How to find a natural number. Material on mathematics "Numbers. Natural numbers"

Integers very familiar and natural to us. And this is not surprising, since acquaintance with them begins from the first years of our life at an intuitive level.

The information in this article creates a basic understanding of natural numbers, reveals their purpose, instills the skills of writing and reading natural numbers. For better assimilation of the material, the necessary examples and illustrations are given.

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Natural numbers are a general representation.

The following opinion is not devoid of sound logic: the appearance of the problem of counting objects (first, second, third object, etc.) and the problem of indicating the number of objects (one, two, three objects, etc.) led to the creation of a tool for its solution, this tool was integers.

This proposal shows main purpose of natural numbers- carry information about the number of any items or the serial number of a given item in the considered set of items.

In order for a person to use natural numbers, they must be accessible in some way, both for perception and for reproduction. If you sound each natural number, then it will become perceptible by ear, and if you depict a natural number, then it can be seen. These are the most natural ways, allowing to convey and perceive natural numbers.

So let's start acquiring the skills of depicting (writing) and the skills of voicing (reading) natural numbers, while learning their meaning.

Decimal notation for a natural number.

First, we should decide on what we will build on when writing natural numbers.

Let's memorize the images of the following characters (we show them separated by commas): 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . The images shown are a record of the so-called numbers. Let's agree right away not to flip, tilt, or otherwise distort the numbers when writing.

Now we agree that only the indicated digits can be present in the notation of any natural number and no other symbols can be present. We also agree that the digits in the notation of a natural number have the same height, are arranged in a line one after another (with almost no indents), and on the left there is a digit that is different from the digit 0 .

Here are some examples correct notation natural numbers: 604 , 777 277 , 81 , 4 444 , 1 001 902 203, 5 , 900 000 (note: the indents between the numbers are not always the same, more on this will be discussed when reviewing). From the above examples, it can be seen that a natural number does not necessarily contain all of the digits 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ; some or all of the digits involved in writing a natural number may be repeated.

Entries 014 , 0005 , 0 , 0209 are not records of natural numbers, since there is a digit on the left 0 .

The record of a natural number, performed taking into account all the requirements described in this paragraph, is called decimal notation of a natural number.

Further we will not distinguish between natural numbers and their notation. Let us clarify this: further in the text, phrases like “given a natural number 582 ", which will mean that a natural number is given, the notation of which has the form 582 .

Natural numbers in the sense of the number of objects.

It's time to deal with the quantitative meaning that the recorded natural number carries. The meaning of natural numbers in terms of numbering objects is considered in the article comparison of natural numbers.

Let's start with natural numbers, the entries of which coincide with the entries of the digits, that is, with the numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 And 9 .

Imagine that we opened our eyes and saw some object, for example, like this. In this case, we can write what we see 1 subject. Natural number 1 reads like " one"(declension of the numeral "one", as well as other numerals, we will give in paragraph), for the number 1 adopted another name - " unit».

However, the term "unit" is multi-valued; in addition to the natural number 1 , are called something that is considered as a whole. For example, any one item from their set can be called a unit. For example, any apple out of many apples is one, any flock of birds out of many flocks of birds is also one, and so on.

Now we open our eyes and see: That is, we see one object and another object. In this case, we can write what we see 2 subject. Natural number 2 , reads like " two».

Likewise, - 3 subject (read " three» subject), - 4 (« four"") of the subject, - 5 (« five»), - 6 (« six»), - 7 (« seven»), - 8 (« eight»), - 9 (« nine”) items.

So, from the considered position, the natural numbers 1 , 2 , 3 , …, 9 indicate number items.

A number whose notation matches the notation of a digit 0 , called " zero". The number zero is NOT a natural number, however, it is usually considered together with natural numbers. Remember: zero means the absence of something. For example, zero items is not a single item.

In the following paragraphs of the article, we will continue to reveal the meaning of natural numbers in terms of indicating the quantity.

single digit natural numbers.

Obviously, the record of each of the natural numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 consists of one sign - one digit.

Definition.

Single digit natural numbers are natural numbers, the record of which consists of one sign - one digit.

