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

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

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Natural numbers – general representation.

The following opinion is not without sound logic: the emergence of the task of counting objects (first, second, third object, etc.) and the task of indicating the number of objects (one, two, three objects, etc.) led to the creation of a tool for solving it, this were the instrument integers.

From this sentence it is clear the main purpose of natural numbers– carry information about the number of any items or the serial number of a given item in the set of items under consideration.

In order for a person to use natural numbers, they must be in some way accessible to both perception and reproduction. If you voice 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 to convey and perceive natural numbers.

So let’s begin to acquire the skills of depicting (writing) and voicing (reading) natural numbers, while learning their meaning.

Decimal notation of a natural number.

First we need to decide what we will start from when writing natural numbers.

Let's remember the images of the following characters (we will show them separated by commas): 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . The images shown are a recording of the so-called numbers. Let's immediately agree not to turn over, tilt, or otherwise distort the numbers when recording.

Now let’s agree that in the notation of any natural number only the indicated digits can be present and no other symbols can be present. Let’s 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 indentation) and on the left is a digit other than the digit 0 .

Here are some examples of correct writing of natural numbers: 604 , 777 277 , 81 , 4 444 , 1 001 902 203, 5 , 900 000 (please note: the indents between numbers are not always the same, more about this will be discussed when reviewing). From the above examples it is clear that the notation of 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.

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

Writing a natural number, made 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 writing. Let us explain this: further in the text we will use 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.

The time has come to understand the quantitative meaning that the written natural number carries. The meaning of natural numbers in terms of numbering of objects is discussed in the article comparison of natural numbers.

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

Let's imagine that we opened our eyes and saw some object, for example, like this. In this case, we can write down what we see 1 item. The natural number 1 is read as " one"(declension of the numeral “one”, as well as other numerals, we will give in paragraph), for the number 1 another name has been adopted - “ unit».

However, the term “unit” is multi-valued, in addition to the natural number 1 , call something considered as a whole. For example, any one item from their many can be called a unit. For example, any apple from a set of apples is a unit, any flock of birds from a set of flocks of birds is also a unit, etc.

Now we open our eyes and see: . That is, we see one object and another object. In this case, we can write down what we see 2 subject. Natural number 2 , reads as " 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, natural numbers 1 , 2 , 3 , …, 9 indicate quantity 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 quantities.

Single digit natural numbers.

Obviously, the recording of each of the natural numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 consists of one character - one number.

Definition.

Single digit natural numbers– these are natural numbers, the writing 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 in total.

Two-digit and three-digit natural numbers.

First, let's define two-digit natural numbers.

Definition.

Two-digit natural numbers– these are natural numbers, the recording of which consists of two signs - two digits (different or the same).

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

Let's figure out what meaning two-digit numbers carry, while we will build on the quantitative meaning of single-digit natural numbers that we already know.

To begin with, let's introduce the concept ten.

Let's imagine this situation - we opened our eyes and saw a set consisting of nine objects and one more object. In this case they talk about 1 ten (one dozen) items. If one ten and another ten are considered together, then they speak of 2 tens (two dozen). 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, let's look at 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 units. Moreover, if there is a digit on the right side 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 quantities.

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 combined into tens.

Let’s answer the question: “How many two-digit natural numbers are there?” Answer: them 90 .

Let's move on to the definition of three-digit natural numbers.

Definition.

Natural numbers whose notation consists of 3 signs – 3 numbers (different or repeating) 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-digit.

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

The set of ten tens is 1 hundred (one hundred). A hundred and a 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, following each other 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, and the next number indicates the number of hundreds. Numbers 0 in recording three-digit number mean the absence of tens and (or) units.

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

Similarly, one can define four-digit, five-digit, six-digit, etc. natural numbers.

Multi-digit natural numbers.

So, let's move on to the definition of multi-valued natural numbers.

Definition.

Multi-digit natural numbers- these are natural numbers, the notation 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 a 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-digit natural numbers?

Let's look at a multi-digit natural number as single-digit natural numbers following one after another 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, then the number of tens of thousands, then hundreds of thousands, then the number of millions, then the number of tens of millions, then hundreds of millions, then – 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 notation of a multi-digit natural number indicate the corresponding number of the above groups.

Reading natural numbers, classes.

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

How are the remaining two-digit numbers read?

