Write down the natural number composed. Studying the exact subject: natural numbers are what numbers, examples and properties

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 base view about 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 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 of the correct notation of 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. 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 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 amount 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.

For this, let's look at 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 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 ones, 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 case), then it will sound like this: "There are no seventy-four apples." Another example. Number 88 - this is 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 is 800 (eight hundred) and 88 (eighty-eight), we read: "Eight hundred and eighty-eight."

Let's move on 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 example, 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 skill of reading multi-digit natural numbers is based on 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 unit digit value, number 3 stands in tens place and number 3 is tens place value, and the number 5 - in hundreds place and number 5 is 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 unit 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 lower (lowest) and highest (highest) category of a multi-valued 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, the binary positional number system is used, and we encounter the sexagesimal system when it comes to measuring time.

Bibliography.

• Maths. Any textbooks for 5 classes of educational 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 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 a 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 there were special characters for the designation of numbers - the predecessors 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.

Natural numbers are familiar to man and intuitive, because they surround us from childhood. In the article below, we will give a basic idea of ​​the meaning of natural numbers, describe the basic skills of writing and reading them. The entire theoretical part will be accompanied by examples.

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General idea of ​​natural numbers

At a certain stage in the development of mankind, the task arose of counting certain objects and designating their quantity, which, in turn, required finding a tool to solve this problem. Natural numbers became such a tool. The main purpose of natural numbers is also clear - to give an idea of ​​the number of objects or the serial number of a particular object, if we are talking about a set.

It is logical that for a person to use natural numbers, it is necessary to have a way to perceive and reproduce them. So, a natural number can be voiced or depicted, which is natural ways transfer of information.

Consider the basic skills of voicing (reading) and images (writing) of natural numbers.

Decimal notation of a natural number

Let's remember how they are portrayed following signs(we specify them separated by commas): 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . These characters are called numbers.

Now let's take as a rule that when depicting (writing) any natural number, only the indicated digits are used without the participation of any other symbols. Let the digits when writing a natural number have the same height, are written one after the other in a line, and there is always a digit on the left that is different from zero.

Let us indicate examples of the correct notation of natural numbers: 703, 881, 13, 333, 1023, 7, 500001. The indents between the digits are not always the same, this will be discussed in more detail below when studying the classes of numbers. The given examples show that when writing a natural number, it is not necessary to have all the digits from the above series. Some or all of them may be repeated.

Definition 1

Records of the form: 065 , 0 , 003 , 0791 are not records of natural numbers, because on the left is the number 0.

The correct notation of a natural number, made taking into account all the described requirements, is called decimal notation of a natural number.

Quantitative meaning of natural numbers

As already mentioned, natural numbers initially carry, among other things, a quantitative meaning. Natural numbers, as a numbering tool, are discussed in the topic of comparing natural numbers.

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

Imagine a certain object, for example, this: Ψ . We can write down what we see 1 subject. The natural number 1 is read as "one" or "one". The term "unit" also has another meaning: something that can be considered as a whole. If there is a set, then any element of it can be denoted by one. For example, out of many mice, any mouse is one; any flower from a set of flowers is a unit.

Now imagine: Ψ Ψ . We see one object and another object, i.e. in the record it will be - 2 items. The natural number 2 is read as "two".

Further, by analogy: Ψ Ψ Ψ - 3 items ("three"), Ψ Ψ Ψ Ψ - 4 ("four"), Ψ Ψ Ψ Ψ Ψ - 5 ("five"), Ψ Ψ Ψ Ψ Ψ Ψ - 6 ("six"), Ψ Ψ Ψ Ψ Ψ Ψ Ψ - 7 ("seven"), Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ - 8 ("eight"), Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ - 9 (" nine").

From the indicated position, the function of a natural number is to indicate quantity items.

Definition 1

If the entry of a number matches the entry of the digit 0, then such a number is called "zero". Zero is not a natural number, but it is considered together with other natural numbers. Zero means no, i.e. zero items means none.

Single digit natural numbers

It is an obvious fact that when writing each of the natural numbers discussed above (1, 2, 3, 4, 5, 6, 7, 8, 9), we use one sign - one digit.

