Any natural number. Natural Numbers - Basics

What are natural and non-natural? integers? How to explain to a child, or maybe not a child, what are the differences between them? Let's figure it out. As far as we know, non-natural and natural numbers are studied in the 5th grade, and our goal is to explain to students so that they really understand and learn what and how.

Story

Natural numbers are one of the old concepts. A long time ago, when people did not yet know how to count and had no idea about numbers, when they needed to count something, for example, fish, animals, they beat out various subjects dots or dashes, as archaeologists later found out. Life was very difficult for them at that time, but civilization developed first to the Roman number system and then to the decimal number system. Nowadays almost everyone uses Arabic numerals

All about natural numbers

Natural numbers are prime numbers that we use in our daily lives to count objects in order to determine quantity and order. Currently, we use the decimal number system to write numbers. In order to write down any number, we use ten digits - from zero to nine.

Natural numbers are those numbers that we use when counting objects or indicating the serial number of something. Example: 5, 368, 99, 3684.

A number series refers to natural numbers that are arranged in ascending order, i.e. from one to infinity. Such a series begins with the smallest number - 1, and there is no largest natural number, since the series of numbers is simply infinite.

In general, zero is not considered a natural number, since it means the absence of something, and there is also no counting of objects

The Arabic number system is modern system, which we use every day. It is a variant of Indian (decimal).

This number system became modern because of the number 0, which was invented by the Arabs. Before this, it was not available in the Indian system.

Unnatural numbers. What is this?

Natural numbers do not include negative numbers or non-integers. This means that they are - unnatural numbers

Below are examples.

Non-natural numbers are:

• Negative numbers, for example: -1, -5, -36.. and so on.
• Rational numbers, which are expressed as decimal fractions: 4.5, -67, 44.6.
• In the form of a simple fraction: 1 / 2, 40 2 /7, etc.

Irrational numbers such as e = 2.71828, √2 = 1.41421 and the like.

We hope that we have greatly helped you understand non-natural and natural numbers. Now it will be easier for you to explain this topic to your baby, and he will learn it as well as the great mathematicians!

Definition

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

Sequence of natural numbers

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

1. Eat smallest number and there is no greatest.
2. Each subsequent number is greater than the previous one by 1 (with the exception of the unit itself).
3. As numbers tend to infinity, they grow without limit.

Sometimes 0 is introduced into a series of natural numbers. This is acceptable, and then they talk about expanded 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 previous 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 number 276: 2 hundreds, 7 tens, 6 ones
• in the number 1098: 1 thousand, 9 tens, 8 ones; The hundreds place is missing here because it is expressed as zero.

For large and very large numbers, you can see a stable 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 you have to deal with the first three classes in everyday life more or less often, then the 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 billion (or billion);
• 5th grade – trillion;
• 6th grade – quadrillion;
• 7th grade – quintillion;
• 8th grade – sextillion;
• 9th grade – septillion.

Addition of natural numbers

Addition of natural numbers is an arithmetic operation that allows you to obtain a number that contains the same number of units as there are in the numbers being added together.

The addition sign is the “+” sign. The numbers added are called addends, and the resulting result is called a sum.

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

Multi-digit numbers that are difficult to add in your head are usually added into a column. To do this, numbers are written one below the other, aligned by the last digit, that is, they write the ones place under the units place, the hundreds place under the hundreds place, 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, the next one) and is summed together with the digits of this digit.

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

Subtracting Natural Numbers

Subtraction is an arithmetic operation, the inverse of addition, which boils down to the fact that using the available sum and one of the terms, you need to find another - an unknown term. The number from which it is subtracted is called the minuend; the number that is being subtracted is subtrahendable. The result of the subtraction is called the difference. The sign used to denote the action of subtraction is “–”.

When moving to addition, the subtrahend and difference turn into addends, and the minuend turns into a sum. Addition is usually used to check the correctness of the subtraction, 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 be greater than the subtrahend. Only in this case the resulting difference will also be a natural number. If the action of subtraction is carried out for an extended natural series, then it is allowed that the minuend be 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. The numbers are written 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 they take one from the previous (located on the left) digit, which, after the transfer, naturally turns into 10. This ten is summed with the number of the given digit being mined and then the subtraction is performed. Then, when subtracting the next digit, be sure to take into account that the one being reduced has become 1 less.

