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Comparing numbers In this lesson we will consolidate the knowledge of comparing numbers. Let us formulate a rule for comparing numbers with respect to their location on the coordinate line. Let's learn how to compare numbers using the concept of "module number". We derive a rule for comparing numbers. Let's consolidate our knowledge by doing exercises to compare numbers. Lesson summary "Comparing numbers" You know that numbers can be compared. Let's remember what numbers you already know how to compare: Therefore, you can compare any positive numbers with each other and with zero. Do you think negative numbers can be compared? Of course! And negative with each other, and negative with positive, and negative with zero. Today in the lesson we will talk about this. Let's draw a coordinate line, mark the origin on it, select a unit segment and indicate the direction. Recall that on a horizontal coordinate line, positive numbers are depicted to the right of zero, and negative numbers to the left of zero. Let's take two numbers, for example, 1 and. Do you know that. Let's mark points A(1) and B() on the coordinate line.

It is clear that point A on the coordinate line is located to the left of point B. Recall the rule: on the horizontal coordinate line, a point with a larger coordinate lies to the right of a point with a smaller coordinate. Accordingly, on a horizontal coordinate line, a point with a smaller coordinate lies to the left of a point with a larger coordinate. And now let's take two negative numbers, for example, - 2 and -. How to compare such numbers? Let us mark points C(–2) and D(–) on the coordinate line. Let's write down the rule for comparing any numbers: Of the two numbers, the one that is depicted on the horizontal coordinate line to the right is greater. And, accordingly, of the two numbers, the one that is depicted on the horizontal coordinate line to the left is less. Example If we consider a vertical coordinate line, then in the formulated comparison rule it is necessary to replace the word "to the right" with "above", and the word "to the left" - for "below". Let us formulate a rule for comparing numbers on a vertical coordinate line.

Of the two numbers, the larger one is the one shown on the vertical coordinate line above. And, accordingly, of the two numbers, the one that is depicted on the vertical coordinate line below is less. I would like to immediately clarify that all positive numbers are greater than zero, and all negative numbers are less than zero. Any negative number is less than a positive number. In general, it is very convenient to compare numbers using the concept of “modulus of a number”. Since the larger of the two positive numbers on the coordinate line is depicted to the right, i.e. farther from the origin, then this number has a larger module. Remember, of two positive numbers, the one whose modulus is greater is greater. Since the larger of the two negative numbers on the coordinate line is depicted to the right, i.e. closer to the origin, then this number has a smaller module. Remember, of two negative numbers, the one whose modulus is less is greater. To learn how to easily compare negative numbers without using a coordinate line, let's reason. When is it warmer - at -25° or at -5°?

Numbers can be compared in various ways:

1) based on the order of naming numbers when counting: the number named earlier will be smaller (this follows from the ordering property of the set of natural numbers);

2) based on the process of counting: three and one will be four, so three is less than four;

3) based on quantitative models of compared numbers:

To fix the comparison process, a comparison sign is introduced.

It should be remembered that the comparison sign is one, but it is read differently depending on the desire of the reader. In accordance with the tradition of reading texts in European scripts from left to right, the first reading of the comparison sign is usually carried out from left to right: 3< 4 (три меньше четырех), но эту же запись при желании можно прочитать и справа налево (четыре больше трех), причем для этого не надо переставлять элементы записи таким образом: 4 >3. Do not instill in the child the wrong idea that there are two signs

comparisons, one labeled "less" and one labeled "greater" because this forms an inflexible, convergent perceptual pattern that will then get in the way of a child in high school when dealing with inequalities. It is useful to offer the child to read each entry of this kind in the two ways given above.

7. Number 10

Ten units is ten.

Ten is the second counting unit in the decimal number system (the decimal number system has the number ten as its base). Ten tens form the next counting unit - a hundred.

The number 10 is the number that completes the first ten.

The number 10 is the first two-digit number in the series of natural numbers.

The number 10 is the first whole ten that the child is introduced to.

In the future, based on the concept of ten, the child gets acquainted with the bit and decimal composition of two-digit and multi-digit numbers. In order not to go into terminological difficulties and not to overload the material with the early introduction of the concept of "digit", it is convenient to fully familiarize yourself with the ten and its notation using numbers on the object model.