Definition 2

Single digit natural number- a natural number, which is written using one sign - one digit.

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

Two-digit and three-digit natural numbers

Two-digit natural numbers- natural numbers, which are written using two signs - two digits. In this case, the numbers used can be either the same or different.

For example, natural numbers 71, 64, 11 are two-digit.

Consider the meaning of two-digit numbers. We will rely on the quantitative meaning of single-valued natural numbers already known to us.

Let's introduce such concept as "ten".

Imagine a set of objects, which consists of nine and one more. In this case, we can talk about 1 dozen ("one dozen") items. If you imagine one dozen and one more, then we will talk about 2 tens (“two tens”). Adding one more tens to two tens, we get three tens. And so on: continuing to add one ten at a time, we get four tens, five tens, six tens, seven tens, eight tens, and finally nine tens.

Let's look at a two-digit number as a set of single-digit numbers, one of which is written on the right, the other on the left. The number on the left will indicate the number of tens in the natural number, and the number on the right will indicate the number of units. In the case when the number 0 is located on the right, then we are talking about the absence of units. The above is the quantitative meaning of natural two-digit numbers. There are 90 of them in total.

Definition 4

Three-digit natural numbers- natural numbers, which are written using three characters - three digits. The numbers can be different or repeated in any combination.

For example, 413, 222, 818, 750 are three-digit natural numbers.

To understand the quantitative meaning of three-valued natural numbers, we introduce the concept "a hundred".

Definition 5

One hundred (1 hundred) is a set of ten tens. One hundred plus one hundred equals two hundred. Add another hundred and get 3 hundreds. Adding gradually one hundred, we get: four hundred, five hundred, six hundred, seven hundred, eight hundred, nine hundred.

Consider the record of a three-digit number itself: the single-digit natural numbers included in it are written one after the other from left to right. far right single digit indicates the number of units; the next one-digit number to the left - by the number of tens; the leftmost single digit is the number of hundreds. If the number 0 is involved in the entry, it indicates the absence of units and / or tens.

So, the three-digit natural number 402 means: 2 units, 0 tens (there are no tens that are not combined into hundreds) and 4 hundreds.

By analogy, the definition of four-digit, five-digit and so on natural numbers is given.

Multivalued natural numbers

From all of the above, it is now possible to proceed to the definition of multivalued natural numbers.

Definition 6

Multivalued natural numbers- natural numbers, which are written using two or more characters. Multi-digit natural numbers are two-digit, three-digit, and so on numbers.

One thousand is a set that includes ten hundred; one million is made up of a thousand thousand; one billion - one thousand million; one trillion is a thousand billion. Even larger sets also have names, but their use is rare.

Similarly to the principle above, we can consider any multi-digit natural number as a set of single-digit natural numbers, each of which, being in a certain place, indicates the presence and number of units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions , hundreds of millions, billions, and so on (from right to left, respectively).

For example, the multi-digit number 4 912 305 contains: 5 units, 0 tens, three hundreds, 2 thousand, 1 tens of thousands, 9 hundreds of thousands and 4 millions.

Summarizing, we examined the skill of grouping units into various sets (tens, hundreds, etc.) and saw that the numbers in the record of a multi-digit natural number are a designation of the number of units in each of such sets.

Reading natural numbers, classes

In the theory above, we denoted the names of natural numbers. In table 1, we indicate how to correctly use the names of single-digit natural numbers in speech and in alphabetic notation:

For competent reading and writing two-digit numbers, you need to learn the data in table 2:

To read other natural two-digit numbers, we will use the data from both tables, consider this with an example. Let's say we need to read a natural two-digit number 21. This number contains 1 unit and 2 tens, i.e. 20 and 1. Turning to the tables, we read the indicated number as “twenty-one”, while the union “and” between the words does not need to be pronounced. Suppose we need to use the specified number 21 in some sentence, indicating the number of items in genitive case: "there are no 21 apples." In this case, the pronunciation will sound like this: “there are no twenty-one apples.”

Let's give another example for clarity: the number 76, which is read as "seventy-six" and, for example, "seventy-six tons."