Product of natural numbers

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

The action of multiplication is indispensable when adding is necessary. 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 performing the following addition: 4+4+4+4+4+4+4.

Numbers that are multiplied are called factors, the result of multiplication is called a product. Accordingly, the term “product” can, depending on the context, express both the process of multiplication and its result.

Multi-digit numbers are multiplied into a column. For this, numbers are written in the same way as for addition and subtraction. It is recommended to write down the longest of the 2 numbers first (above). 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 product, write down the units digit, and keep the tens digit in mind. When multiplying the digit of the 2nd number by the next digit of the 1st number, the digit that is kept in mind is added to the product. And again, write down the units number of the result obtained, and remember the tens number. When multiplied by the last digit of the 1st number, the number obtained in this way is written down in full.

The results of multiplying the digit 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 resulting rows of numbers should be added (according to the rule of column addition). 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 an arithmetic operation with the help of which, given a known product and one of the factors, another – unknown – factor can be found. Division is the inverse of multiplication and is used to check whether a multiplication has been performed correctly (and vice versa).

The number that is divided is called the dividend; the number being divided by is the divisor; the result of division is called the quotient. The division sign is “:” (sometimes, less commonly, “÷”).

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

Not all natural numbers can be divided among themselves. In this case, divide with a remainder. It consists in the fact that a factor is selected for the divisor such 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 and a factor is called an incomplete quotient. Attention: the balance must be less than the selected multiplier! If the remainder is greater, this means that the multiplier was chosen incorrectly and should be increased.

We select a multiplier for 7. V in this case this number is 5. Find the incomplete quotient: 7·5=35. We 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, write the dividend and divisor side by side, separating the divisor with a vertical and horizontal line. In the dividend, the first digit or first few digits (on the right) are isolated, which must represent a number that is minimally sufficient to divide by the divisor (that is, this number must be greater than the divisor). For this number, an incomplete quotient is selected, as described in the rule for division with a remainder. The digit of the multiplier used to find the partial quotient is written under the divisor. The incomplete quotient is written below the number being divided, aligned to the right. Find their difference. Take down the next digit of the dividend by writing it next to this difference. For the resulting number, the partial quotient is again found by writing down the digit of the selected multiplier next to the previous one under the divisor. And so on. Such actions are carried out until the digits of the dividend run out. After this, the division is considered complete. If the dividend and the divisor are divided by a whole (without remainder), then the last difference will give zero. Otherwise, the remainder number will be obtained.

Exponentiation

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

Such expressions are written in the form: a x,

Where a– a number multiplied by itself, x– the number of such factors.

Prime and composite natural numbers

Every natural number, except 1, can be divided into 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 divisible only by 1 and themselves. Numbers that are divisible by more than these 2 numbers are called composite numbers. A unit divisible solely by itself is neither simple nor composite.

Prime numbers are: 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).

Every composite number can be factorized into prime factors. By prime factors we mean its divisors, which are prime numbers.

Example of prime factorization:

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, prime natural numbers have 2 divisors, composite numbers have more than 2 divisors.

Many numbers have common factors. A common divisor is a number by which the given numbers are divided without a remainder.

• The numbers 12 and 15 have a common divisor of 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, you need to decompose the given numbers into prime factors and represent it as the product of their common prime factors, taken in their smallest powers.

You need to find the gcd of numbers 36 and 48.

Divisibility of natural numbers

It is not always possible to determine by eye whether one number is divisible by another without a remainder. In such cases, the corresponding test of divisibility turns out to be useful, that is, a rule by which in a matter of seconds you can determine whether numbers can be divided without a remainder. The sign “” is used to indicate divisibility.

Least common multiple

This quantity (denoted LOC) 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.

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

The LCM is determined by factoring given numbers into prime factors. To form it, take a product consisting of each of the occurring (at least for 1 number) prime factors, represented to the maximum degree.

You need 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 numerical set.

The numbers given are 2,84,53,176,17,28. You need to find their arithmetic mean.

