Presentation on the topic of real numbers. Relevance of the selected topic
“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 in yellow, and the concept itself is highlighted in red. This will help students better concentrate on this concept and remember it better visually.
Further, the 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 to” 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 number and an irrational number to a positive integer. As can be seen from the solution, during transformations an irrational number in the form of a square root is written through an infinite non-periodic fraction.
The third example is the simplest. After all, it is proposed to compare a negative number with a positive one. 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 work and tests.
The set of real numbers can be described as the set of all finite and infinite decimal fractions. All finite and infinite decimal periodic fractions are rational numbers, and infinite decimal non-periodic fractions are irrational numbers. Every real number can be represented by a point on a coordinate line; each point M on a coordinate line has a real coordinate. 2+2=? 2+2=4
Let's draw a straight line and mark point O on it, which we will take as the origin. Let's choose a direction and a unit segment. They say that a coordinate line is given. Each natural number corresponds to one single point on the coordinate line. Let there be a point M(x) on a segment of the coordinate line. Divide the segment into 10 equal parts (segments of the 1st rank). Let's assume that M Δ4, that is, x=0.4.... Let's divide Δ4 into 10 segments of the 2nd rank. Let's assume that M Δ40. That is, x=0, Δ0 Δ1 Δ2 Δ3 Δ4 Δ5 Δ6 Δ7 Δ8 Δ9 M(x) Δ40
The coordinate line, or number line, is a geometric model of the set of real numbers. For real numbers a, b, c, the usual laws are satisfied: 1)a+b=b+a 2)a*b=b*a 3)a+(b+c)=(a+b)+c 4)a* (b*c)=(a*b)*c 5)(a+b)*c=a*c+b*c as well as the usual rules: The quotient of 2 positive numbers is a positive number.
Presentation for class “Real numbers. The set of real, rational and irrational numbers"
Target: recall basic concepts related to real numbers.
1 slide
Subject: Sets of numbers
Prepared the work
Teacher at Rzhev College
Sergeeva T.A.
2 slide.
“Numbers rule the world,” said the Pythagoreans. But numbers make it possible for a person to control the world, and the entire course of development of science and technology of our days convinces us of this.
(A. Dorodnitsyn)
3 slide.
Let's recall the basic concepts associated with real numbers.
What sets of numbers do you know?
4 slide.
Integers – numbers that are used to count objects: 1,2,3,4,5……
Denote the set of natural numbers by a letter N
For example:“5 belongs to the set of natural numbers” and writes -
5 slide
Integers , which are divisible by 1 and by itself (for example, 2, 3, 5, 7, 11) are called prime numbers .
All other numbers are called composite and can be factorized into prime factors (for example,)
Any natural number in the decimal number system is written using digits
(For example)
6 slide
Example
Number, i.e. number consists of 1 thousand, 2 hundreds, 3 tens and 7 ones
This means that if a is the digit of thousands, b is the digit of hundreds, d is the digit of tens and c is the digit of units then we have a 1000+b 100+ c 10+d .
Slide 7
The natural numbers, their opposites and the number zero make up the set whole numbers.
The set of integers is denoted by the letter Z.
For example:“-5 belongs to the set of integers” and then write -
8 slide
Fractional numbers of the form (where n is a natural number, m is an integer), decimals (0.1, 3.5) and integers (positive and negative) together make up the set rational numbers.
Denote the set of rational numbers by the letter Q.
For example:“-4,3 belongs to rational integers” then write
Slide 9
Fractional numbers of the form, decimals (0.1, 3.5) and integers (positive and negative) together make up the set rational numbers.
Any rational number can be represented as a simple fraction, (where n is a natural number, m is an integer)
For example:
Any rational number can be represented as an infinite periodic decimal fraction.
For example:
10 slide
The set of rational numbers includes whole numbers and fractions, and the set of real numbers includes rational and irrational numbers. This leads to the definition of real numbers.
Definition: Real numbers are the set of rational and irrational numbers.
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Historical reference
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A bunch of valid numbers are also called number line.
Each point on the coordinate line corresponds to some real number, and each real number corresponds single point on the coordinate line.
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Homework.