To read in full three-digit number, we also use the data of all specified tables. For example, given a natural number 305 . This number corresponds to 5 units, 0 tens and 3 hundreds: 300 and 5. Taking the table as a basis, we read: "three hundred and five" or in declension by cases, for example, like this: "three hundred and five meters."

Let's read one more number: 543. According to the rules of the tables, the indicated number will sound like this: “five hundred and forty-three” or in case declension, for example, like this: “no five hundred and forty-three rubles.”

Let's move on to general principle reading multi-digit natural numbers: to read a multi-digit number, you need to split it from right to left into groups of three digits, and the leftmost group can have 1, 2 or 3 digits. Such groups are called classes.

The extreme right class is the class of units; then the next class, to the left - the class of thousands; further - the class of millions; then comes the class of billions, followed by the class of trillions. The following classes also have a name, but natural numbers consisting of a large number of characters (16, 17 and more) are rarely used in reading, it is quite difficult to perceive them by ear.

For convenience of perception of the record, the classes are separated from each other by a small indent. For example, 31 013 736 , 134 678 , 23 476 009 434 , 2 533 467 001 222 .

To read a multi-digit number, we call in turn the numbers that make it up (from left to right, by class, adding the name of the class). The name of the class of units is not pronounced, and those classes that make up the three digits 0 are also not pronounced. If one or two digits 0 are present on the left in one class, then they are not used in any way when reading. For example, 054 is read as "fifty-four" or 001 as "one".

Example 1

Let us examine in detail the reading of the number 2 533 467 001 222:

We read the number 2, as a component of the class of trillions - "two";

Adding the name of the class, we get: "two trillion";

We read the following number, adding the name of the corresponding class: “five hundred thirty-three billion”;

We continue by analogy, reading the next class to the right: “four hundred and sixty-seven million”;

In the next class, we see two digits 0 located on the left. According to the above read rules, the digits 0 are discarded and do not participate in reading the record. Then we get: "one thousand";

Reading last class units, without adding its name - "two hundred twenty-two".

Thus, the number 2 533 467 001 222 will sound like this: two trillion five hundred thirty-three billion four hundred sixty-seven million one thousand two hundred twenty-two. Using this principle, we can also read the other given numbers:

31 013 736 - thirty one million thirteen thousand seven hundred thirty six;

134 678 - one hundred thirty-four thousand six hundred seventy-eight;

23 476 009 434 - twenty-three billion four hundred seventy-six million nine thousand four hundred thirty-four.

Thus, the basis correct reading multi-digit numbers is the skill to break a multi-digit number into classes, knowledge of the corresponding names and understanding of the principle of reading two- and three-digit numbers.

As it already becomes clear from all of the above, its value depends on the position on which the digit stands in the record of the number. That is, for example, the number 3 in the natural number 314 denotes the number of hundreds, namely, 3 hundreds. The number 2 is the number of tens (1 ten), and the number 4 is the number of units (4 units). In this case, we will say that the number 4 is in the ones place and is the value of the units place in the given number. The number 1 is in the tens place and serves as the value of the tens place. The number 3 is located in the hundreds place and is the value of the hundreds place.

Definition 7

Discharge is the position of a digit in the notation of a natural number, as well as the value of this digit, which is determined by its position in a given number.

The discharges have their own names, we have already used them above. From right to left, the digits follow: units, tens, hundreds, thousands, tens of thousands, etc.

For convenience of memorization, you can use the following table (we indicate 15 digits):

Let's clarify this detail: the number of digits in a given multi-digit number is the same as the number of characters in the number entry. For example, this table contains the names of all digits for a number with 15 characters. Subsequent discharges also have names, but are used extremely rarely and are very inconvenient for listening.

With the help of such a table, it is possible to develop the skill of determining the rank by writing a given natural number in the table so that the rightmost digit is written in the units digit and then in each digit by digit. For example, let's write a multi-digit natural number 56 402 513 674 like this:

Pay attention to the number 0, located in the discharge of tens of millions - it means the absence of units of this category.

We also introduce the concepts of the lowest and highest digits of a multi-digit number.