Integers– numbers that are used to count objects . Any natural number can be written using ten numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This type of number is called decimal

The sequence of all natural numbers is called natural next to .

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ...

The most small natural number is one (1). In the natural series, each next number is 1 greater than the previous one. Natural series endless, there is no largest number in it.

The meaning of a digit depends on its place in the number record. For example, the number 4 means: 4 units if it is in the last place in the number record (in units place); 4 ten, if she is in second to last place (in the tens place); 4 hundreds, if she is in third place from the end (V hundreds place).

The number 0 means absence of units of this category in the decimal notation of a number. It also serves to designate the number “ zero" This number means "none". The score 0:3 in a football match means that the first team did not score a single goal against the opponent.

Zero do not include to natural numbers. And indeed, counting objects never starts from scratch.

If the notation of a natural number consists of one sign – one digit, then it is called unambiguous. Those. unambiguousnatural number– a natural number, the notation of which consists of one sign – one digit. For example, the numbers 1, 6, 8 are single digits.

Double digitnatural number– a natural number, the notation of which consists of two characters – two digits.

For example, the numbers 12, 47, 24, 99 are two-digit numbers.

Also, based on the number of characters in a given number, names are given to other numbers:

numbers 326, 532, 893 – three-digit;

numbers 1126, 4268, 9999 – four-digit etc.

Two-digit, three-digit, four-digit, five-digit, etc. numbers are called multi-digit numbers .

To read multi-digit numbers, they are divided, starting from the right, into groups of three digits each (the leftmost group may consist of one or two digits). These groups are called classes.

Million– this is a thousand thousand (1000 thousand), it is written 1 million or 1,000,000.

Billion- that's 1000 million. It is written as 1 billion or 1,000,000,000.

The first three digits on the right make up the class of units, the next three – the class of thousands, then come the classes of millions, billions, etc. (Fig. 1).

Rice. 1. Millions class, thousands class and units class (from left to right)

The number 15389000286 is written in the bit grid (Fig. 2).

Rice. 2. Bit grid: number 15 billion 389 million 286

This number has 286 units in the units class, zero units in the thousands class, 389 units in the millions class, and 15 units in the billions class.

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

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General understanding of natural numbers

At a certain stage in the development of mankind, the task of counting certain objects and designating their quantity arose, which, in turn, required finding a tool to solve this problem. Natural numbers became such a tool. It is also clear that the main purpose of natural numbers is to give an idea of ​​the number of objects or the serial number of a specific 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. Thus, a natural number can be voiced or depicted, which are natural ways of transmitting information.

Let's look at the basic skills of voicing (reading) and representing (writing) natural numbers.

Decimal notation of a natural number

Let us remember how the following characters are represented (we will indicate them separated by commas): 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . We call these signs numbers.

Now let's take it as a rule that when depicting (recording) any natural number, only the indicated numbers 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 another in a line and there is always a digit other than zero on the left.

Let us indicate examples of the correct recording of natural numbers: 703, 881, 13, 333, 1,023, 7, 500,001. The spacing between numbers is 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, all the digits from the above series do not have to be present. 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 recording 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 a quantitative meaning, among other things. Natural numbers, as a numbering tool, are discussed in the topic on comparing natural numbers.

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

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

Now imagine: Ψ Ψ . We see one object and another object, i.e. in the recording 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 quantities items.

Definition 1

If the record of a number coincides with the record of the number 0, then such a number is called "zero". Zero is not a natural number, but it is considered along with other natural numbers. Zero denotes absence, 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, when writing which two signs are used - two digits. In this case, the numbers used can be either the same or different.

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

Let's consider what meaning is contained in two-digit numbers. We will rely on the quantitative meaning of single-digit natural numbers that is already known to us.

Let's introduce such a concept as “ten”.

Let's imagine a set of objects that consists of nine and one more. In this case, we can talk about 1 ten (“one dozen”) objects. If you imagine one ten and one more, then we are talking about 2 tens (“two tens”). Adding one more to two tens, we get three tens. And so on: continuing to add one ten at a time, we will 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 a natural number, and the number on the right will indicate the number of units. In the case where the number 0 is located on the right, then we are talking about the absence of units. The above is the quantitative meaning of two-digit natural numbers. There are 90 of them in total.