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The presentation on the topic “Real numbers” (8th grade) can be downloaded absolutely free on our website. Project subject: Mathematics. Colorful slides and illustrations will help you engage your classmates or audience. To view the content, use the player, or if you want to download the report, click on the corresponding text under the player. The presentation contains 11 slide(s).
Presentation slides
Slide 1
Prepared by 8th grade student Anastasia Karpova.
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Stages of development of the concept of number.
The geometric idea of numbers as segments leads to the expansion of the set Q to the set of real (or real) numbers R: N ⊂ Z ⊂ Q ⊂ R.
Using rational numbers, you can 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 and 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 form 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.
Slide 4
More about real numbers:
The real numbers include the numbers of the rational and irrational set.
Real numbers can be added, subtracted, multiplied, divided, and compared by magnitude. Let us list the main properties that these operations have. We will denote the set of all real numbers by R, and its subsets will be called number sets.
Slide 5
I. Addition operation. For any pair of real numbers a and b, a unique number is defined, called their sum and denoted 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 a number called zero and denoted 0 such 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 -a, for which a + (-a) = 0. The number a + (-b) = 0, a, b∈R, is called the difference of the numbers a and b and is denoted a - b.
Real numbers.
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II. Multiplication operation. For any pair of real numbers a and b, a unique number is defined, called their product and denoted 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 a number called unity and denoted 1 such 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 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, we get a set of numbers called real numbers.
The set of real numbers consists of rational and irrational numbers
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Slide captions:
Real numbers 09/02/13
Text Numerical sets Designation Set name N Set of natural numbers Z Set of integers Q=m/n Set of rational numbers I=R/Q Set of irrational numbers R Set of real numbers
The set of natural numbers Natural numbers are counting numbers. N=(1,2,…n,…). Note that the set of natural numbers is closed under addition and multiplication, i.e. addition and multiplication are always performed, but subtraction and division are generally not performed
A set of integers. Let us introduce new numbers into consideration: 1) the number 0 (zero), 2) the number (- n), the opposite of the natural n. In this case, we assume: n+(-n)=(-n)+n=0, -(-n)=n. Then the set of integers can be written as follows: Z =(…,-n,…-2,-1,0,1,2,…,n,…). Note also that: This set is closed under addition, subtraction and multiplication, i.e. From the set of integers we select two subsets: 1) the set of even numbers 2) the set of odd numbers
The set of rational numbers. The set of rational numbers can be represented as: In particular, Thus, the Set of rational numbers is closed under addition, subtraction, multiplication and division (except for the case of division by 0).
But in the set of rational numbers it is impossible, for example, to measure the hypotenuse of a right triangle with legs. According to the Pythagorean theorem, the hypotenuse will be equal. But the number will not be rational, since for any m and n. The equation cannot be solved. You cannot measure circumference, etc. Note that any rational number can be represented as a finite or infinite periodic decimal fraction.
Lots of irrational numbers. Numbers that are represented by an infinite non-periodic fraction will be called irrational. Let's denote the set of irrational numbers by I. There is no single form of notation for irrational numbers. Let us note two irrational numbers, which are denoted by letters - these are numbers and e.
Number "pi" The ratio of the circumference to the diameter is a constant value equal to the number d
Number e. If we consider a number sequence: with a common member of the sequence, then as n increases, the values will increase, but will never be greater than 3. This means that the sequence is limited. Such a sequence has a limit, which is equal to the number e.
It is known that the power of irrational numbers is greater than the power of rational numbers, i.e. There are “more” irrational numbers than rational numbers. In addition, no matter how close two rational numbers are, there is always an irrational between them, i.e.
The set of real numbers. The set of real numbers is the union of the set of rational numbers. Conclusion:
Determination of the modulus of a real number Let point A on the number axis have coordinate a. The distance from the origin point O to point A is called the modulus of the real number a and is denoted by | a | . | a | = | OA | R’ a a A A O 2) The module is revealed according to the rule:
For example: Note. The definition of a module can be expanded: Example. Expand the module sign. where f (x) is a function of the argument x
Basic properties of the module 1) 2) 3) 4) 5) 6)
Solving examples using the properties of the module Example 1. Calculate Example 2. Expand the sign of the module Example 3. Calculate 1) 2) 3)
