natural value. What is a natural number

Mathematics emerged from general philosophy around the sixth century BC. e., and from that moment began her victorious march around the world. Each stage of development introduced something new - elementary counting evolved, transformed into differential and integral calculus, centuries changed, formulas became more and more confusing, and the moment came when "the most complex mathematics began - all numbers disappeared from it." But what was the basis?

The beginning of time

Natural numbers appeared along with the first mathematical operations. Once a spine, two spines, three spines ... They appeared thanks to Indian scientists who deduced the first positional

The word "positionality" means that the location of each digit in a number is strictly defined and corresponds to its category. For example, the numbers 784 and 487 are the same numbers, but the numbers are not equivalent, since the first includes 7 hundreds, while the second only 4. The Arabs picked up the innovation of the Indians, who brought the numbers to the form that we know Now.

In ancient times, numbers were given mystical meaning, Pythagoras believed that the number underlies the creation of the world along with the basic elements - fire, water, earth, air. If we consider everything only from the mathematical side, then what is a natural number? The field of natural numbers is denoted as N and is an infinite series of numbers that are integer and positive: 1, 2, 3, … + ∞. Zero is excluded. It is mainly used for counting items and indicating order.

What is in mathematics? Peano's axioms

The field N is the base field on which elementary mathematics relies. Over time, the fields of integers, rational,

The work of the Italian mathematician Giuseppe Peano made possible the further structuring of arithmetic, achieved its formality and paved the way for further conclusions that went beyond the field N.

What is a natural number was clarified earlier in simple language, below we will consider a mathematical definition based on Peano's axioms.

• One is considered a natural number.
• The number that follows a natural number is a natural number.
• There is no natural number before one.
• If the number b follows both the number c and the number d, then c=d.
• The axiom of induction, which in turn shows what a natural number is: if some statement that depends on a parameter is true for the number 1, then we assume that it also works for the number n from the field of natural numbers N. Then the statement is also true for n =1 from the field of natural numbers N.

Basic operations for the field of natural numbers

Since the field N became the first for mathematical calculations, both the domains of definition and the ranges of values ​​of a number of operations below refer to it. They are closed and not. The main difference is that closed operations are guaranteed to leave a result within the set N, no matter what numbers are involved. It is enough that they are natural. The outcome of the remaining numerical interactions is no longer so unambiguous and directly depends on what kind of numbers are involved in the expression, since it may contradict the main definition. So, closed operations:

• addition - x + y = z, where x, y, z are included in the field N;
• multiplication - x * y = z, where x, y, z are included in the N field;
• exponentiation - x y , where x, y are included in the N field.

The remaining operations, the result of which may not exist in the context of the definition "what is a natural number", are the following:

Properties of numbers belonging to the field N

All further mathematical reasoning will be based on the following properties, the most trivial, but no less important.

• The commutative property of addition is x + y = y + x, where the numbers x, y are included in the field N. Or the well-known "the sum does not change from a change in the places of the terms."
• The commutative property of multiplication is x * y = y * x, where the numbers x, y are included in the field N.
• The associative property of addition is (x + y) + z = x + (y + z), where x, y, z are included in the field N.
• The associative property of multiplication is (x * y) * z = x * (y * z), where the numbers x, y, z are included in the field N.
• distribution property - x (y + z) = x * y + x * z, where the numbers x, y, z are included in the field N.

Pythagorean table

One of the first steps in the knowledge of the entire structure of elementary mathematics by schoolchildren, after they have understood for themselves which numbers are called natural, is the Pythagorean table. It can be considered not only from the point of view of science, but also as a valuable scientific monument.

This multiplication table has undergone a number of changes over time: zero has been removed from it, and numbers from 1 to 10 denote themselves, without taking into account orders (hundreds, thousands ...). It is a table in which the headings of rows and columns are numbers, and the contents of the cells of their intersection is equal to their product.

In the practice of teaching in recent decades, there has been a need to memorize the Pythagorean table "in order", that is, memorization went first. Multiplication by 1 was excluded because the result was 1 or greater. Meanwhile, in the table with the naked eye, you can see a pattern: the product of numbers grows by one step, which is equal to the title of the line. Thus, the second factor shows us how many times we need to take the first one in order to get the desired product. This system is much more convenient than the one practiced in the Middle Ages: even understanding what a natural number is and how trivial it is, people managed to complicate their everyday counting using a system based on powers of two.

