Numbers 1, 2, 3... - natural numbers Natural numbers are numbers that arise naturally when counting. There are two approaches to determining natural numbers; numbers used in: listing (numbering) objects (first, second, third, ...); designation of the number of items (no items, one item, two items, ...). 2
3
4
,
9 Mathematicians Ancient Greece more than twenty centuries ago they came to the conclusion that there is neither an integer nor a fractional number expressing the diagonal of a square with side 1. This caused a crisis in mathematical science: a square has a diagonal, but it has no length! Mathematicians have found a way out of this situation: since the available supply of numbers - integers and fractions - is not enough to express the lengths of segments, it means that some new numbers are needed. This is how irrational numbers appeared.
10 Measuring the lengths of segments on a coordinate line Work with the textbook pp. 63 – 64 p. 11. Answer the questions orally: 1. How can you measure the length of any segment? 2. How can you get a more accurate result (with an accuracy of 0.1, 0.01 and 0.001? 3. What numbers will be the result of the measurements?
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Let's compare the numbers 2.36366... and 2.37011... coincide in the hundredths place; the first fraction has fewer units than the second, therefore 2.36366... 
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Irrational numbers Natural numbers Natural numbers Integers Integers Rational numbers Rational numbers –6(3) 7, … 345 π π 1.24(53) 21
1. 276, 277, 281 (a, c, d) ,
1.Algebra. 8th grade. Textbook. Federal State Educational Standard. Yu.N. Makarychev, N.G. Mindyuk, K.I. Neshkov, S.B. Suvorov. Ed. S.A.Telyakovsky, 2.Algebra, 8th grade, Lesson plans, Dyumina T.Yu., Makhonina A.A., 2012: CD; 3. html 4. gifhttp://img1.liveinternet.ru/images/attach/c/4/80/35/ _ _skola1. gif 5. jp jpghttp:// jpg 7. Literature and Internet resources: 27
Definition of an irrational number
Irrational numbers are those numbers that decimal notation represent infinite non-periodic decimal fractions.
So, for example, numbers obtained by taking the square root of natural numbers are irrational and are not squares of natural numbers. But not all irrational numbers are obtained by extraction square roots, because the number “pi” obtained by division is also irrational, and you are unlikely to get it when trying to extract the square root of a natural number.
Properties of irrational numbers
Unlike numbers written as infinite decimals, only irrational numbers are written as non-periodic infinite decimals.
The sum of two non-negative irrational numbers can ultimately be rational number.
Irrational numbers define Dedekind sections in the set of rational numbers, in the lower class which do not have the large number, and in the upper there is no less.
Any real transcendental number is irrational.
All irrational numbers are either algebraic or transcendental.
The set of irrational numbers on a line is densely located, and between any two of its numbers there is sure to be an irrational number.
The set of irrational numbers is infinite, uncountable and is a set of the 2nd category.
When performing any arithmetic operation on rational numbers, except division by 0, the result will be a rational number.
When adding a rational number to an irrational number, the result is always an irrational number.
When adding irrational numbers, we can end up with a rational number.
The set of irrational numbers is not even.
Numbers are not irrational
Sometimes it is quite difficult to answer the question of whether a number is irrational, especially in cases where the number is in the form of a decimal fraction or in the form numerical expression, root or logarithm.
Therefore, it will not be superfluous to know which numbers are not irrational. If we follow the definition of irrational numbers, then we already know that rational numbers cannot be irrational.
Irrational numbers are not:
First, all natural numbers;
Secondly, integers;
Third, ordinary fractions;
Fourthly, different mixed numbers;
Fifthly, these are infinite periodic decimal fractions.
In addition to all of the above, an irrational number cannot be any combination of rational numbers that is performed by the signs arithmetic operations, as +, -, , :, since in this case the result of two rational numbers will also be a rational number.
Now let's see which numbers are irrational:
Do you know about the existence of a fan club where fans of this mysterious mathematical phenomenon are looking for more and more information about Pi, trying to unravel its mystery? Any person who knows by heart a certain number of Pi numbers after the decimal point can become a member of this club;
Did you know that in Germany, under the protection of UNESCO, there is the Castadel Monte palace, thanks to the proportions of which you can calculate Pi. King Frederick II dedicated the entire palace to this number.
It turns out that they tried to use the number Pi during construction Tower of Babel. But unfortunately, this led to the collapse of the project, since at that time the exact calculation of the value of Pi was not sufficiently studied.
Singer Kate Bush in her new disc recorded a song called “Pi”, in which one hundred and twenty-four numbers from the famous number series 3, 141…..
Math lesson in 8th grade
Lesson topic: Irrational numbers. Real numbers.
