Target :

• • • FamiliarizationstudentsWithconceptinequalities anddecisioninequalities, performanceexercisesonfindingsolutionsprotozoainequalities.
      Developmentlogicalthinkingstudents.

      Cultivating accuracy in work.

Move lesson

I . Update supporting knowledge

Mathematical dictation

Students write down answers to questions in their notebooks.

Which of the numbers is 3; 12; 14 are the roots of the equation?

• X+21=24

  49's=47

  2x-10=18

Write an equation for the problem, taking the unknown number to beX. Find this number. (Can be found orally by writing down only the answer.)

Vanya thought of a number. If to this number atadd 12 and subtract 19 from the resulting amount,it will be 31. What number did Vanya have in mind??

II . Learning new material

With respect to two different natural numbers, alwaysyou can tell which one is larger and which is smaller. This means,that natural numbers can be compared.

The result of the comparison is written in the form of inequalitiesusing signs<; (меньше) и >(more). Eg.2<5 (read: two less than five) or5>2 (read: five is more than two).

Rules:

If two natural numbers have different number characters (digits), then the number with more characters is larger.

For example,

3421 >803; 5703<21844.

2. If two natural numbers have the same number signs, then the number that has more units in the highest rank. If the number of units in this digit is the same, then digits one level lower are compared etc.

The smallest natural number is one (1).

There is no greatest natural number: for anyoneof a given natural number can be called a natural numbera number that is greater than a given one. Therefore they say that the seriesnatural numbers 1, 2, 3, ...is not limited.

The number 0 is less than any natural number. Any natural number is greater than 0.

Coordinate ray property:

On the coordinate ray, the larger number is located to the right, and the smaller number is located to the left.

A B

0 1 2 3 4 5 6 7 8 Then he collects his homework notebooks for checking.

VI . Homework

It is clear that 5 is less than 7, and 171 is greater than 19. This comparison result is written using (greater than) signs: 5 19 Such records are called inequalities 19 Such entries are called inequalities"> 19 Such entries are called inequalities"> 19 Such entries are called inequalities" title="It is clear that 5 is less than 7, and 171 is greater than 19. This comparison result is written using (greater than) signs: 5 19 Such records are called inequalities"> title="It is clear that 5 is less than 7, and 171 is greater than 19. This comparison result is written using (greater than) signs: 5 19 Such records are called inequalities"> !}

You can compare three numbers at the same time. For example, the number 17 is greater than 15, but less than 20. This is written using a double inequality: 15

1. Count the number of digits in each number. The number that has more digits is greater: > 99 124 396"> 99 124 396"> 99 124 396" title="1. Count the number of digits in each number. The number with more digits is greater: 594 321 505 > 99 124 396"> title="1. Count the number of digits in each number. The number with more digits is greater: 594,321,505 > 99,124,396"> !}

2. If two multi-digit numbers have the same number of digits, then they need to be compared by digits: 7256 > 7249 582 647 7249 582 647 7249 582 647 7249 582 647 title="2. If two multi-digit numbers have the same number of digits, then you need to compare them by digit: 7256 > 7249 582 647

We use comparisons in life all the time. For example, a long or short road, a tall or short person, a lot of toys or few, a large container or a small one. So, what is comparing natural numbers?

Comparison of natural numbers– this is the determination of which is greater and which is less.

Ways to compare natural numbers.

1, 2, 3, 4, 5, 6, 7 , 8, 9 ,10, 11, 12, 13, 14, 15, …

1) The numbers on the right are always greater than the numbers on the left.
For example, let's compare the numbers 7 and 9. The number 9 is to the right of the number 7, therefore the number 9 is greater than 7.

One is the smallest natural number.

Any natural number is greater than zero.

2) The natural number that has more is always greater.

Let's compare two numbers 45 and 190. It is immediately clear that the number 190 is greater than the number 45. We made this conclusion because the number 190 is a three-digit number, and 45 is a two-digit number. The number 190 has a hundreds, tens and ones place, while the number 45 only has a tens and ones place.

3) If the number of digits is the same, then we will compare the values ​​of the digits of the digits, starting from (from left to right).
For example, let's compare the numbers 478 and 399. Both numbers are three-digit numbers, so let's look at hundreds in detail. The first number, 478, has a hundreds place of 4, and the second number, 399, has a hundreds place of 3. Therefore, the first number, 478, is greater than the second number, 399, because 4 is greater than 3.

If they are the same, we compare the next smaller digit.
Let's compare the numbers 7890 and 7860. We begin to compare the highest digit of units of thousands; for both numbers it is equal to 7. The next digit of hundreds is also equal to 8 for both numbers. But the digit of tens is different. The first number 7890 has a tens place of 9, and the second number 7860 has a 6. Next we conclude that the first number 7890 is greater than 7860, because the tens place of the first number is greater than that of the second. To put it simply, 9 is greater than 6.

