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At the first stages of the existence of human society, the numbers discovered in the process of practical activity served for the 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 large numbers. This process continued for many centuries and required intense intellectual work.
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Hypothesis:
There is no need to study real numbers in detail.
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Purpose: to trace the process of the appearance of real numbers and their further study.
Research objectives: To trace the process of the appearance of real numbers; To study the development of the theory of real numbers; Find out why you need to study real numbers;
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Relevance of the chosen topic
The concept of number originated in ancient times. Over the centuries, this concept has been expanded and generalized.
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Research progress:
Studied various sources of information; She traced the process of the appearance of real numbers; After analyzing the work done, I came to the conclusion.
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Research results:
At the first stage, the concepts of “greater than”, “less than” or “equal to” 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, the division of numbers was considered very difficult and served as a sign of an extremely high education of a person.
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With the discovery of actions with numbers or operations on them, the science of arithmetic arose. After some time, Pythagoras discovered immeasurable segments, the lengths of which could not be expressed either by an integer or a fractional number. In the future, the concept of "geometric expression" arises. Thanks to the first discoveries, the mathematicians of 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 "Geometry" by Descartes.
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After 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 any infinite periodic decimal fraction is a rational number. The construction of real numbers based on infinite decimal fractions was given by the German mathematician K. Weirstrass.
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ALGEBRA and the beginning of analysis Grade 10 Sh.A. Alimov, Yu.M. Kolyagin et al. 15th ed. M.: Enlightenment, 2007 Mathematics teacher Pivovarenok N.N. GOU School No. 247 Chapter I. Real numbers Lesson 2 "Algebra is nothing but a mathematical language adapted to denote relationships between quantities." I. Newton
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have concepts about: irrational numbers; set of real numbers; modulus of a real number; be able to perform: calculations with irrational expressions; compare numerical values of irrational expressions §2 Real numbers Knowledge and skills of students:
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1. The need for further expansion of the set of numbers is mainly due to two reasons: an irrational number is an infinite decimal non-periodic fraction 1) Rational numbers are not enough to express the results of measurements (the length of the diagonal of a square with a side of 1) 2) Such numerical expressions are not rational numbers
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A real number is an infinite decimal fraction, i.e. a fraction of the form + a0, a1a2a3 ... or - a0, a1a2a3 ..., where a0 is a non-negative integer, and each of the letters a1, a2, a3, ... is one of ten digits: 0,1,2,3,4,5, 6,7,8,9 1) π = 3.1415… a0 = 3 a1=1 a2= 4 a3=1 a4=5… 2)- √234 = - 15.297058… a0 = 15 a1=2 a2= 9 a3=7 a4=0 ... 3)37,19 a0 = 37 a1=1 a2= 9 an=0 for n≥3 The union of the set of rational numbers and the set of irrational numbers (infinite decimal non-periodic fractions) gives the set R of real numbers For example: A real number can be positive, negative, or zero.
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2. Arithmetic operations on real numbers are usually replaced by operations on their approximations. accurate to one: accurate to tenth: accurate to hundredth: Compute the sum of Number 3; 3.1; 3.15 etc. are successive approximations of the value of the sum
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3. All basic operations on rational numbers are preserved for real numbers as well. 4. The modulus of a real number x is denoted by |x| and is defined in the same way as the modulus of a rational number:
Purpose: To systematize knowledge about natural, integer, rational numbers, periodic fractions. To learn how to write an infinite decimal fraction in the form of an ordinary fraction, to form the skill of performing actions with decimal and ordinary fractions. Have a concept of irrational numbers, the set of real numbers. Have a concept of irrational numbers, the set of real numbers. Learn to perform calculations with irrational expressions, compare the numerical values of irrational expressions.
Numbers do not rule the world, but they show how to rule it. Numbers do not rule the world, but they show how to rule it. I. Goethe. I. Goethe. Numbers do not rule the world, but they show how to rule it. Numbers do not rule the world, but they show how to rule it. I. Goethe. I. Goethe. natural. N Naturalis To count objects, numbers are used, which are called naturals. 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? What is the set of natural numbers?
Rational numbers QQuotient The set of numbers that can be represented in the form is called the set of rational numbers and is denoted - Q by the first letter of the French word Quotient - "relationship". integers Zahl The natural numbers, their opposites and the number zero form a set of integers, which is denoted by Z - the first letter of the German word Zahl - "number". What numbers are called integers? What is the set of integers? What numbers are called rational? What is the set of rational numbers?
