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 knocked out dots or dashes on various objects, 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 in Indian system she was absent.
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 that are expressed as decimals: 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!
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
Definition 3Two-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 us 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 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:
| Number | Masculine | Feminine | Neuter gender |
|
1 |
One |
One |
One |
| Number | Nominative case | Genitive | Dative | Accusative | Instrumental case | Prepositional |
| 1 2 3 4 5 6 7 8 9 |
One Two Three Four Five Six Seven Eight Nine |
One Two Three Four Five Six Semi Eight Nine |
Alone Two Three Four Five Six Semi Eight Nine |
One Two Three Four Five Six Seven Eight Nine |
One Two Three Four Five Six Family Eight Nine |
About one thing About two About three About four Again About six About seven About eight About nine |
To correctly read and write two-digit numbers, you need to memorize the data in Table 2:
| Number |
Masculine, feminine and neuter gender |
| 10 11 12 13 14 15 16 17 18 19 20 30 40 50 60 70 80 90 |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety |
| Number | Nominative case | Genitive | Dative | Accusative | Instrumental case | Prepositional |
| 10 11 12 13 14 15 16 17 18 19 20 30 40 50 60 70 80 90 |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety |
Ten |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Magpie Fifty Sixty Seventy Eighty Ninety |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety |
Ten Eleven twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Magpie Fifty sixty Seventy Eighty nineteen |
About ten About eleven About twelve About thirteen About fourteen About fifteen About sixteen About seventeen About eighteen About nineteen About twenty About thirty Oh magpie About fifty About sixty About seventy About eighty Oh ninety |
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 specified number 21 in a certain sentence, indicating the number of objects in genitive case: “there are no 21 apples.” sound in in this case the pronunciation will be as follows: “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.”
| Number | Nominative | Genitive | Dative | Accusative | Instrumental case | Prepositional |
| 100 200 300 400 500 600 700 800 900 |
One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
hundred Two hundred Three hundred Four hundred Five hundred Six hundred Semistam Eight hundred Nine hundred |
One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
Oh hundred About two hundred About three hundred About four hundred About five hundred About six hundred About the seven hundred About eight hundred About nine hundred |
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 general principle 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.
| Class trillion |
Class billions |
Class millions |
Class of thousands | Unit class |
| 134 | 678 | |||
| 31 | 013 | 736 | ||
| 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 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 will 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 correct 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
Definition 9Notation- 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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Definition
Natural numbers are 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 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:
- Eat smallest number and there is no greatest.
- Each subsequent number is greater than the previous one by 1 (with the exception of the unit itself).
- 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 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 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 subtraction action is carried out for the 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. 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
- 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.
- 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 that is used to famous work and one of the factors may find another – unknown – factor. 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 factor for 7. In this case, it is the number 5. We 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, 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 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 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 (recording) and the skills of 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 notation. 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 " two».
Likewise, - 3
subject (read " three» subject), - 4
(« four") 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 coincides with 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 writing a three-digit number means 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 figure 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.
Integers
Natural numbers definition are positive integers. Natural numbers are used to count objects and for many other purposes. These are the numbers:
This is a natural series of numbers.
Is zero a natural number? No, zero is not a natural number.
How many natural numbers are there? There is an infinite number of natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It is impossible to specify it, because there is an infinite number of natural numbers.
The sum of natural numbers is a natural number. So, adding natural numbers a and b:
The product of natural numbers is a natural number. So, the product of natural numbers a and b:
c is always a natural number.
Difference of natural numbers There is not always a natural number. If the minuend is greater than the subtrahend, then the difference of the natural numbers is a natural number, otherwise it is not.
The quotient of natural numbers is not always a natural number. If for natural numbers a and b
where c is a natural number, this means that a is divisible by b. In this example, a is the dividend, b is the divisor, c is the quotient.
The divisor of a natural number is a natural number by which the first number is divisible by a whole.
Every natural number is divisible by one and itself.
Prime natural numbers are divisible only by one and themselves. Here we mean divided entirely. Example, numbers 2; 3; 5; 7 is only divisible by one and itself. These are simple natural numbers.
One is not considered a prime number.
Numbers that are greater than one and that are not prime are called composite numbers. Examples of composite numbers:
One is not considered a composite number.
The set of natural numbers is one, prime numbers and composite numbers.
The set of natural numbers is denoted by the Latin letter N.
Properties of addition and multiplication of natural numbers:
commutative property of addition
associative property of addition
(a + b) + c = a + (b + c);
commutative property of multiplication
associative property of multiplication
(ab) c = a (bc);
distributive property of multiplication
A (b + c) = ab + ac;
Whole numbers
Integers are the natural numbers, zero and the opposites of the natural numbers.
The opposite of natural numbers are negative integers, for example:
1; -2; -3; -4;...
The set of integers is denoted by the Latin letter Z.
Rational numbers
Rational numbers are whole numbers and fractions.
Any rational number can be represented as a periodic fraction. Examples:
1,(0); 3,(6); 0,(0);...
From the examples it is clear that any integer is a periodic fraction with period zero.
Any rational number can be represented as a fraction m/n, where m is an integer and n is a natural number. Let's imagine the number 3,(6) from the previous example as such a fraction.