Definition 8

Lowest (junior) rank any multi-valued natural number is the units digit.

Highest (senior) category of any multi-digit natural number - the digit corresponding to the leftmost digit in the notation of the given number.

So, for example, in the number 41,781: the lowest rank is the rank of units; the highest rank is the tens of thousands digit.

It follows logically that it is possible to talk about the seniority of the digits relative to each other. Each subsequent digit when moving from left to right is lower (younger) than the previous one. And vice versa: when moving from right to left, each next digit is higher (older) than the previous one. For example, the thousands digit is older than the hundreds digit, but younger than the millions digit.

Let us clarify that when solving some practical examples not the natural number itself is used, but the sum of the bit terms of the given number.

Briefly about the decimal number system

Notation- a method of writing numbers using signs.

Positional number systems- those in which the value of a digit in the number depends on its position in the notation of the number.

According to this definition, we can say that, while studying natural numbers and the way they are written above, we used the positional number system. Special place the number 10 plays here. We keep counting in tens: ten units make ten, ten tens unite into a hundred, and so on. The number 10 serves as the base of this number system, and the system itself is also called decimal.

In addition to it, there are other number systems. For example, computer science uses the binary system. When we keep track of time, we use the sexagesimal number system.

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Natural numbers and their properties

Natural numbers are used to count objects in life. Any natural number uses the digits $0,1,2,3,4,5,6,7,8,9$

A sequence of natural numbers, each next number in which is $1$ greater than the previous one, forms a natural series that starts with one (because one is the smallest natural number) and does not have the greatest value, i.e. endless.

Zero is not considered a natural number.

Following relationship properties

All properties of natural numbers and operations on them follow from the four properties of sequence relations, which were formulated in $1891$ by D. Peano:

One is a natural number that does not follow any natural number.

Every natural number is followed by one and only one number

Every natural number other than $1$ follows one and only one natural number

The subset of natural numbers containing the number $1$, and together with each number the number following it, contains all the natural numbers.

If the record of a natural number consists of one digit, it is called single-digit (for example, $2.6.9$, etc.), if the record consists of two digits, it is called double-digit (for example, $12.18.45$), etc. Similarly. Two-digit, three-digit, four-digit, etc. numbers are called multivalued in mathematics.

Addition property of natural numbers

Commutative property: $a+b=b+a$

The sum does not change when the terms are rearranged

Associative property: $a+ (b+c) =(a+b) +c$

To add the sum of two numbers to a number, you can first add the first term, and then, to the resulting sum, the second term

Adding zero does not change the number, and if you add any number to zero, you get the added number.

subtraction properties

The property of subtracting the sum from the number $a-(b+c) =a-b-c$ if $b+c ≤ a$

In order to subtract the sum from a number, you can first subtract the first term from this number, and then from the resulting difference, the second term

The property of subtracting a number from the sum $(a+b) -c=a+(b-c)$ if $c ≤ b$

To subtract a number from the sum, you can subtract it from one term, and add another term to the resulting difference

If you subtract zero from a number, the number will not change.

If you subtract it from the number itself, you get zero

Multiplication Properties

Displacement $a\cdot b=b\cdot a$

The product of two numbers does not change when the factors are rearranged

Associative $a\cdot (b\cdot c)=(a\cdot b)\cdot c$

To multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor

When multiplied by one, the product does not change $m\cdot 1=m$

When multiplied by zero, the product is zero

When there are no brackets in the product notation, the multiplication is performed in order from left to right

Properties of multiplication with respect to addition and subtraction

Distributive property of multiplication with respect to addition

$(a+b)\cdot c=ac+bc$

In order to multiply the sum by a number, you can multiply each term by this number and add the resulting products

For example, $5(x+y)=5x+5y$

The distributive property of multiplication with respect to subtraction

$(a-b)\cdot c=ac-bc$

In order to multiply the difference by a number, multiply the minuend and subtracted by this number and subtract the second from the first product

For example, $5(x-y)=5x-5y$

Comparison of natural numbers

For any natural numbers $a$ and $b$, only one of the three relations $a=b$, $a

The smaller number is the one that appears earlier in the natural series, and the larger one that appears later. Zero is less than any natural number.