Subset as the cradle of mathematics

On this moment the field of natural numbers N is considered only as one of the subsets of complex numbers, but this does not make them less valuable in science. The natural number is the first thing a child learns by studying himself and the world. One finger, two fingers ... Thanks to him, a person develops logical thinking, as well as the ability to determine the cause and deduce the effect, paving the way for great discoveries.

Question to the scientist: I heard that the sum of all natural numbers is −1/12. Is this some kind of trick, or is it true?

MIPT press service response- Yes, such a result can be obtained using a technique called the expansion of a function in a series.

The question posed by the reader is rather complicated, and therefore we are answering it not with the usual text for the “Question to a Scientist” rubric for several paragraphs, but with some greatly simplified semblance of a mathematical article.

IN scientific articles in mathematics, where it is required to prove some complex theorem, the story is divided into several parts, and various auxiliary statements can be proved one by one in them. We assume that readers are familiar with the mathematics course within the nine grades, so we apologize in advance to those who find the story too simple - graduates can immediately turn to http://en.wikipedia.org/wiki/Ramanujan_summation .

Total sum

Let's start by talking about how you can add all natural numbers. Natural numbers are numbers that are used to count solid objects - they are all whole and non-negative. It is natural numbers that children learn first of all: 1, 2, 3 and so on. The sum of all natural numbers will be an expression of the form 1+2+3+... = and so on ad infinitum.

The series of natural numbers is infinite, it is easy to prove: after all, one can always be added to an arbitrarily large number. Or even multiply this number by itself, or even calculate its factorial - it is clear that an even larger value will turn out, which will also be a natural number.

All operations with infinitely large values ​​are dealt with in detail in the course of mathematical analysis, but now, in order to be understood by those who have not yet passed this course, we will somewhat simplify the essence. Let's say that infinity, to which one was added, infinity, which was squared or factorial of infinity, is all also infinity. We can assume that infinity is such a special mathematical object.

And according to all the rules of mathematical analysis within the framework of the first semester, the sum 1+2+3+...+infinity is also infinite. This is easy to understand from the previous paragraph: if you add something to infinity, it will still be infinity.

However, in 1913, the brilliant self-taught Indian mathematician Srinivasa Ramanujan Iyengor came up with a way to add natural numbers in a slightly different way. Despite the fact that Ramanujan did not receive special education, his knowledge was not limited to today's school course - the mathematician knew about the existence of the Euler-Maclaurin formula. Since she plays an important role in the further narrative, she will also have to be told in more detail.

Euler-Maclaurin formula

Let's start by writing this formula:

As you can see, it is quite complex. Some readers can skip this section entirely, some can read the relevant textbooks or at least the Wikipedia article, and for the rest we will give a brief commentary. The key role in the formula is played by an arbitrary function f(x), which can be almost anything, as long as it has a sufficient number of derivatives. For those who are not familiar with this mathematical concept (and still decided to read what was written here!), let's say even simpler - the function graph should not be a line that breaks sharply at any point.

The derivative of a function, if its meaning is extremely simplified, is a value that shows how quickly the function grows or decreases. From a geometric point of view, the derivative is the tangent of the slope of the tangent to the graph.

On the left side of the formula is the sum of the form “the value of f(x) at point m + the value of f(x) at point m+1 + the value of f(x) at point m+2 and so on until point m+n”. Moreover, the numbers m and n are natural, this should be emphasized especially.

On the right, we see several terms, and they seem very cumbersome. The first one (ends with dx) is the integral of the function from point m to point n. At the risk of incurring the wrath of all

The third term is the sum of the Bernoulli numbers (B 2k) divided by the factorial of the double value of the number k and multiplied by the difference between the derivatives of the function f(x) at the points n and m. Moreover, what complicates matters even more, here is not just a derivative, but a derivative of order 2k-1. That is, the whole third term looks like this:

The Bernoulli number B 2 (“2” since 2k is in the formula, and we start adding from k=1) we divide by the factorial 2 (this is just a two for now) and multiply by the difference of the first order derivatives (2k-1 with k=1) functions f(x) at points n and m

The Bernoulli number B 4 (“4” since 2k is in the formula, and k is now equal to 2) is divided by the factorial 4 (1 × 2x3 × 4 \u003d 24) and multiplied by the difference of third-order derivatives (2k-1 with k \u003d 2) functions f(x) at points n and m

We divide the Bernoulli number B 6 (see above) by the factorial 6 (1 × 2x3 × 4x5 × 6 \u003d 720) and multiply by the difference of the fifth order derivatives (2k-1 for k \u003d 3) of the function f (x) at points n and m

The summation continues up to k=p. The numbers k and p are obtained by some arbitrary values, which we can choose in different ways, together with m and n - natural numbers, by which the section with the function f (x) we are considering is limited. That is, there are as many as four parameters in the formula, and this, coupled with the arbitrariness of the function f (x), opens up a lot of scope for research.