When introducing a child to the number 10 (the first two-digit number and the first whole ten), it is very important to consider it from different positions: both as a new number in a series (following nine and therefore subject to the general principle of constructing a set of natural numbers), and as the first number, in entries which used two characters; and as a new counting unit (ten), for which they use a bunch of ten sticks as a counting unit: one ten; two tens, three tens...

You should not rush to enter the standard names of these dozens (twenty, thirty, etc.), it is more useful to use bundles of 10 sticks for counting for one or two lessons in order to form an idea of ​​\u200b\u200bten as a counting unit.

Zero in such an analogy symbolizes a "bundle" covering a ring. To assimilate this analogy, it is useful to immediately offer children tasks of the reverse type: show on sticks the number 30 (three bundles), the number 40 (four bundles), etc.

Counting by tens (10,20,30,40,50,60,70,80,90) is a process "technically" similar to counting by ones within 10. It is useful to teach a child to count and count tens in the same way as he did with ones. In the future, this skill will help the child to more easily master the computational techniques of addition and subtraction within 100.

When introducing a child to the numbering of single digits, we recommend that the teacher use the following types of tasks:

1) on the method of forming each next number by adding one to the previous one:

How to get 4 from number 3? (Add one to three.)

2) to determine the place of a number in a row:

What is the number behind the 5? (Behind the number 4.) Where is the place of the number 8? (Between numbers 7 and 9.)

3) to compare both two neighboring and non-neighboring numbers:

Compare numbers: 5...4 7.„2

4) on the composition of the number:

5) to memorize the reverse sequence of numbers in the series:

Comparison of natural numbers among themselves is the topic of this article. Let us analyze the comparison of two natural numbers and study the concept of equal and unequal natural numbers. Let's find out the larger and smaller of the two numbers with examples. Let's talk about the natural series of numbers and their comparison. The results of comparisons of three or more numbers will be shown.

Comparison of natural numbers

Let's look at this with an example. When there is a flock of 7 birds on a tree, and 5 dozen birds on another, the flocks are considered different, since they do not resemble each other. From this we can conclude that this dissimilarity is a comparison.

When comparing natural numbers, such a check for similarity is carried out.

• Equality. This case is possible when the numbers are equal.
• Inequality. When the numbers are not equal.

When we get an inequality, it means that one of these numbers is greater or less than the other, which increases the range of use of natural numbers.

Consider the definitions of equal and unequal numbers. Let's see how this is determined.

Equal and unequal natural numbers

Consider the definition of equal and unequal numbers.

Definition 1

In the case when the entries of two natural numbers are the same, they are considered equal between themselves. When entries have differences, then these numbers unequal.

Based on the definition, the numbers 402 and 402 are considered equal, as well as 7 and 7, since they are written the same way. But numbers such as 55283 and 505283 are not equal, since their records are not the same and have differences, 582 and 285 are different, since they differ in records.

Such equalities have a short notation. Equal sign "=" and not equal sign "≠" . Their location is directly between the numbers, for example, 47 = 47. Means that these numbers are equal. Or 56 ≠ 65. This means that the numbers are different and differ in writing.

In a record that has two natural numbers with the sign “=”, they are called equality. They are either true or false. For example, 45 = 45 , which is considered a true equality. If 465 = 455 , which is considered an invalid equality.

Comparison of single-digit natural numbers

Single-digit numbers are considered a series from 1 to 9. Of the two single-digit numbers written down, the one to the left is considered less, and the one to the right is greater.

Numbers can be more or less than several at the same time. For example, if 1 is less than 2 , then so is 8 , and 5 is less than all numbers starting from 6 . This applies to every number in the given series from 1 to 9.

The shorthand for the less-than sign is "< », а знака больше – « >". Their location between two compared numbers. When there is an entry where 3 > 1 , it means that 3 is greater than one if the entry is of the form 6< 9 , тогда 6 меньше 9 .

Definition 3

If the entry contains two natural numbers with signs "< » и « >', then it is called inequality. Inequalities can be true or false.

Entry 4< 7 – верная, а 3 >9 is incorrect.

Comparison of single-valued and multi-valued natural numbers

If we accept as a rule that all single-digit numbers are less than two-digit ones, then we get:

5 < 10 , 6 < 42 , 303 >3 , 32043 > 7 . This entry is believed to be correct. Here is an example of an incorrect record of inequality: 3 > 11 , 733< 5 и 2 > 1 020 .