Example 1

Compare the numbers $a$ and $555$, if it is known that there is some number $b$, and the following relations hold: $a

Solution: Based on the specified property, because by condition $a

in any subset of natural numbers containing at least one number, there is smallest number

A subset in mathematics is a part of a set. A set is said to be a subset of another if each element of the subset is also an element of the larger set.

Often, to compare numbers, they find their difference and compare it with zero. If the difference is greater than $0$, but the first number is greater than the second, if the difference is less than $0$, then the first number is less than the second.

Rounding natural numbers

When full precision is not needed, or not possible, the numbers are rounded off, that is, they are replaced by close numbers with zeros on the end.

Natural numbers are rounded up to tens, hundreds, thousands, etc.

When rounding a number to tens, it is replaced by the nearest number consisting of whole tens; such a number has the digit $0$ in the units place

When rounding a number to hundreds, it is replaced by the nearest number consisting of whole hundreds; such a number should have the digit $0$ in the tens and ones place. Etc

The numbers to which the given is rounded are called the approximate value of the number with an accuracy of the specified digits. For example, if you round the number $564$ to tens, then we get that it can be rounded with a disadvantage and get $560$, or with an excess and get $570$.

Rounding rule for natural numbers

If to the right of the digit to which the number is rounded is the figure $5$ or a figure greater than $5$, then $1$ is added to the digit of this digit; otherwise, this figure is left unchanged.

All digits located to the right of the digit to which the number is rounded are replaced by zeros

Definition

Natural numbers are called numbers intended for counting objects. To record natural numbers, 10 Arabic numerals (0–9) are used, which form the basis of the decimal number system generally accepted for mathematical calculations.

Sequence of natural numbers

The natural numbers make up a series starting at 1 and covering the set of all positive integers. Such a sequence consists of numbers 1,2,3, ... . This means that in the natural series:

1. There is a smallest number and no largest.
2. Each next number is greater than the previous one by 1 (the exception is the unit itself).
3. As the numbers go to infinity, they grow indefinitely.

Sometimes 0 is also introduced into a series of natural numbers. This is permissible, and then they talk about extended natural series.

Classes of natural numbers

Each digit of a natural number expresses a certain digit. The last one is always the number of units in the number, the one before it is the number of tens, the third from the end is the number of hundreds, the fourth is the number of thousands, and so on.

• in the number 276: 2 hundreds, 7 tens, 6 units
• in the number 1098: 1 thousand, 9 tens, 8 ones; the hundreds place is absent here, since it is expressed as zero.

For large and very large numbers, you can see a steady trend (if you examine the number from right to left, that is, from the last digit to the first):

• the last three digits in the number are units, tens and hundreds;
• the previous three are units, tens and hundreds of thousands;
• the three in front of them (i.e. the 7th, 8th and 9th digits of the number, counting from the end) are units, tens and hundreds of millions, etc.

That is, every time we are dealing with three digits, meaning units, tens and hundreds of a larger name. Such groups form classes. And if with the first three classes in Everyday life have to deal with more or less often, then others should be listed, because not everyone remembers their names by heart.

• The 4th class, following the class of millions and representing numbers of 10-12 digits, is called a billion (or a billion);
• 5th grade - trillion;
• 6th grade - quadrillion;
• 7th grade - quintillion;
• 8th grade - sextillion;
• 9th grade - septillion.

Addition of natural numbers

The addition of natural numbers is arithmetic operation, which allows you to get a number that contains as many units as there are in the numbers added together.

The sign of addition is the "+" sign. Added numbers are called terms, the result is called the sum.

Small numbers are added (summed up) orally, in writing such actions are written in a line.

Multi-digit numbers, which are difficult to add in the mind, are usually added in a column. For this, the numbers are written one under the other, aligned with the last digit, that is, they write the units digit under the units digit, the hundreds digit under the hundreds digit, and so on. Next, you need to add the digits in pairs. If the addition of digits occurs with a transition through a ten, then this ten is fixed as a unit above the digit on the left (that is, following it) and is added together with the digits of this digit.