Let's list all single-digit natural numbers: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . There are nine single-digit natural numbers.

Two-digit and three-digit natural numbers.

First, we give a definition of two-digit natural numbers.

Definition.

Two-digit natural numbers- these are natural numbers, the record of which is two characters - two digits (different or the same).

For example, a natural number 45 - two-digit, numbers 10 , 77 , 82 also two-digit 5 490 , 832 , 90 037 - not double digit.

Let's figure out what meaning two-digit numbers carry, while we will start from the quantitative meaning of single-digit natural numbers already known to us.

First, let's introduce the concept ten.

Let's imagine such a situation - we opened our eyes and saw a set consisting of nine objects and one more object. In this case, one speaks of 1 ten (one dozen) items. If one considers together one ten and one more ten, then one speaks of 2 tens (two tens). If we add another ten to two tens, we will have three tens. Continuing this process, we will get four tens, five tens, six tens, seven tens, eight tens, and finally nine tens.

Now we can move on to the essence of two-digit natural numbers.

To do this, consider a two-digit number as two single-digit numbers - one is on the left in the notation of a two-digit number, the other is on the right. The number on the left indicates the number of tens, and the number on the right indicates the number of ones. Moreover, if there is a digit on the right in the record of a two-digit number 0 , then this means the absence of units. This is the whole point of two-digit natural numbers in terms of indicating the amount.

For example, a two-digit natural number 72 corresponds 7 dozens and 2 units (that is, 72 apples is a set of seven dozen apples and two more apples), and the number 30 answers 3 dozens and 0 there are no units, that is, units that are not united in tens.

Let's answer the question: "How many two-digit natural numbers exist"? Answer: them 90 .

We turn to the definition of three-digit natural numbers.

Definition.

Natural numbers whose notation consists of 3 signs - 3 digits (different or repeated) are called three-digit.

Examples of natural three-digit numbers are 372 , 990 , 717 , 222 . Integers 7 390 , 10 011 , 987 654 321 234 567 are not three digits.

To understand the meaning inherent in three-digit natural numbers, we need the concept hundreds.

A set of ten tens is 1 one hundred (one hundred). Hundred and hundred is 2 hundreds. Two hundred and another hundred is three hundred. And so on, we have four hundred, five hundred, six hundred, seven hundred, eight hundred, and finally nine hundred.

Now let's look at a three-digit natural number as three single-digit natural numbers, going one after another from right to left in the notation of a three-digit natural number. The number on the right indicates the number of units, the next number indicates the number of tens, the next number the number of hundreds. Numbers 0 in the record of a three-digit number means the absence of tens and (or) ones.

Thus, a three-digit natural number 812 corresponds 8 hundreds 1 top ten and 2 units; number 305 - three hundred 0 tens, that is, tens not combined into hundreds, no) and 5 units; number 470 - four hundred and seven tens (there are no units that are not combined into tens); number 500 - five hundred (tens not combined into hundreds, and units not combined into tens, no).

Similarly, one can define four-digit, five-digit, six-digit, and so on. natural numbers.

Multivalued natural numbers.

So, we turn to the definition of multi-valued natural numbers.

Definition.

Multivalued natural numbers- these are natural numbers, the record of which consists of two or three or four, etc. signs. In other words, multi-digit natural numbers are two-digit, three-digit, four-digit, etc. numbers.

Let's say right away that the set consisting of ten hundred is one thousand, a thousand thousand is one million, a thousand million is one billion, a thousand billion is one trillion. A thousand trillion, a thousand thousand trillion, and so on can also be given their own names, but there is no particular need for this.

So what is the meaning behind multi-valued natural numbers?

Let's look at a multi-digit natural number as single-digit natural numbers following one after the other from right to left. The number on the right indicates the number of units, the next number is the number of tens, the next is the number of hundreds, then the number of thousands, the next is the number of tens of thousands, the next is hundreds of thousands, the next is the number of millions, the next is the number of tens of millions, the next is hundreds of millions, the next - the number of billions, then - the number of tens of billions, then - hundreds of billions, then - trillions, then - tens of trillions, then - hundreds of trillions, and so on.

