Presentation on mathematics for the lesson “Real numbers. Set of real, rational and irrational numbers. The set of real numbers can be described as the set of all finite and infinite decimal fractions. All finite and infinite

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.

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“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)

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Let's recall the basic concepts associated with real numbers.

What sets of numbers do you know?

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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 -

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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)

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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 .

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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 -

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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” and writes

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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:

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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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Numerical sets.

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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

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A set of integers.

Let's 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 non-carrier numbers

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Division with remainder.

In the general case, the operation of division in a set of integers is not performed, but it is known that division with a remainder can always be performed, except for division by 0. Definition of division with a remainder. An integer m is said to be divisible by an integer n with a remainder if there are two numbers q and p such that: (*) The division with remainder algorithm is well known. Note: if r=0, then we will say that m is divisible by n. m=nq+r, where 0≤r

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EXAMPLES:

Divide with remainder m by n. 1). m=190, n=3 190 3 18 6 3 10 9 1 q=63, r=1, 1 q=2, r=3 (3 q=-4, r=1 -15=4*(-4) +1 4). M=6, n=13 By formula(*): 6=13q+r =>q=0, r=6 6=13*0+6

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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).

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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 no 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.

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Lots of irrational numbers.

Numbers that are represented by an infinite non-periodic fraction will be called irrational. We denote the set of irrational numbers. 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.

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Pi"

The ratio of the circumference to the diameter is a constant value equal to the number d

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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.

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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. Examples of irrational numbers: (golden ratio), etc.

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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.

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)

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 decimal 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 the numerical values of 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, the first letter of the French word Quotient - “ratio”. integers Zahl 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? 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