The remaining modest R, alas, is not a constant here, but also a rather cumbersome construction, expressed in terms of the Bernoulli numbers already mentioned above. Now is the time to explain what it is, where it came from and why in general mathematicians began to consider such complex expressions.

Bernoulli numbers and series expansions

In mathematical analysis, there is such a key concept as series expansion. This means that you can take some function and write it not directly (for example, y = sin(x^2) + 1/ln(x) + 3x), but as an infinite sum of a set of terms of the same type. For example, many functions can be represented as a sum of power functions multiplied by some coefficients - that is, a graph of complex shape will be reduced to a combination of linear, quadratic, cubic ... and so on - curves.

In the theory of electrical signal processing, the so-called Fourier series plays a huge role - any curve can be expanded into a series of sines and cosines of different periods; such a decomposition is necessary to convert the signal from the microphone into a sequence of zeros and ones inside, say, the electronic circuit of a mobile phone. Series expansions also make it possible to consider non-elementary functions, and a number of the most important physical equations, when solved, give expressions in the form of a series, and not in the form of some finite combination of functions.

In the 17th century, mathematicians began to work closely with the theory of series. Somewhat later, this allowed physicists to efficiently calculate the heating processes of various objects and solve many other problems that we will not consider here. We only note that in the MIPT program, as in the mathematical courses of all leading physics universities, at least one semester is devoted to equations with solutions in the form of one or another series.

Jacob Bernoulli investigated the problem of summation of natural numbers to the same degree (1^6 + 2^6 + 3^6 + ... for example) and obtained numbers that can be used to expand other functions into the power series mentioned above - for example, tg(x). Although, it would seem, the tangent is not very similar to at least a parabola, at least to any power function!

Bernoulli polynomials later found their application not only in the equations of mathematical physics, but also in probability theory. This, in general, is predictable (after all, a number of physical processes, such as Brownian motion or the decay of nuclei, are precisely due to various kinds of accidents), but it still deserves special mention.

The cumbersome Euler-Maclaurin formula has been used by mathematicians for various purposes. Since, on the one hand, it contains the sum of the values ​​of functions at certain points, and on the other hand, there are both integrals and series expansions, using this formula it is possible (depending on what we know) how to take a complex integral, and determine the sum of the series.

Srinivasa Ramanujan came up with another application for this formula. He modified it a little and got the following expression:

As a function of f(x), he considered simply x - let f(x) = x, this is a completely legitimate assumption. But for this function, the first derivative is simply equal to one, and the second and all subsequent derivatives are equal to zero: if everything is carefully substituted into the above expression and the corresponding Bernoulli numbers are determined, then exactly −1/12 will turn out.

This, of course, was taken by the Indian mathematician himself as something out of the ordinary. Since he was not just self-taught, but a talented self-taught person, he did not tell everyone about the discovery that corrected the foundations of mathematics, but instead wrote a letter to Godfrey Hardy, a recognized expert in the field of both number theory and mathematical analysis. By the way, the letter contained a note that Hardy would probably want to point out to the author the nearest psychiatric hospital: however, the result, of course, was not a hospital, but a joint work.

Paradox

Summarizing all the above, we get the following: the sum of all natural numbers is equal to −1/12 using a special formula that allows you to expand an arbitrary function into a series with coefficients called Bernoulli numbers. However, this does not mean that 1+2+3+4 turns out to be greater than 1+2+3+... and so on ad infinitum. In this case, we are dealing with a paradox, which is due to the fact that expansion into a series is a kind of approximation and simplification.

One can give an example of a much simpler and more obvious mathematical paradox associated with the expression of one thing in terms of something else. Let's take a sheet of paper in a box and draw a stepped line with the width and height of the step in one cell. The length of such a line, obviously, is equal to twice the number of cells - but the length of the diagonal straightening the "ladder" is equal to the number of cells multiplied by the root of two. If you make the ladder very small, it will still be the same length and the broken line, which is almost indistinguishable from the diagonal, will be at the root of two times the same diagonal! As you can see, for paradoxical examples it is not necessary to write long complex formulas.