For example, a multi-digit natural number 7 580 521 corresponds 1 unit, 2 dozens, 5 hundreds 0 thousands 8 tens of thousands 5 hundreds of thousands and 7 millions.

Thus, we learned to group units into tens, tens into hundreds, hundreds into thousands, thousands into tens of thousands, and so on, and found out that the numbers in the record of a multi-digit natural number indicate the corresponding number of the above groups.

Reading natural numbers, classes.

We have already mentioned how one-digit natural numbers are read. Let's learn the contents of the following tables by heart.

And how are the other two-digit numbers read?

Let's explain with an example. Reading a natural number 74 . As we found out above, this number corresponds to 7 dozens and 4 units, that is, 70 And 4 . We turn to the tables just written, and the number 74 we read as: “Seventy-four” (we do not pronounce the union “and”). If you want to read a number 74 in the sentence: "No 74 apples" ( Genitive), then it will sound like this: "There are no seventy-four apples." Another example. Number 88 - this 80 And 8 , therefore, we read: "Eighty-eight." And here is an example of a sentence: "He is thinking about eighty-eight rubles."

Let's move on to reading three-digit natural numbers.

To do this, we will have to learn a few more new words.

It remains to show how the remaining three-digit natural numbers are read. In this case, we will use the already acquired skills in reading single-digit and double-digit numbers.

Let's take an example. Let's read the number 107 . This number corresponds 1 hundred and 7 units, that is, 100 And 7 . Turning to the tables, we read: "One hundred and seven." Now let's say the number 217 . This number is 200 And 17 , therefore, we read: "Two hundred and seventeen." Likewise, 888 - this 800 (eight hundred) and 88 (eighty-eight), we read: "Eight hundred and eighty-eight."

We turn to reading multi-digit numbers.

For reading, the record of a multi-digit natural number is divided, starting from the right, into groups of three digits, while in the leftmost such group there may be either 1 , or 2 , or 3 numbers. These groups are called classes. The class on the right is called unit class. The next class (from right to left) is called class of thousands, the next class is class of millions, next - class of billions, then goes trillion class. You can give the names of the following classes, but natural numbers, the record of which consists of 16 , 17 , 18 etc. signs are usually not read, since they are very difficult to perceive by ear.

Look at examples of splitting multi-digit numbers into classes (for clarity, classes are separated from each other by a small indent): 489 002 , 10 000 501 , 1 789 090 221 214 .

Let's put the recorded natural numbers in a table, according to which it is easy to learn how to read them.

To read a natural number, we call from left to right the numbers that make it up by class and add the name of the class. At the same time, we do not pronounce the name of the class of units, and also skip those classes that make up three digits 0 . If the class record has a digit on the left 0 or two digits 0 , then ignore these numbers 0 and read the number obtained by discarding these digits 0 . For instance, 002 read as "two", and 025 - like "twenty-five".

Let's read the number 489 002 according to the given rules.

We read from left to right,

• read the number 489 , representing the class of thousands, is "four hundred and eighty-nine";
• add the name of the class, we get "four hundred eighty-nine thousand";
• further in the class of units we see 002 , zeros are on the left, we ignore them, therefore 002 read as "two";
• the unit class name need not be added;
• as a result we have 489 002 - four hundred and eighty-nine thousand two.

Let's start reading the number 10 000 501 .

• On the left in the class of millions we see the number 10 , we read "ten";
• add the name of the class, we have "ten million";
• next we see the record 000 in the thousands class, since all three digits are digits 0 , then we skip this class and move on to the next one;
• unit class represents number 501 , which we read "five hundred and one";
• thus, 10 000 501 ten million five hundred and one.

Let's do it without detailed explanations: 1 789 090 221 214 - "one trillion seven hundred eighty-nine billion ninety million two hundred twenty-one thousand two hundred fourteen."

So, the basis of the skill of reading multi-digit natural numbers is the ability to break multi-digit numbers into classes, knowledge of the names of classes and the ability to read three-digit numbers.