Consider comparisons of multi-digit numbers.

Comparison of multivalued natural numbers

Consider a comparison of two unequal multi-valued natural numbers with an equal number of signs. First, you should repeat the section that studies the digits of a natural number and the value of the digit.

In this case, a bitwise comparison is performed, that is, from left to right. The smaller number is the one that has the smaller value of the corresponding digit and vice versa.

To solve the example, you need to understand that 0 is always less than any natural number and that it is equal to itself. The number zero belongs to the category of natural numbers.

Example 1

Compare the numbers 35 and 63.

Solution

It can be visually seen that the numbers are unequal, since they differ in writing. First, let's compare the tens of a given number. It can be seen that 3< 6 , а это означает, что заданные числа 35 и 63 не равны, а первое число меньше второго. Это решение записывается так: 35 < 63 .

Answer: 35 < 63 .

Example 2

Make comparison given numbers 301 and 308.

Solution

It is visually obvious that the numbers are not equal, as their notation is different. They are both three-digit, which means that the comparison must begin with hundreds, after which a dozen and then units. We get that 3 = 3 , then 0 = 0 . Units differ from each other, we have: 1< 8 . Отсюда имеем, что 301 < 308 .

Answer: 301 < 308 .

Multivalued natural numbers are compared in a different way. A large number is one that has a smaller number of characters and vice versa.

Example 3

Compare the given natural numbers 40391 and 92248712 .

Solution

Visually note that the number 40391 has 5 digits, and 92248712 has 8 digits.

This means that the number of characters equal to 5 is less than 8 . Hence we have that the first number is less than the second.

Answer: 40 391 < 92 248 712 .

Example 4

Reveal more natural number from the given: 50 933 387 or 10 000 011 348?

Solution

Note that the first number 50 933 387 has 8 digits, and the second 10 000 011 348 has 11 digits. It follows that 8 is less than 11 . So the number 50 933 387 is less than 10 000 011 348 .

Answer: 10000011348 > 50933387 .

Example 5

Compare multi-valued natural given numbers: 9 876 545 678 and 987 654 567 811 .

Solution

Consider that the first number has 10 digits, the second - 12 . We conclude that the second number is greater than the first, since 10 is less than 12. The comparison of 10 and 12 is done bit by bit. We get that 1 = 1 , but 0 is less than 2 . Hence we get that 0< 2 . Это говорит о том, что 10 < 12 .

Answer: 9 876 545 678 < 987 654 567 811 .

Natural series of numbers, numbering, counting

Let's record natural numbers so that the next one is greater than the previous one. Let's write this series: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . This sequence continues with two-digit numbers: 1 , 2 , . . , 10 , 11 , . . .99 A series with three-digit numbers has the form 1 , 2 , . . , 10 , 11 , . . , 99 , 100 , 101 , . . .999

This entry continues ad infinitum. Such an infinite sequence of numbers is called natural side by side numbers.

There is another process - the account. During the counting, the numbers are called one after the other, that is, in the way they are fixed in a row. This process is applicable to determine the number of items.

If available certain number items, but we need to know the quantity, use the account. It is produced starting from one. If during the recalculation we shift objects into a heap, then it can be called a natural series of numbers. The last item will be the number of their number. When the process is over, we know their number, that is, the items are recounted.

During counting, the natural number that is found earlier and called earlier is smaller. The application of numbering is used to specifically identify an item, that is, by assigning it a specific number. For example, we have a certain number of items. On each of them we fix their serial number. This is how the numbering is done. It is used to distinguish the same objects.

First you need to repeat the definition of the coordinate ray.

When viewed from left to right, we see dashes that represent a specific sequence of numbers, ranging from 0 to infinity. These strokes are called dots. The points to the left are smaller than the points to the right. It follows that the point with the smaller coordinate on the coordinate ray is located to the left of the point with the larger coordinate.

Consider the example of two numbers 2 and 6 . Let's put two points A and B on the coordinate beam, placing them on the values ​​2 and 6.

It follows that point A is to the left, which means that it is less than point B, since the location of point B is to the right of point A. We write it as an inequality: 2< 6 . Иначе можно озвучить, как «точка В лежит правее точки А, значит число 6 на координатном луче more number 2".