Natural numbers Numbers opposite to them 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 -rational r - rational
Find a period in the record of numbers and write down each number briefly: 0.55555….4.133333…3, …7, ….3, …3.727272…21, …
0, Let x \u003d 0.4666 ... 10 x \u003d 4.666 ... 10 x \u003d 4.666 ... 100 x \u003d 46.666 ... 100 x - 10 x \u003d 46.666 ... - 4, x \u003d 42
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The presentation on the topic "Real numbers" (grade 8) can be downloaded absolutely free of charge on our website. Project subject: Mathematics. Colorful slides and illustrations will help you keep your classmates or audience interested. To view the content, use the player, or if you want to download the report, click on the appropriate text under the player. The presentation contains 11 slide(s).
Presentation slides
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Prepared by 8th grade student Anastasia Karpova.
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Stages of development of the concept of number.
The geometric representation of numbers as segments leads to an extension of the set Q to the set of real (or real) numbers R: N ⊂ Z ⊂ Q ⊂ R.
Rational numbers can be used to solve equations of the form nx = m, n ≠ 0, where m and n are integers.
The root of any equation is ax + b = c, where a, b, c are rational numbers, a ≠ 0, is a rational number.
Rational numbers can be written as fractions of the form, where m is an integer, n is a natural number.
The set of rational numbers is denoted by Q; N ⊂ Z ⊂ Q.
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Chapter 6, Conversation 7
The natural numbers are part of the integers: N ⊂ Z.
Natural numbers: 1, 2, 3, ...
The set of all integers is denoted by Z.
Negative integers: -1, -2, -3, ...
Negative integers arise when solving equations of the form x + m = n, where m and n are natural numbers.
The set of natural numbers is usually denoted N.
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More about real numbers:
The real numbers include the numbers of rational and irrational sets.
Real numbers can be added, subtracted, multiplied, divided, and compared in magnitude. We list the main properties that these operations have. The set of all real numbers will be denoted by R, and its subsets will be called numerical sets.
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I. Operation of addition. For any pair of real numbers a and b, a single number is defined, called their sum and denoted by a + b, so that the following conditions are satisfied: 1. a + b = b + a, a,b∈ R. 2. a + (b + c) = (a + b) + c, a, b, c ∈R. 3 There is such a number, called zero and denoted by 0, that for any a R the condition a + 0 = a is satisfied. 4. For any number a ∈R, there is a number, called its opposite and denoted by -a, for which a + (-a) = 0. The number a + (-b) = 0, a, b∈R, is called the difference of numbers a and b and is denoted by a - b.
Real numbers.
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II. multiplication operation. For any pair of real numbers a and b, a single number is defined, called their product and denoted by ab, such that the following conditions are satisfied: II1. ab = ba, a, b∈R. II2. a(bc) = (ab)c, a, b, c ∈R. II3. There is such a number, called one and denoted by 1, that for any a∈R the condition a*1= a is satisfied. II4. For any number a≠0, there is a number called its inverse and denoted by or 1/a, for which a*1/a=1 The number a*1/b, b≠0, is called the quotient of a divided by b and is denoted by a: b or or a/b.
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If we add their opposite numbers and the number zero to positive infinite decimal fractions, then we get a set of numbers, which are called real numbers.
The set of real numbers consists of rational and irrational numbers
Tips on how to make a good presentation or project report
- Try to involve the audience in the story, set up interaction with the audience using leading questions, the game part, do not be afraid to joke and smile sincerely (where appropriate).
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- No need to overload your project slides with text blocks, more illustrations and a minimum of text will better convey information and attract attention. Only the key information should be on the slide, the rest is better to tell the audience orally.
- The text must be well readable, otherwise the audience will not be able to see the information provided, will be greatly distracted from the story, trying to make out at least something, or completely lose all interest. To do this, you need to choose the right font, taking into account where and how the presentation will be broadcast, and also choose the right combination of background and text.
- It is important to rehearse your report, think over how you will greet the audience, what you will say first, how you will finish the presentation. All comes with experience.
- Choose the right outfit, because. The speaker's clothing also plays a big role in the perception of his speech.
- Try to speak confidently, fluently and coherently.
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