\(\left(\begin(array)(c)78 \color(blue) (9)0\\ 78\color(red) (6)0\end(array)\right)\)

4) If, when comparing, all the digits of the digits of two natural numbers are the same, then the numbers are equal.
For example, let's compare the numbers 4890765 and 4890765. It can be seen that both numbers have the same digits, therefore they are equal.

\(\left(\begin(array)(c)4890765\\ 4890765\end(array)\right)\)

Inequality and inequality signs.

In order not to write with words greater than, less than or equal to, notations were invented in mathematics. More (>), less (<), равно (=) . For example, 3 is greater than 2, the mathematical notation would be 3>2. Or 6 is less than 10, we write it as 6<10. 8 равно 8, запишем 8=8.

Expressions 3>2, 6<10 и 8=8 называются в математики inequalities.

Such entry 2<3<4 называется double inequality.

Questions to the topic:
What is the smallest natural number?
Answer: one.

What is the largest natural number?
Answer: The natural series of numbers is infinite, so there is no largest natural number.

Which number is greater, a six-digit number or a seven-digit number?
Answer: A seven-digit number is greater than a six-digit number.

Examples with answers to typical tasks of the topic are analyzed.
Example #1:
Read the inequality: a) 5<12 б) 6>1 c) 7=7
Answer: a) five is less than twelve b) six is ​​more than one c) seven is equal to seven.

Example #2:
Write down the inequality: a) 4 is less than 8 b) 10 is more than 9 c) 11 is equal to 11.
Answer: a) 4<8 б) 10>9 c) 11=11.

Example #3:
Are the inequalities true? Check the comparison signs: a) 5<6 б) 7<3 в) 22>23 g) 5=55
Answer: a) true b) false c) false d) false.

Example #4:
Compare the numbers, put the inequality signs correctly (<, >, =): a) 3 and 3 b) 4 and 9 c) 8 and 3
Answer: a) 3=3 b) 4<9 в) 8>3

Example #5:

Look at the picture and make up the inequality.

When counting, natural numbers are called in order: 1, 2, 3, 4, 5, 6, 7, 8, 9... .

Of two natural numbers, the smaller is the one that is called earlier when counting, and the larger is the one that is called later when counting. Unit– the smallest natural number. The number 4 is less than. 7, and the number 8 is greater than 7.

The point with the smaller coordinate lies on the coordinate ray to the left of the point with the larger coordinate.

For example, point A(4) lies to the left of point B(7) (Fig. 16). Zero is less than any natural number.

Rice. 16. Coordinate beam

The result of comparing two numbers is written in the form inequalities, using signs< (меньше) и >(more). For example, 4< 7, 8 >7. The number 3 is less than 6 and greater than 2. This is written as double inequality 2 < 3 < 6. Так как нуль меньше, чем единица, то записывают 0 < 1.

Multi-digit numbers are compared like this. The number 2305 is greater than 984 because 2305 is a four-digit number and 984 is a three-digit number. The numbers 2305 and 1178 are four-digit numbers, but 2305>1178 because the first number has more thousands than the second. The four-digit numbers 2305 and 2186 have equal numbers of thousands, but the first number has more hundreds, and therefore 2305 > 2186.

Signs< и >also denote the result of comparing segments. If segment AB is shorter than segment CD, then write:

If segment AB is longer than segment CD, then write:

Inequalities are read like this: the left side is in the nominative case, and the right side is in the genitive case.

For example: 55<128 – пятьдесят пять меньше ста двадцати восьми.

Many different ways of writing numbers have been created by people. In Ancient Rus', numbers were designated by letters with a special sign “~” (title), which was written above the letter (Fig. 17).

Rice. 17. Recording numbers in Ancient Rus'

The first nine letters of the alphabet represent units, the next nine letters represent tens, and the last nine letters represent hundreds. The number ten thousand was called the word “darkness” (and now we say: “to the people - darkness”).

The modern, fairly simple and convenient decimal system for writing numbers was borrowed by Europeans from the Arabs, who in turn adopted it from the Indians. Therefore, the numbers we now use are called “Arab” by Europeans, and “Indian” by Arabs. This system was introduced to Europe around 1120 by an English explorer. Adelard . By 1600 it had been accepted in most countries of the world.

Russian names of numbers are closely related to the decimal number system. For example, seventeen means “seven times ten,” seventy means “seven tens,” and seven hundred means “seven hundred.”

