 13 14

Goal: Systematize knowledge about natural, integer, rational numbers, periodic fractions. Learn to write an infinite decimal fraction in the form of an ordinary fraction, develop the skill of performing operations with decimals and ordinary fractions. Have an understanding of irrational numbers, the set of real numbers. Have an understanding of irrational numbers, the set of real numbers. Learn to perform calculations with irrational expressions, compare numeric values irrational expressions.

Numbers don't rule the world, but they show how to rule it. Numbers don't rule the world, but they show how to rule it. I. Goethe. I. Goethe. Numbers don't rule the world, but they show how to rule it. Numbers don't rule the world, but they show how to rule it. I. Goethe. I. Goethe. natural. N Naturalis Numbers called naturals are used to count objects. To denote the set of natural numbers, the letter N is used - the first letter of the Latin word Naturalis, “natural”, “natural”. What numbers are called natural? How is the set of natural numbers denoted?

Rational numbers QQuotient The set of numbers that can be represented in the form is called the set of rational numbers and is denoted by Q as the first letter French word Quotient - “attitude”. integers Zahl Natural numbers, their opposites and the number zero form a set of integers, which is denoted by Z - the first letter German word Zahl - "number". What numbers are called integers? How is the set of integers denoted? What numbers are called rational? How is the set of rational numbers denoted?

Natural numbers Numbers, their opposites Integers 0

Sum, product, difference The sum, product, difference and quotient of rational numbers is a rational number. Sum, product, difference The sum, product, difference and quotient of rational numbers is a rational number. Rational numbers rrational r - rational

Find the period in the notation of numbers and write down each number briefly: 0.55555....4.133333...3, ...7, ....3, ...3.727272...21, ...

0, Let x = 0.4666... ​​10 x = 4.666... ​​10 x = 4.666... ​​100 x = 46.666... ​​100 x – 10 x = 46.666...- 4, x = 42

Slide 2

In the first stages of existence human society numbers discovered in the process practical activities, served for primitive counting of objects, days, steps. In primitive society, a person needed only the first few numbers. But with the development of civilization, he needed to invent larger numbers. This process continued over many centuries and required intense intellectual work.

Slide 3

Hypothesis:

There is no need to study real numbers in detail.

Slide 4

Purpose: to trace the process of the emergence of real numbers and their further study.

Research objectives: To trace the process of the emergence of real numbers; Study the development of the theory of real numbers; Find out why you need to study real numbers;

Slide 5

Relevance of the selected topic

The concept of number originated in ancient times. Over the centuries, this concept has been expanded and generalized.

Slide 6

Progress of the study:

Studied various sources of information; I traced the process of the emergence of real numbers; After analyzing the work done, I came to a conclusion.

Slide 7

Research results:

At the first stage, the concepts of “more,” “less,” or “equal” arose. Probably, at the same stage of development, people began to add numbers. Much later they learned to subtract numbers, then multiply and divide them. Even in the Middle Ages, dividing numbers was considered very complex and served as a sign of an extremely highly educated person.

Slide 8

With the discovery of operations with numbers or operations on them, the science of ARITHMETICS arose. After some time, Pythagoras discovered immeasurable segments, the lengths of which could not be expressed either by a whole or fractional number. Subsequently, the concept of “geometric expression” arises. Thanks to the first discoveries, mathematicians in India, the Near and Middle East, and later Europe used irrational quantities. However, for a long time they were not recognized as equal numbers. Their recognition was facilitated by the appearance of Descartes' Geometry.

Slide 9

Later it became known that any number can be represented as an infinite decimal fraction. In the 18th century L. Euler and I. Lambert showed that every infinite periodic decimal fraction is a rational number. Constructing real numbers from infinite numbers decimals was given by the German mathematician K. Weirstrass.