If the column adds up not 2, but more numbers, then when summing the digits of the category, not 1 dozen, but several may be redundant. In this case, the number of such tens is transferred to the next digit.

Subtraction of natural numbers

Subtraction is an arithmetic operation, the reverse of addition, which boils down to the fact that, given the amount and one of the terms, you need to find another - an unknown term. The number that is being subtracted from is called the minuend; the number that is being subtracted is the subtrahend. The result of the subtraction is called the difference. The sign that denotes the operation of subtraction is "-".

In the transition to addition, the subtrahend and the difference turn into terms, and the reduced into the sum. Addition usually checks the correctness of the subtraction performed, and vice versa.

Here 74 is the minuend, 18 is the subtrahend, 56 is the difference.

A prerequisite for subtracting natural numbers is the following: the minuend must necessarily be greater than the subtrahend. Only in this case the resulting difference will also be a natural number. If the subtraction action is carried out for an extended natural series, then it is allowed that the minuend is equal to the subtrahend. And the result of subtraction in this case will be 0.

Note: if the subtrahend is equal to zero, then the subtraction operation does not change the value of the minuend.

Subtraction of multi-digit numbers is usually done in a column. Write down the numbers in the same way as for addition. Subtraction is performed for the corresponding digits. If it turns out that the minuend is less than the subtrahend, then one is taken from the previous (located on the left) digit, which, after the transfer, naturally turns into 10. This ten is summed up with the figure of the reduced given digit and then subtracted. Further, when subtracting the next digit, it is necessary to take into account that the reduced has become 1 less.

Product of natural numbers

The product (or multiplication) of natural numbers is an arithmetic operation, which is finding the sum of an arbitrary number of identical terms. To record the operation of multiplication, use the sign "·" (sometimes "×" or "*"). For example: 3 5=15.

The action of multiplication is indispensable when necessary to add a large number of terms. For example, if you need to add the number 4 7 times, then multiplying 4 by 7 is easier than doing this addition: 4+4+4+4+4+4+4.

The numbers that are multiplied are called factors, the result of multiplication is the product. Accordingly, the term "work" can, depending on the context, express both the process of multiplication and its result.

Multi-digit numbers are multiplied in a column. For this number is written in the same way as for addition and subtraction. It is recommended to write first (above) which of the 2 numbers, which is longer. In this case, the multiplication process will be simpler, and therefore more rational.

When multiplying in a column, the digits of each of the digits of the second number are sequentially multiplied by the digits of the 1st number, starting from its end. Having found the first such work, they write down the number of units, and keep the number of tens in mind. When multiplying the digit of the 2nd number by the next digit of the 1st number, the number that is kept in mind is added to the product. And again they write down the number of units of the result obtained, and remember the number of tens. When multiplying by the last digit of the 1st number, the number obtained in this way is written down in full.

The results of multiplying the digits of the 2nd digit of the second number are written in the second row, shifting it 1 cell to the right. And so on. As a result, a "ladder" will be obtained. All the resulting rows of numbers should be added (according to the rule of addition in a column). Empty cells should be considered filled with zeros. The resulting sum is the final product.

Note

1. The product of any natural number by 1 (or 1 by a number) is equal to the number itself. For example: 376 1=376; 1 86=86.
2. When one of the factors or both factors are equal to 0, then the product is equal to 0. For example: 32·0=0; 0 845=845; 0 0=0.

Division of natural numbers

Division is called an arithmetic operation, with the help of which, according to a known product and one of the factors, it can be found another - unknown - factor. Division is an action reciprocal of multiplication, and is used to check the correctness of the performed multiplication (and vice versa).

The number that is being divided is called the divisible; the number by which it is divided is the divisor; the result of a division is called a quotient. The division sign is ":" (sometimes, less often - "÷").

Here 48 is the dividend, 6 is the divisor, and 8 is the quotient.