Sinichenkova Galina Alekseevna
mathematic teacher
Municipal educational institution Gribanovskaya secondary school
Goals:- introduce the concept of an irrational number, a real number; - teach how to find approximate values of roots using a microcalculator; - introduce four-digit mathematical tables; - consolidate the skill of conversion common fraction into decimal and decimal infinite periodic fractions into ordinary fractions; - develop memory and thinking.During the classes
I Update background knowledge.
Homework check: a) Present as a decimal fraction: 38/11 =
b) Present as an ordinary fraction: 1,(3) = 0.3(17) =
c) Card: Present as an ordinary fraction: 1 option 2 option 3 option 7.4 (31) 1.3 (4) 4.7 (13)
II Oral exercises 1) Read the fractions:0,(5); 3,(24); 15.2(57); -3.51(3)2) Calculate:
3) Round these numbers: 3.45; 10.59; 23.263; 0.892A) to units; B) to tenths.
III Learning new material1. Communicate the topic and objectives of the lesson2. Teacher's explanation Along with infinite periodic fractions, infinite non-periodic fractions are also considered in mathematics. In the last lesson you were introduced to the concept of rational numbers. And you know that any rational number can be represented as a decimal fraction, finite or infinite. For example, fractions 0.1010010001...0.123456...2.723614...Infinite decimal non-periodic fractions are called irrational numbers.
Rational and irrational numbers form a set real numbers.
Arithmetic operations and comparison rules for real numbers are defined in such a way that the properties of these operations, as well as the properties of equalities and inequalities, are the same as for rational numbers.
When do you get irrational numbers?
1) When extracting square roots. In the know higher mathematics it is proved that from any non-negative number you can take the square root.
For example
2) Irrational numbers are obtained not only by taking roots. For example 
4. Message “From the history of irrational numbers”
5. In practice, tables, microcalculators and other computing tools are used to find approximate values of roots with the required accuracy. 1). Introduction to four-digit math tables (page 35)
For those who are interested in learning more about finding square roots using a table, you can read the explanations to the table.
2). Currently, a microcalculator is most often used to find approximate values of roots.
Example
IV Consolidation of the studied material
No. 322(1,3,5) Disassemble and write on the board.
6. Working with cardsCalculate on a microcalculator with an accuracy of 0.001
7. Geometrically real numbers are represented by points on the number axis Page 89 (Fig. 30)
V Assimilation of the studied materialIndependent work
Option 1
- Compare numbers
Option 2
- Compare numbers
VI Homework : item 21, No. 322 (2,4,6), No. 323, additional task(cards)
VII Lesson summary and grading.- What numbers are called irrational? - What numbers form the set of real numbers?
? Yu.N.Makarychev Algebra. 8th grade: textbook for educational institutions-M.: Enlightenment, 2014
? N.G. Mindyuk Didactic materials. Algebra. 8th grade - M.: Education, 2014.
? N.G. Mindyuk Workbook. Part 1 Algebra. 8th grade - M.: Education, 2014.
- Projector
- Computer
During the classes
- Organizing time
- Oral work
- m/ n, where m- integer, n-natural. Example 3/5 can be imagined different ways: 3/5=6/10=9/15=…….)
- What sets do you already know? (natural numbers -N, integers -Z, rational ones -Q,
- Task on the board: Determine which set each number belongs to? Fill the table. ; 0.2020020002…; -p.
Natural -N
Rational - Q
7; 19; 235; -90
7; 19; 3/8; -5,7; 235; -90; -1(4/11)
And these numbers are 0.2020020002...; -p where should I put it?
“NOT” will be replaced by the prefix “IR”.
Irrational number- decimal infinite periodic fraction.
Where T - integer, P- natural.
Let's return to our table. (Let’s add irrational numbers and 0.2020020002…; -p
Consolidation
1st - tasks to determine belonging to different numerical sets.
2nd - tasks for comparing real numbers.
Test followed by verification
13) The number p is real.
14) Number 3.1(4) less number p.
15 correct answers - score “5”
12-14 correct answers - score “4”
Reflection
Homework
№278; 281; 282
Lesson grades.
Thank you for the lesson!
"Plan"
Municipal budget educational institution
"Turgenevskaya Secondary School"
Teacher: Loiko Galina Alekseevna
Lesson plan on topic
"Irrational Numbers"
"Numbers don't rule the world"
LESSON OBJECTIVES:
Learning Objectives:
development cognitive interest through application entertaining tasks and examples.
2. The purpose of education:
nurturing conscious motives for learning and a positive attitude towards knowledge.
Educational and methodological support
● Yu.N.Makarychev Algebra. 8th grade: textbook for general education institutions - M.: Prosveshchenie, 2014.