The smallest and largest natural number

It is believed that 1 is the smallest natural number from the set of all natural numbers. All numbers located to the right of it are considered greater than the previous one. This series is infinite, so there is no largest number from this set of numbers.

We can select the largest number from a series of single-digit natural numbers. It is equal to 9 . This is easy to do since the number of single digits is limited. Similarly, we find the largest number from the set of two-digit numbers. It equals 99 . In the same way, a larger number of three-digit and so on numbers are searched.

When comparing a pair of numbers, we note that it is possible to search for a smaller and larger number. If 4 is the smallest number, then 40 is the largest of the given series: 4 , 6 , 34 , 34 , 67 , 18 , 40 .

Double, triple inequalities

It is known that 5< 12 , а 12 < 35 . Два неравенства можно представить в виде одного двойного. Такая запись имеет вид: 5 < 12 < 35 . Отсюда видно, что при записи двойного неравенства получаем три неравенства, которые запишем 5 < 12 , 12 < 35 и 5 < 35 .

Recording in the form of a double inequality is applicable for comparing three numbers. When it is necessary to compare 76 , 512 and 10 , we get three inequalities 76< 512 , 76 >10, 512 > 10. They, in turn, can be written as one, but double 10< 76 < 512 .

In the same way, triple, quadruple, and so on inequalities are fulfilled.

If it is known that 5< 16 , 16 < 305 , 305 < 1 001 , 1 001 < 3 214 , тогда запись может быть представлена в виде 5 < 16 < 305 < 1 001 < 3 214 .

It is necessary to be careful when compiling double inequalities, since it is possible to produce it incorrectly, which will entail an incorrect solution of the problem.

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After received full view about integers, we can talk about their comparison. To do this, it turns out which numbers are equal and unequal. We will understand the rules, thanks to which we find out which of the two unequal ones is more or less. This rule is based on comparing natural numbers. Comparison of three or more integers, finding the smallest and largest integer from a given set will be considered.

Equal and unequal integers

Comparing two numbers results in them being either equal or not equal . Let's look at the definitions.

Definition 1

Two integers are called equal, when their record is exactly the same. Otherwise they are considered unequal.

A separate place for discussion has 0 and - 0 . The opposite number - 0 is 0 , in this case these two numbers are equivalent.

The definition will help compare the given two numbers. Take, for example, the numbers - 95 and - 95. Their record completely coincides, that is, they are considered equal. If you take the numbers 45 and - 6897 , then you can visually see that they are different and are not considered equal. They have different signs.

If the numbers are equal, this is written using the “=” sign. Its location goes between the numbers. If we take the numbers - 45 and - 45, then they are equal. The entry becomes - 45 = - 45 . In case the numbers are unequal, then the sign "≠" is used. Consider the example of two numbers: 57 and - 69. These numbers are integers, but not equal, since the notation is different from each other.

When comparing numbers, the number modulus rule is used .

Definition 2

If two numbers have the same signs and their moduli are equal, then these two numbers considered equal. Otherwise they are called not equal.

Consider this definition as an example.

Example 1

For example, given two numbers - 709 and - 712. Find out if they are equal.

It can be seen that the numbers have the same sign, but this does not mean that they are equal. The modulus of the number is used for comparison. The modulo of the first number is less than the second. They are not equal either modulo or without it.

So we conclude that the numbers are not equal.

Let's consider another example.

Example 2

If two numbers are taken 11 and 11 . They are both equal. Modulus is also the same numbers. These natural numbers can be considered equal, since their entries coincide completely.

If we get unequal numbers, then it is necessary to clarify which of them is less and which is more.

Comparing Arbitrary Integers to Zero

In the previous paragraph, it was noted that zero is equal to itself even with a minus sign. In this case, the equalities 0 = 0 and 0 = - 0 are equivalent and valid. When comparing natural numbers, we have that all natural numbers are greater than zero. All positive integers are natural, and therefore greater than 0.

When comparing negative numbers with zero, the situation is different. All numbers less than zero are considered negative. From this we conclude that any negative number is less than zero, zero is equal to zero, and any positive integer is greater than zero. The essence of the rule is that zero is greater than negative numbers, but less than all positive ones.