Not all natural numbers can be divided among themselves. In this case, division is performed with a remainder. It consists in the fact that for the divisor such a factor is selected so that its product by the divisor would be a number that is as close as possible in value to the dividend, but less than it. The divisor is multiplied by this factor and subtracted from the dividend. The difference will be the remainder of the division. The product of a divisor by a factor is called an incomplete quotient. Attention: the remainder must be less than the selected multiplier! If the remainder is larger, then this means that the multiplier is chosen incorrectly, and it should be increased.

We select a factor for 7. In this case, this number is 5. We find an incomplete quotient: 7 5 \u003d 35. Calculate the remainder: 38-35=3. Since 3<7, то это означает, что число 5 было подобрано верно. Результат деления следует записать так: 38:7=5 (остаток 3).

Multi-digit numbers are divided into a column. To do this, the dividend and divisor are written side by side, separating the divisor with a vertical and horizontal line. In the dividend, the first digit or the first few digits (on the right) are selected, which should be a number that is minimally sufficient for dividing by a divisor (that is, this number must be greater than the divisor). For this number, an incomplete quotient is selected, as described in the rule of division with a remainder. The number of the multiplier used to find the partial quotient is written under the divisor. The incomplete quotient is written under the number that was divided, right-aligned. Find their difference. The next digit of the dividend is demolished by writing it next to this difference. For the resulting number, an incomplete quotient is again found by writing down the figure of the selected factor, next to the previous one under the divisor. And so on. Such actions are performed until the numbers of the dividend run out. After that, the division is considered complete. If the dividend and the divisor are divided entirely (without a remainder), then the last difference will give zero. Otherwise, the remainder number will be returned.

Exponentiation

Exponentiation is a mathematical operation that consists in multiplying an arbitrary number of identical numbers. For example: 2 2 2 2.

Such expressions are written as: a x,

where a is a number multiplied by itself x is the number of such factors.

Prime and composite natural numbers

Any natural number, except 1, can be divided by at least 2 numbers - one and itself. Based on this criterion, natural numbers are divided into prime and composite.

Prime numbers are numbers that are only divisible by 1 and itself. Numbers that are divisible by more than these 2 numbers are called composite numbers. A unit divisible solely by itself is neither prime nor compound.

Numbers are prime: 2,3,5,7,11,13,17,19, etc. Examples of composite numbers: 4 (divisible by 1,2,4), 6 (divisible by 1,2,3,6), 20 (divisible by 1,2,4,5,10,20).

Any composite number can be decomposed into prime factors. In this case, prime factors are understood to be its divisors, which are prime numbers.

An example of factorization into prime factors:

Divisors of natural numbers

A divisor is a number by which a given number can be divided without a remainder.

In accordance with this definition, simple natural numbers have 2 divisors, composite numbers have more than 2 divisors.

Many numbers have common divisors. The common divisor is the number by which the given numbers are divisible without a remainder.

• The numbers 12 and 15 have a common divisor 3
• The numbers 20 and 30 have common divisors 2,5,10

Of particular importance is the greatest common divisor (GCD). This number, in particular, is useful to be able to find for reducing fractions. To find it, it is required to decompose the given numbers into prime factors and present it as the product of their common prime factors, taken in their smallest powers.

It is required to find the GCD of the numbers 36 and 48.

Divisibility of natural numbers

It is far from always possible to determine “by eye” whether one number is divisible by another without a remainder. In such cases, the corresponding divisibility test is useful, that is, the rule by which in a matter of seconds you can determine whether it is possible to divide numbers without a remainder. The sign "" is used to indicate divisibility.

Least common multiple

This value (denoted LCM) is the smallest number that is divisible by each of the given ones. The LCM can be found for an arbitrary set of natural numbers.

LCM, like GCD, has a significant applied meaning. So, it is the LCM that needs to be found by reducing ordinary fractions to a common denominator.

The LCM is determined by factoring the given numbers into prime factors. For its formation, a product is taken, consisting of each of the occurring (at least for 1 number) prime factors represented to the maximum degree.

It is required to find the LCM of the numbers 14 and 24.

Average

The arithmetic mean of an arbitrary (but finite) number of natural numbers is the sum of all these numbers divided by the number of terms:

The arithmetic mean is some average value for a number set.

The numbers 2,84,53,176,17,28 are given. It is required to find their arithmetic mean.