Let's explain with an example. Let's read the 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 we just recorded, and the number 74 we read it as: “Seventy-four” (we do not pronounce the conjunction “and”). If you need to read a number 74 in the sentence: "No 74 apples" (genitive case), 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 skills we have already acquired in reading single-digit and double-digit numbers.

Let's look at 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 eighty eight.”

Let's move on to reading multi-digit numbers.

To read, the record of a multi-digit natural number is divided, starting from the right, into groups of three digits, and 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 class of units. The class following it (from right to left) is called class of thousands, next class - million class, next - billion class, next comes trillion class. You can give the names of the following classes, but natural numbers, the notation 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 dividing 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 written down natural numbers in a table that makes it easy to learn how to read them.

To read a natural number, we call its constituent numbers by class from left to right 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 entry has a number on the left 0 or two digits 0 , then we ignore these numbers 0 and read the number obtained by discarding these numbers 0 . Eg, 002 read as “two”, and 025 - as in “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 eighty-nine”;
• add the name of the class, we get “four hundred eighty nine thousand”;
• further in the class of units we see 002 , there are zeros on the left, we ignore them, therefore 002 read as "two";
• there is no need to add the name of the unit class;
• in the end we have 489 002 - “four hundred 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 , read “ten”;
• add the name of the class, we have “ten million”;
• then we see the entry 000 in the thousand class, since all three digits are digits 0 , then we skip this class and move on to the next one;
• class of units represents number 501 , which we read “five hundred and one”;
• Thus, 10 000 501 - ten million five hundred one.

Let's do this without detailed explanation: 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 divide 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 meaning of each digit depends on its position. For example, a natural number 539 corresponds 5 hundreds, 3 dozens and 9 units, therefore, the figure 5 in writing the number 539 determines the number of hundreds, digit 3 – the number of tens, and the digit 9 - number of units. At the same time they say that the figure 9 costs in units digit and number 9 is unit digit value, number 3 costs in tens place and number 3 is tens place value, and the number 5 - V hundreds place and number 5 is hundreds place value.

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

The categories are given names. If you look at the numbers in the notation of a natural number from right to left, then they will correspond to the following digits: units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, and so on.

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

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 recording of which contains up to 15 characters. The following ranks also have their own names, but they are very rarely used, so there is no point in mentioning them.

Using a 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 give an example. Let's write down a natural number 67 922 003 942 into the table, and the digits and meanings of these digits will become clearly visible.

The number in this number is 2 stands in the units place, digit 4 – in the tens place, digit 9 – in the hundreds place, etc. You should pay attention to the numbers 0 , located in the tens of thousands and hundreds of thousands categories. Numbers 0 in these digits means the absence of units of these digits.

It is also worth mentioning the so-called lowest (junior) and highest (most significant) digit of a multi-digit natural number. Lowest (junior) rank of any multi-digit natural number is the units digit. The highest (most significant) digit of a natural number is the digit corresponding to the rightmost digit in the recording of this number. For example, the low-order digit of the natural number 23,004 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 subsequent digit lower (younger) previous one. For example, the rank of thousands is lower than the rank of tens of thousands, and even more so the rank of thousands is lower than the rank of hundreds of thousands, millions, tens of millions, etc. If in the notation of a natural number we move by digits from right to left, then each subsequent digit taller (older) previous one. For example, the hundreds digit is older than the tens digit, and even more so, older than the units digit.

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

Briefly about the decimal number system.

So, we got acquainted with natural numbers, 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 meaning of a digit in a number notation may or may not depend on its position. Number systems in which the value of a digit in a number depends on its position are called positional.

Thus, the natural numbers we examined and the method of writing them indicate that we use a positional number system. It should be noted that the number has a special place in this number system 10 . Indeed, counting is done in tens: ten ones are combined into a ten, a dozen tens are combined into a hundred, a dozen 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 the binary positional number system is used, and we encounter the sexagesimal system when it comes to measuring time.

Bibliography.

• Mathematics. Any textbooks for 5th grade of general education institutions.

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 fingers on a hand, and they said: “I have as many nuts as there are fingers on my hand.”

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

Remember!

Integers- these are numbers, starting from 1, obtained by 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 depict one with one stick, then with two sticks - the number 2, with three - the number 3.

| — 1, || — 2, ||| — 3, ||||| — 5 …

Then they appeared special signs for denoting numbers - the predecessors of modern numbers. The numerals we use to write numbers originated in India approximately 1,500 years ago. The Arabs brought them to Europe, which is why they are called Arabic numerals.