Roman numerals, which were used in Ancient Rome about 2600 years ago, are still used.

I - 1, V - 5, X - 10, L - 50, C - 100, D - 500, M - 1000.

The rest of the numbers are written using these numbers using addition and subtraction. So, for example, the number XXVII means 27, since

10 + 10 + 5 + 1 + 1 = 27.

If a smaller number (I, X, C) comes before a larger one, then its value is subtracted.

For example, IV means 4(5 - 1 = 4), IX means 9(10 – 1 = 9), XC means 90. Thus, the number MCMLXXXIX means 1989. since:

1000 + (1000 - 100) + 50 + 10 + 10 + 10 + (10 - 1) = 1989.

Currently, Roman numerals are usually used when numbering chapters and sections of books, months of the year, to designate dates of significant events, and anniversaries.

For calculations, writing numbers using Roman numerals is inconvenient. You can see this for yourself if you try, for example, adding the numbers CCXCVII and ХLIХ or dividing the number CCXCVII by the number IX.

Handbook - Mathematics

Comparing natural numbers is very easy. You can always tell which of two different natural numbers is smaller and which is larger. Let's say: “7 is less than 12” or “12 is more than 7.”

For example, if in a drawing lesson Olya had 12 colored pencils, and Igor had 7, then it is clear that Olya has more pencils than Igor, and Igor has fewer than Olya.

When comparing two numbers in a recording, the word less is replaced with the sign “<», а слово больше — знаком «>" Let's write down what was said using comparison signs: 7< 12 или 12 > 7.

Please note: the sharp “beak” of the “more than” and “less than” icons is always directed towards the smaller of the two numbers.

If both Olya and Igor had 12 or 7 pencils, we would say that they have an equal number of pencils, because 12 is equal to 12, and 7 is equal to 7.

When writing, the word equal is replaced with the “=” sign.

Two friends Nastya and Anya decided to count which of them received more A's in a week at school. Nastya counted: “1,2, 3, 4, 5, 6, 7.” Nastya has 7 A's in total. Then Anya counted: “1, 2, 3, 4, 5, 6, 7, 8, 9.” In total, Anya has 9 A's. It is clear that Anya received more A’s in a week than Nastya: 9 > 7.

When comparing two natural numbers, the one to the right in the natural series is greater.

When the numbers are large, it is sometimes difficult to immediately determine which one is to the right in the natural series.

When comparing two natural numbers with different numbers of digits, the number with more digits is greater.

For example: 93< 256, потому что в первом числе две цифры, а во втором — три.

Multi-digit natural numbers with the same number of digits are compared bitwise, starting with the most significant digit.

First, the units of the most significant digit are compared, then the next, the next, and so on. For example, let's compare the numbers 5791 and 5319.

Think about it this way:

5 791 =5 t. 7 s. 9 days 1 unit

5 319-5 t. Z. 1 d. 9 units.

I compare units of thousands. In the place of units of thousands, the number 5,791 is 5 units, in the place of units of thousands, the number 5,319 is 5 units. Having compared the units of thousands, I still do not get an answer to the question of which number is larger. I'll discuss further. I compare hundreds. In the hundreds place the number 5791 is 7 units, in the hundreds place the number 5319 is 3 units, comparing, I get 7 > 3, therefore 5791 > 5319.

Numbers can be arranged in descending or ascending order. If in a record of several natural numbers each next number is less than the previous one, then the numbers are said to be written in descending order.

Let's write down the numbers 7,11,21, 791, 2 in descending order. Think about it this way:

I'll find a larger number. The numbers 7 and 2 are single-digit, 11 and 21 are two-digit, 791 is a three-digit number and therefore the largest. I write 791 in the first place. Of the two-digit numbers 11 and 21, the greater is 21. After the number 791, I write the number 21, and then 11. Of the numbers 7 and 2, the greater is 7. After the number 11, I write 7, and then 2.

791, 21, 11, 7, 2 - recording these numbers in descending order.

If in a record of several natural numbers each next number is greater than the previous one, then the numbers are said to be written in ascending order.

Now let's write down the numbers 12, 5, 31, 279, 268 in ascending order. Think about it this way:

Among the numbers 12, 5, 31, 279, 268 I will find the smaller one. The numbers 279 and 268 are three-digit, 12 and 31 are two-digit, 5 is single-digit. The smaller number is 5. In the first place I write the number 5. Of the two-digit numbers, 12 are less, 31 are greater. After the number 5 I write 12, then 31. 5, 12, 31 3. Of the three-digit numbers, 268 is the smaller, 279 is the larger. After the number 31 I write 268, then 279. 5, 12, 31, 268, 279 - write these numbers in ascending order.