For example, the numbers 4 , 57666 , 677848 are greater than 0 because they are positive. It follows that zero is less than the indicated numbers, since they are signed with + .

When comparing negative numbers, things are different. The number - 1 is an integer and less than 0 because it has a minus sign. So -50 is also less than zero. But zero is greater than all numbers with a minus sign.

Certain notations are accepted for writing using less or more signs, that is< и >. An entry such as - 24< 0 имеет значение, что - 24 меньше нуля. Если необходимо записать, что одно число больше, чем другое, применяют знак >, for example, 45 > 0 .

Comparing positive integers

All positive integers are natural. This means that the comparison of positive numbers is similar to the comparison of natural numbers.

Example 3

If we look at the example of comparing 34001 and 5999. Visually we see that the first number has 5 digits, and the second 4 . It follows that 5 is greater than 4 , that is, 34001 is greater than 5999 .

Answer: 34001 > 5999 .

Let's consider one more example.

Example 4

If there are positive numbers 357 and 359 , then it is clear that they are not equal, although both are three-digit. A bitwise comparison is made. First hundreds, then tens, then units.

We get that the number 357 is less than 359 .

Answer: 357< 359 .

Comparison of integer negative and positive numbers

Any negative integer is less than a positive integer and vice versa.

Let's compare some numbers and look at an example.

Compare given numbers - 45 and 23 . We see that 23 is a positive number, and 45 is negative. Note that 23 is greater than 45

If we compare - 1 and 511 , then it is visually clear that - 1 is less, since it has a minus sign, and 511 has a + sign.

Comparing negative integers

Consider the comparison rule:

Definition 5

Of two negative numbers, the smaller is the one whose modulus is greater and vice versa.

Let's look at an example.

Example 5

If you compare - 34 and - 67, then you should compare them modulo.

We get that 34 is less than 67 . Then the modulo - 67 is greater than the modulus - 34, which means that the number - 34 is greater than the number - 67.

Answer: - 34 > - 67 .

Consider integers located on the coordinate line.

From the rules discussed above, we obtain that on the horizontal coordinate line, the points that correspond to large integers, that is, lie to the right of those that correspond to smaller ones.

From the numbers - 1 and - 6 it is clear that - 6 lies to the left, and therefore is less than - 1. Point 2 is located to the right - 7, which means it is larger.

The starting point is zero. It is greater than all negative and less than all positive. The same is true for points on a coordinate line.

Largest negative and smallest positive integer

In the previous paragraphs, the comparison of two integers was discussed in detail. In this paragraph, we will talk about comparing three or more numbers, consider situations.

When comparing three or more numbers, all sorts of pairs are made to begin with. For example, consider for the numbers 7 , 17 , 0 and − 2 . It is necessary to compare them in pairs, that is, the record will take the form 7< 17 , 7 >0 , 7 > − 2 , 17 > 0 , 17 > − 2 and 0 > − 2 . The results can be combined into a chain of inequalities. The number is written in ascending order. AT this case the chain will look like − 2< 0 < 7 < 17 .

When several numbers are compared, the definition of the largest and smallest value of the number appears.

Definition 6

The number of the given set is considered least if it is less than any other of the given numbers in the set.

Definition 7

The number of the given set is greatest if it is greater than any other of the given numbers in the set.

If the set consists of 6 integers, then we write it like this: − 4 , − 81 , − 4 , 17 , 0 and 17 . It follows from this that − 81< − 4 = − 4 < 0 < 17 = 17 . Видно, что - 81 – наименьшее число из данного множества, а 17 – наибольшее. Это значит, что эти числа наибольшее и наименьшее только в заданном множестве.

All numbers in the set must be written in ascending order. The chain can be infinite, as in this case: … , − 5 , − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , 4 , 5 , … . This series will be written as...< − 5 < − 4 < − 3 < − 2 < − 1 < 0 < 1 < 2 < 3 < 4 < 5 < … .

Obviously, the set of integers is huge and infinite, so it is impossible to specify the smallest or largest number. This can only be done in a given set of numbers. The number located to the right on the coordinate line is always considered greater than the one to the left.

The set of positive numbers has the smallest natural number, which is 1 . Zero is considered the smallest non-negative number. All numbers to the left of it are negative and less than 0 .

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