There are ten numbers in total: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Using these numbers you can write any natural number.

Remember!

Natural series is a 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 greatest 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 meaning of a digit depends on its place in the number record, 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 subsequent 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 exceeds the number of all atoms (the smallest particles of matter) in the entire Universe.

This number received special name — googol. Googol is a number with 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, since we encounter them when counting and when searching, designating the number of objects.

In contact with

What numbers are called natural numbers?

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

• When counting any objects (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 because the set of integer values ​​is unlimited.

Attention! Natural numbers are obtained when counting objects or when indicating their quantity.

Absolutely any number can be decomposed and presented in the form of digit 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. On the diagram of sets, they would be located in each other, since the set of natural ones 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

In most math schools, the smallest value of N is considered a unit, since the absence of objects is considered emptiness.

But in foreign mathematical schools, for example in French, it is considered natural. The presence of zero in the series makes the proof easier some theorems.

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

Series of natural numbers

N series 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 ease 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 you can find two intervals to which the sequence N belongs:

from zero to plus infinity, including ends, and from one to plus infinity, including ends, that is, everything positive integer answers.

N sets of digits can be either even or odd. Let's consider the concept of oddity.

Odd (any odd number ends 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 even N is divided by 2, there will be no remainder, that is, the result is the whole answer. For example, 50:2=25, 100:2=50, 3456:2=1728.

Important! A number series of N cannot consist only of even or odd values, since they must alternate: even is always followed by odd, followed by even again, etc.

Properties N

Like all other sets, N has its own special properties. Let's 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 represent 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 it always turns out natural meaning.
• Permutation and combination can be used in calculations.
• 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 the number series there will always be a C for which the equality holds: 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 = 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 reverse direction.

What are the components of multiplication called?

In many simple and even complex problems, finding the answer depends on the skills of schoolchildren.

In order to multiply quickly and correctly and be able to solve inverse problems, you need to know the components of multiplication.

15. 10=150. In this expression there are 15 and 10 are multipliers, and 150 is a product.

Multiplication has properties that are necessary when solving problems, equations and inequalities:

• Rearranging the factors will not change the final product.
• To find an unknown factor, you need to divide the product by a known factor (true for all factors).

For example: 15 . X=150. Let's divide the product by a known factor. 150:15=10. Let's do a check. 15 . 10=150. According to this principle, they even decide complex linear equations(to simplify them).

Important! A product can consist of more than just two factors. For example: 840=2 . 5. 7. 3. 4

What are natural numbers in mathematics?

Places and classes of natural numbers

Conclusion

Let's summarize. N is used when counting or indicating the number of items. The series of natural sets of numbers is infinite, but it includes only integer and positive sums of digits and classes. Multiplication is also necessary in order to to count objects, as well as for solving problems, equations and various inequalities.

Numbers are an abstract concept. They are a quantitative characteristic of objects and can be real, rational, negative, integer and fractional, as well as natural.

The natural series is usually used when counting, in which quantity notations naturally arise. Acquaintance with counting begins in early childhood. What kid avoided funny rhymes that used elements of natural counting? "One, two, three, four, five... The bunny went out for a walk!" or "1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the king decided to hang me..."

For any natural number, you can find another one greater than it. This set is usually denoted by the letter N and should be considered infinite in the direction of increase. But this set has a beginning - it is one. Although there are French natural numbers, the set of which also includes zero. But the main distinctive features of both sets is the fact that they do not include either fractional or negative numbers.

The need to count a variety of objects arose in prehistoric times. Then the concept of “natural numbers” was supposedly formed. Its formation occurred throughout the entire process of changing a person’s worldview and the development of science and technology.

However, they could not yet think abstractly. It was difficult for them to understand what the commonality of the concepts of “three hunters” or “three trees” was. Therefore, when indicating the number of people, one definition was used, and when indicating the same number of objects of a different kind, a completely different definition was used.

And it was extremely short. It contained only the numbers 1 and 2, and the count ended with the concepts of “many”, “herd”, “crowd”, “heap”.

Later, a more progressive and broader account was formed. An interesting fact is that there were only two numbers - 1 and 2, and the next numbers were obtained by adding.