●N.G. Mindyuk Didactic materials. Algebra. 8th grade - M.: Education, 2014.
● N.G. Mindyuk Workbook. Part 1 Algebra. 8th grade - M.: Education, 2014.
Necessary equipment and materials for classes :
Projector
Computer
During the classes
What topic did we study in the last lesson? (Rational numbers)
What numbers are called rational? (Numbers that can be represented as fractions m / n, where m is an integer, n is a natural number. Example 3/5 can be represented in different ways: 3/5=6/10=9/15=……..)
What sets do you already know? (natural numbers – N, integers – Z, rational – Q,
Task on the board: Determine which set each number belongs to? Fill the table. -7; 19; 3/8; -5.7; 235; -90; -1(4/11); 0.2020020002…; -.
Organizing time
Oral work
| Natural –N | Integer-Z | Rational – Q | |
| 7; 19; 235; -90 | 7; 19; 3/8; -5,7; 235; -90; -1(4/11) |
And these numbers are 0.2020020002...; - where should it be attributed?
Our knowledge is not enough to say anything about them. And now we are moving on to studying new material, and the topic of the lesson is “Irrational numbers”, you will find out what numbers are called irrational and give examples.
Consider the infinite decimal fraction
This endless decimal by definition is not rational.
This means that this fraction is not a rational number.
“NOT” will be replaced by the prefix “IR”.
We get an “irrational” number.
Irrational number
Let's look at examples of irrational numbers.
The irrational cannot be represented as a fraction
WhereT -
integer,P
– natural.
Real numbers can be added, subtracted, multiplied, divided, and compared.
Let's return to our table. (Let’s add irrational numbers and 0.2020020002…; -
Let's generalize knowledge about all sets of numbers
Consolidation
All tasks from the textbook can be divided into 2 groups.
1st – tasks to determine membership in various numerical sets.
2nd – tasks for comparing real numbers.
Let's do numbers: No. 276, 277, 279, 287. (orally)
Let's do the numbers: No. 280, 283, 288 (at the board)
Test followed by verification
“+” - I agree with the statement; “-” - I do not agree with the statement.
1) Every integer is natural.
2) Every natural number is rational.
3) The number -7 is rational.
4) The sum of two natural numbers is always natural number.
5) The difference of two natural numbers is always a natural number.
6) The product of two integers is always an integer.
7) The quotient of two integers is always an integer.
8) The sum of two rational numbers is always a rational number.
9) The quotient of two rational numbers is always a rational number.
10) Every irrational number is real.
11) A real number cannot be natural.
12) The number 2.7(5) is irrational.
15) The number - 10 belongs simultaneously to the set of integers, rational and real numbers.
8-11 correct answers - score “3”
less than 8 you should learn the theory.
Reflection
What numbers are called rational and irrational?
What numbers does the set of real numbers consist of?
Homework
№278; 281; 282
Lesson grades.
Thank you for the lesson!
View document contents
"Test followed by verification"
Test followed by verification
“+” - I agree with the statement;
“-” - I do not agree with the statement.
1) Every integer is natural.
2) Every natural number is rational.
3) The number -7 is rational.
4) The sum of two natural numbers is always a natural number.
5) The difference of two natural numbers is always a natural number.
6) The product of two integers is always an integer.
7) The quotient of two integers is always an integer.
8) The sum of two rational numbers is always a rational number.
9) The quotient of two rational numbers is always a rational number.
10) Every irrational number is real.
11) A real number cannot be natural.
12) The number 2.7(5) is irrational.
13) The number is real.
14) The number 3.1(4) is less than the number .
15) The number - 10 belongs simultaneously to the set of integers, rational and real numbers.
Answers
| "Irrational Numbers" "Numbers don't rule the world" but they show how to manage it"  LESSON OBJECTIVES 1 Learning objectives:
2. The purpose of education:
 Consider the infinite decimal fraction This infinite decimal is by definition not rational. This means that this fraction is not a rational number. "NOT" replace it with a prefix "IR" . We get an “irrational” number. Irrational number – decimal infinite periodic fraction.  Let's look at examples of irrational numbers. The irrational cannot be represented as a fraction Where T – integer, P – natural.  Valid numbers Rational numbers Irrational numbers Fractional numbers Endless non-periodic fractions Whole numbers Negative numbers Ordinary fractions Zero Decimal fractions Positive numbers Final Endless periodic  Key to the test  Grade 15 correct answers – score “5” 12-14 correct answers – score “4” 8-11 correct answers - score “3” less than 8 you should learn the theory.  Homework. № 278 № 281 № 282  |