The digits of a natural number, the value of the digit.

In writing a natural number, the value of each digit depends on its position. For example, a natural number 539 corresponds 5 hundreds 3 dozens and 9 units, hence the figure 5 in the number entry 539 defines the number of hundreds, a digit 3 is the number of tens, and the digit 9 - number of units. It is said that the number 9 stands in units digit and number 9 is an unit digit value, number 3 stands in tens place and number 3 is an tens place value, and the number 5 - in hundreds place and number 5 is an hundreds place value.

In this way, discharge- this is, on the one hand, the position of the digit in the notation of a natural number, and on the other hand, the value of this digit, determined by its position.

The ranks have been given names. If you look at the numbers in the record of a natural number from right to left, then the following digits will correspond to them: units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, and so on.

The names of the categories are convenient to remember when they are presented in the form of a table. Let's write a table containing the names of 15 digits.

Note that the number of digits of a given natural number is equal to the number of characters involved in writing this number. Thus, the recorded table contains the names of the digits of all natural numbers, the record of which contains up to 15 characters. The following digits also have their own names, but they are very rarely used, so it makes no sense to mention them.

Using the table of digits, it is convenient to determine the digits of a given natural number. To do this, you need to write this natural number into this table so that there is one digit in each digit, and the rightmost digit is in the units digit.

Let's take an example. Let's write a natural number 67 922 003 942 in the table, and the digits and the values ​​​​of these digits will become clearly visible.

In the record of this number, the digit 2 stands in the units place, digit 4 - in the tens place, digit 9 - in the hundreds place, etc. Pay attention to the numbers 0 , which are in the digits of tens of thousands and hundreds of thousands. Numbers 0 in these digits means the absence of units of these digits.

We should also mention the so-called lowest (lowest) and highest (highest) category of a multivalued natural number. Lower (junior) rank any multi-valued natural number is the units digit. The highest (highest) digit of a natural number is the digit corresponding to the rightmost digit in the record of this number. For example, the least significant digit of the natural number 23004 is the units digit, and the highest digit is the tens of thousands digit. If in the notation of a natural number we move by digits from left to right, then each next digit lower (younger) the previous one. For example, the digit of thousands is less than the digit of tens of thousands, especially the digit of thousands is less than the digit of hundreds of thousands, millions, tens of millions, etc. If, in the notation of a natural number, we move in digits from right to left, then each next digit higher (older) the previous one. For example, the hundreds digit is older than the tens digit, and even more so, it is older than the ones digit.

In some cases (for example, when performing addition or subtraction), not the natural number itself is used, but the sum of the bit terms of this natural number.

Briefly about the decimal number system.

So, we got acquainted with natural numbers, with the meaning inherent in them, and the way to write natural numbers using ten digits.

In general, the method of writing numbers using signs is called number system. The value of a digit in a number entry may or may not depend on its position. Number systems in which the value of a digit in a number entry depends on its position are called positional.

Thus, the natural numbers we have considered and the method of writing them indicate that we are using a positional number system. It should be noted that a special place in this number system has the number 10 . Indeed, the score is kept in tens: ten units are combined into a ten, ten tens are combined into a hundred, ten hundreds into a thousand, and so on. Number 10 called basis given number system, and the number system itself is called decimal.

In addition to the decimal number system, there are others, for example, in computer science, a binary positional number system is used, and we encounter a sexagesimal system when we are talking about the measurement of time.

Bibliography.

• Maths. Any textbooks for 5 classes of educational institutions.

Integers

Natural numbers are those numbers that are used to count various items or in order to indicate the serial number of any object among similar or homogeneous.

Natural numbers can be written using the first ten digits:

To write simple natural numbers, it is customary to use a positional decimal calculus, where the value of any digit is determined by its place in the record.

Natural numbers are the simplest numbers that we often use in Everyday life. With the help of these numbers, we make calculations, count objects, determine their quantity, order and number.

With natural numbers, we begin to get acquainted with the very early childhood, so they are familiar and natural for each of us.