Definition 4

Three-digit natural numbers- natural numbers, when writing which three signs are used - 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-digit natural numbers, we introduce the concept "a hundred".

Definition 5

One hundred (1 hundred) is a set consisting of ten tens. A hundred and another hundred make 2 hundreds. Add one more hundred and get 3 hundreds. By gradually adding one hundred at a time, we get: four hundred, five hundred, six hundred, seven hundred, eight hundred, nine hundred.

Let's consider the notation of a three-digit number itself: the single-digit natural numbers included in it are written one after another from left to right. The rightmost single digit number indicates the number of units; the next single-digit number to the left is by the number of tens; the leftmost single digit number is in the number of hundreds. If the entry contains the number 0, it indicates the absence of units and/or tens.

Thus, 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.

Multi-digit natural numbers

From all of the above, it is now possible to move on to the definition of multi-valued natural numbers.

Definition 6

Multi-digit natural numbers– natural numbers, when writing which two or more characters are used. Multi-digit natural numbers are two-digit, three-digit, and so on numbers.

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

Similar 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 ten thousand, 9 hundred thousand and 4 million.

To summarize, we looked at the skill of grouping units into various sets (tens, hundreds, etc.) and saw that the numbers in the notation 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 indicated 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 letter writing:

To correctly read and write two-digit numbers, you need to memorize the data in Table 2:

To read other two-digit natural numbers, we will use the data from both tables; we will consider this with an example. Let's say we need to read the two-digit natural 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 conjunction “and” between the words does not need to be pronounced. Let's say we need to use the indicated number 21 in a certain sentence, indicating the number of objects in the genitive case: “there are no 21 apples.” In this case, the pronunciation will sound like this: “there are not twenty-one apples.”

Let us give another example for clarity: the number 76, which is read as “seventy-six” and, for example, “seventy-six tons.”

To fully read a three-digit number, we also use the data from all of the indicated tables. For example, given the 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 case, 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 forty-three” or in declension according to cases, for example, like this: “there are no five hundred forty-three rubles.”

Let's move on to the general principle of reading multi-digit natural numbers: to read a multi-digit number, you need to divide 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 rightmost 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, and it is quite difficult to perceive them by ear.

To make the recording easier to read, classes are separated from each other by a small indentation. For example, 31,013,736, 134,678, 23,476,009,434, 2,533,467,001,222.

To read a multi-digit number, we call the numbers that make it up one by one (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 three digits 0 are also not pronounced. If one class contains one or two digits on the left, then they are not used in any way when reading. For example, 054 would be read as “fifty-four” or 001 as “one”.

Example 1

Let's look at the reading of the number 2,533,467,001,222 in detail:

We read the number 2 as a component of the class of trillions - “two”;

By adding the name of the class, we get: “two trillion”;

We read the next 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 sixty-seven million”;

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

We read the last class of 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 will 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 for correctly reading multi-digit numbers is the skill of dividing a multi-digit number into classes, knowledge of the corresponding names and understanding of the principle of reading two- and three-digit numbers.

As is already clear from all of the above, its value depends on the position at which the digit appears in the notation of a number. That is, for example, the number 3 in the natural number 314 indicates 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 ones 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- this 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 categories have their own names, we have already used them above. From right to left there are digits: units, tens, hundreds, thousands, tens of thousands, etc.

For ease of remembering, 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’s notation. For example, this table contains the names of all digits for a number with 15 digits. Subsequent discharges also have names, but are used extremely rarely and are very inconvenient to hear.

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

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

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

Definition 8

Lowest (junior) rank of any multi-digit natural number – the units digit.

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

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

Logically it follows 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 place is older than the hundreds place, but younger than the millions place.

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

Briefly about the decimal number system

Notation– a method of writing numbers using signs.

Positional number systems– those in which the meaning of a digit in a number depends on its position in the number record.

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. The number 10 plays a special place here. We count in tens: ten units make a ten, ten tens will unite into a hundred, etc. 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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