The Euler-Maclaurin formula, if you do not go into the wilds of mathematical analysis, is the same approximation as a broken line instead of a straight line. Using this approximation, you can get the same −1/12, but this is far from always appropriate and justified. In a number of problems of theoretical physics, such calculations are used for calculations, but this is the very cutting edge of research, where it is too early to talk about the correct representation of reality by mathematical abstractions, and discrepancies between different calculations with each other are quite common.

Thus, estimates of the vacuum energy density based on quantum field theory and based on astrophysical observations differ by more than 120 orders of magnitude. That is 10^120 power times. This is one of the unsolved problems of modern physics; here clearly shines through a gap in our knowledge of the universe. Or the problem is the lack of suitable mathematical methods for describing the world around us. Theoretical physicists, together with mathematicians, are trying to find ways to describe physical processes in which there will be no divergent (going to infinity) series, but this is far from an easy task.

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

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

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Natural numbers are a general representation.

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

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

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

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

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

single digit natural numbers.

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

Definition.

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

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

Two-digit and three-digit natural numbers.

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

Definition.

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

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

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

First, let's introduce the concept ten.

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

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

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

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

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

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

Definition.

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

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

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

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

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

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, the next is 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 80 And 8 , therefore, we read: "Eighty-eight." And here is an example of a sentence: "He is thinking about eighty-eight rubles."

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

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

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

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

We turn to reading multi-digit numbers.

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

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

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

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

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

We read from left to right,

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

Let's start reading the number 10 000 501 .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Briefly about the decimal number system.

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

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

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

Apart from decimal system there are others, for example, in computer science, a binary positional number system is used, and we encounter a sexagesimal system when it comes to measuring time.

Bibliography.

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

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

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What numbers are called natural

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

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

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

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

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

Set N

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

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

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

smallest natural number

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

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

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

Series of natural numbers

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

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

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

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

All N

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

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

For example:

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

Sequence in N

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

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

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

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

What does even N mean?

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

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

N properties

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

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

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

What are the components of a multiplication called?

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

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

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

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

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

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

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

What are natural numbers in mathematics?

Discharges and classes of natural numbers

Conclusion

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

The history of natural numbers began in primitive times. Since ancient times, people have counted objects. For example, in trade, a commodity account was needed, or in construction, a material account. Yes, even in everyday life, too, I had to count things, products, livestock. At first, numbers were used only for counting in life, in practice, but later, with the development of mathematics, they became part of science.

Integers are the numbers that we use when counting objects.

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

Zero is not a natural number.

All natural numbers, or let's call the set of natural numbers, is denoted by the symbol N.

Table of natural numbers.

natural row.

Natural numbers written in ascending order in a row form natural series or series of natural numbers.

Properties of the natural series:

• The smallest natural number is one.
• In the natural series, the next number is greater than the previous one by one. (1, 2, 3, …) Three dots or three dots are used if it is impossible to complete the sequence of numbers.
• The natural series has no maximum number, it is infinite.

Example #1:
Write the first 5 natural numbers.
Solution:
Natural numbers start with one.
1, 2, 3, 4, 5

Example #2:
Is zero a natural number?
Answer: no.

Example #3:
What is the first number in natural series?
Answer: the natural number starts with one.

Example #4:
What is the last number in the natural series? What is the largest natural number?
Answer: The natural number starts from one. Each next number is greater than the previous one by one, so the last number does not exist. There is no largest number.

Example #5:
Does the unit in the natural series have a previous number?
Answer: no, because one is the first number in the natural series.

Example #6:
Name the next number in the natural series after the numbers: a) 5, b) 67, c) 9998.
Answer: a) 6, b) 68, c) 9999.

Example #7:
How many numbers are in the natural series between the numbers: a) 1 and 5, b) 14 and 19.
Solution:
a) 1, 2, 3, 4, 5 - three numbers are between the numbers 1 and 5.
b) 14, 15, 16, 17, 18, 19 - four numbers are between the numbers 14 and 19.

Example #8:
Name the previous number after the number 11.
Answer: 10.

Example #9:
What numbers are used to count objects?
Answer: natural numbers.