General idea of ​​natural numbers

Natural numbers are designed to carry information about the number of objects, their serial number and the set of objects.

A person uses natural numbers, since they are available to him both at the level of perception and at the level of reproduction. When voicing any natural number, we can easily catch it by ear, and having depicted a natural number, we see it.

All natural numbers are arranged in ascending order and form a number series starting with the smallest natural number, which is one.

If we have decided on the smallest natural number, then it will be more difficult with the largest, since such a number does not exist because the series of natural numbers is infinite.

When we add one to a natural number, we end up with the number that follows the given number.

A number such as 0 is not a natural number, but only serves to denote the number "zero" and means "none". 0 means the absence of numbers of units of this series in the decimal notation.

All natural numbers are denoted by the capital Latin letter N.

Historical reference for the designation of natural numbers

In ancient times, people did not yet know what a number is and how to count the number of objects. But even then there was a need for an account, and a person figured out how to count the caught fish, harvested berries etc.

A little bit later, ancient man came to the conclusion that it is easier to write down the amount he needs. For these purposes primitive people they began to use pebbles, and then sticks, which were preserved in Roman numerals.

The next moment in the development of the calculus system was the use of letters of the alphabet in the notation of some numbers.

The first systems of calculation include the decimal Indian system and the sexagesimal Babylonian.

The modern system of calculus, although called Arabic, is, in fact, one of the variants of the Indian one. True, in its system of calculation there is no number zero, but the Arabs added it, and the system acquired its current form.

Decimal system

We have already met natural numbers and learned how to write them using ten digits. You also already know that writing numbers using signs is called a number system.

The value of a digit in a number entry depends on its position and is called positional. That is, when writing natural numbers, we use the positional calculus.

This system is based on bit depth and decimal. In the decimal system, the basis for its construction will be the numbers from 0 to 9.

A special place in such a system is given to the number 10, since, basically, the account is kept in tens.

Table of classes and categories:

So, for example, 10 units are combined into tens, then into hundreds, thousands, and the like. Therefore, the number 10 is the base of the calculus system and is called the decimal calculus system.

Natural numbers are one of the oldest mathematical concepts.

In the distant past, people did not know numbers, and when they needed to count objects (animals, fish, etc.), they did it differently than we do now.

The number of objects was compared with parts of the body, for example, with the fingers on the hand, and they said: "I have as many nuts as there are fingers on the hand."

Over time, people realized that five nuts, five goats and five hares have common property- their number is five.

Remember!

Integers are numbers, starting with 1, obtained when counting objects.

1, 2, 3, 4, 5…

smallest natural number — 1 .

largest natural number does not exist.

When counting, the number zero is not used. Therefore, zero is not considered a natural number.

People learned to write numbers much later than to count. First of all, they began to represent the unit with one stick, then with two sticks - the number 2, with three - the number 3.

| — 1, || — 2, ||| — 3, ||||| — 5 …

Then special signs appeared for designating numbers - the forerunners of modern numbers. The numbers we use to write numbers originated in India about 1,500 years ago. The Arabs brought them to Europe, so they are called Arabic numerals.

There are ten digits in total: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These digits can be used to write any natural number.

Remember!

natural series is the sequence of all natural numbers:

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

In the natural series, each number is greater than the previous one by 1.

The natural series is infinite, there is no largest natural number in it.

The counting system we use is called decimal positional.

Decimal because 10 units of each digit form 1 unit of the most significant digit. Positional because the value of a digit depends on its place in the notation of a number, that is, on the digit in which it is written.

Important!

The classes following the billion are named according to the Latin names of numbers. Each next unit contains a thousand previous ones.

• 1,000 billion = 1,000,000,000,000 = 1 trillion (“three” is Latin for “three”)
• 1,000 trillion = 1,000,000,000,000,000 = 1 quadrillion (“quadra” is Latin for “four”)
• 1,000 quadrillion = 1,000,000,000,000,000,000 = 1 quintillion (“quinta” is Latin for “five”)

However, physicists have found a number that surpasses the number of all atoms (the smallest particles of matter) in the entire universe.