An example of this was the information that has reached us about the numerical series of the Australian tribe. They had 1 for the word “Enza”, and 2 for the word “petcheval”. The number 3 therefore sounded like “petcheval-Enza”, and 4 sounded like “petcheval-petcheval”.

Most peoples recognized fingers as the standard of counting. Further development of the abstract concept of “natural numbers” followed the path of using notches on a stick. And then it became necessary to designate a dozen with another sign. The ancient people found our way out - they began to use another stick, on which notches were made to indicate tens.

The ability to reproduce numbers expanded enormously with the advent of writing. At first, numbers were depicted as lines on clay tablets or papyrus, but gradually other writing icons began to be used. This is how Roman numerals appeared.

Much later, they appeared that opened up the possibility of writing numbers with a relatively small set of characters. Today it is not difficult to write down such huge numbers as the distance between planets and the number of stars. You just have to learn to use degrees.

Euclid in the 3rd century BC in the book “Elements” establishes the infinity of the numerical set, and Archimedes in “Psamita” reveals the principles for constructing the names of arbitrarily large numbers. Almost until the middle of the 19th century, people did not face the need for a clear formulation of the concept of “natural numbers”. The definition was required with the advent of the axiomatic mathematical method.

And in the 70s of the 19th century he formulated a clear definition of natural numbers, based on the concept of set. And today we already know that natural numbers are all integers, starting from 1 to infinity. Young children, taking their first step in becoming acquainted with the queen of all sciences - mathematics - begin to study these very numbers.

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Definition. Integers- these are the numbers that are used for counting: 1, 2, 3, ..., n, ...

The set of natural numbers is usually denoted by the symbol N(from lat. naturalis- natural).

Natural numbers in decimal system Numbers are written using ten digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

The set of natural numbers is ordered set, i.e. for any natural numbers m and n one of the following relations holds true:

• or m = n (m equals n),
• or m > n (m greater than n ),
• or m< n (m меньше n ).

• Least natural number - one (1)
• There is no greatest natural number.
• Zero (0) is not a natural number.

The set of natural numbers is infinite, since for any number n there is always a number m that is greater than n

Of the neighboring natural numbers, the number that is to the left of n is called previous number n, and the number that is to the right is called next after n.

Operations on natural numbers

Closed operations on natural numbers (operations resulting from natural numbers) include the following arithmetic operations:

• Addition
• Multiplication
• Exponentiation a b , where a is the base and b is the exponent. If the base and exponent are natural numbers, then the result will be a natural number.

Additionally, two more operations are being considered. From a formal point of view, they are not operations on natural numbers, since their result will not always be a natural number.

• Subtraction(In this case, the Minuend must be greater than the Subtrahend)
• Division

Classes and ranks

Place is the position (position) of a digit in a number record.

The lowest rank is the one on the right. The most significant digit is the one on the left.

Example:

5 - units, 0 - tens, 7 - hundreds,
2 - thousands, 4 - tens of thousands, 8 - hundreds of thousands,
3 - million, 5 - tens of millions, 1 - hundred million

For ease of reading, natural numbers are divided into groups of three digits each, starting from the right.

Class- a group of three digits into which the number is divided, starting from the right. The last class can consist of three, two or one digits.

• The first class is the class of units;
• The second class is the class of thousands;
• The third class is the class of millions;
• The fourth class is the class of billions;
• Fifth class - class of trillions;
• Sixth class - class of quadrillions (quadrillions);
• The seventh class is the class of quintillions (quintillions);
• Eighth class - sextillion class;
• Ninth class - septillion class;

Example:

34 - billion 456 million 196 thousand 45

Comparison of natural numbers

1. Comparing natural numbers with different numbers of digits

   Among natural numbers, the one with more digits is greater

2. Comparing natural numbers with an equal number of digits

   Compare numbers bit by bit, starting with the most significant digit. The one that has more units in the highest rank of the same name is greater

Example:

3466 > 346 - since the number 3466 consists of 4 digits, and the number 346 consists of 3 digits.

34666 < 245784 - since the number 34666 consists of 5 digits, and the number 245784 consists of 6 digits.

Example:

346 667 670 52 6 986

346 667 670 56 9 429

The second natural number with an equal number of digits is greater, since 6 > 2.

Dream Interpretation