This number has a special name - googol. A googol is a number that has 100 zeros.

In mathematics, there are several different sets of numbers: real, complex, integer, rational, irrational, ... In our Everyday life we most often use natural numbers, as we encounter them when counting and when searching, indicating the number of objects.

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What numbers are called natural

From ten digits, you can write down absolutely any existing sum of classes and ranks. Natural values ​​are those which are used:

• When counting any items (first, second, third, ... fifth, ... tenth).
• When indicating the number of items (one, two, three ...)

N values ​​are always integer and positive. There is no largest N, since the set of integer values ​​is not limited.

Attention! Natural numbers are obtained by counting objects or by designating their quantity.

Absolutely any number can be decomposed and presented as bit terms, for example: 8.346.809=8 million+346 thousand+809 units.

Set N

The set N is in the set real, integer and positive. In the set diagram, they would be in each other, since the set of naturals is part of them.

The set of natural numbers is denoted by the letter N. This set has a beginning but no end.

There is also an extended set N, where zero is included.

smallest natural number

Most math schools the smallest value N counted as a unit, since the absence of objects is considered empty.

But in foreign mathematics schools, for example in French, is considered natural. The presence of zero in the series facilitates the proof some theorems.

A set of values ​​N that includes zero is called extended and is denoted by the symbol N0 (zero index).

Series of natural numbers

An N row is a sequence of all N sets of digits. This sequence has no end.

The peculiarity of the natural series is that the next number will differ by one from the previous one, that is, it will increase. But the meanings cannot be negative.

Attention! For the convenience of counting, there are classes and categories:

• Units (1, 2, 3),
• Tens (10, 20, 30),
• Hundreds (100, 200, 300),
• Thousands (1000, 2000, 3000),
• Tens of thousands (30.000),
• Hundreds of thousands (800.000),
• Millions (4000000) etc.

All N

All N are in the set of real, integer, non-negative values. They are theirs integral part.

These values ​​go to infinity, they can belong to the classes of millions, billions, quintillions, etc.

For example:

• Five apples, three kittens,
• Ten rubles, thirty pencils,
• One hundred kilograms, three hundred books,
• A million stars, three million people, etc.

Sequence in N

In different mathematical schools, one can find two intervals to which the sequence N belongs:

from zero to plus infinity, including the ends, and from one to plus infinity, including the ends, that is, all positive whole answers.

N sets of digits can be either even or odd. Consider the concept of oddness.

Odd (any odd ones end in the numbers 1, 3, 5, 7, 9.) with two have a remainder. For example, 7:2=3.5, 11:2=5.5, 23:2=11.5.

What does even N mean?

Any even sums of classes end in numbers: 0, 2, 4, 6, 8. When dividing even N by 2, there will be no remainder, that is, the result is a whole answer. For example, 50:2=25, 100:2=50, 3456:2=1728.

Important! A numerical series of N cannot consist only of even or odd values, since they must alternate: an even number is always followed by an odd number, then an even number again, and so on.

N properties

Like all other sets, N has its own special properties. Consider the properties of the N series (not extended).

• The value that is the smallest and that does not follow any other is one.
• N are a sequence, that is, one natural value follows another(except for one - it is the first).
• When we perform computational operations on N sums of digits and classes (add, multiply), then the answer always comes out natural meaning.
• In calculations, you can use permutation and combination.
• Each subsequent value cannot be less than the previous one. Also in the N series, the following law will apply: if the number A is less than B, then in number series there will always be a C for which the equality is true: A+C=B.
• If we take two natural expressions, for example, A and B, then one of the expressions will be true for them: A \u003d B, A is greater than B, A is less than B.
• If A is less than B and B is less than C, then it follows that that A is less than C.
• If A is less than B, then it follows that: if we add the same expression (C) to them, then A + C is less than B + C. It is also true that if these values ​​are multiplied by C, then AC is less than AB.
• If B is greater than A but less than C, then: B-A less S-A.

Attention! All of the above inequalities are also valid in the opposite direction.

What are the components of a multiplication called?

In many simple and even complex tasks, finding the answer depends on the ability of students