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

Yandex.RTB R-A-339285-1

General idea of ​​natural numbers

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

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

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

Decimal notation of a natural number

Recall how the following characters are displayed (we indicate them separated by commas): 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . These characters are called numbers.

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

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

Definition 1

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

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

Quantitative meaning of natural numbers

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

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

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

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

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

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

Definition 1

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

Single digit natural numbers

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

Definition 2

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

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

Two-digit and three-digit natural numbers

Definition 3

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

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

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

Let's introduce such concept as "ten".

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

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

Definition 4

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

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

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

Definition 5

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

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

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

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

Multivalued natural numbers

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

Definition 6

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

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

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

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

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

Reading natural numbers, classes

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

Number	masculine	Feminine gender	Neuter gender
1 2 3 4 5 6 7 8 9	One Two Three Four Five Six Seven Eight Nine	One Two Three Four Five Six Seven Eight Nine	One Two Three Four Five Six Seven Eight Nine

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	to one two Trem 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 About two About three About four Again About six About seven About eight About nine

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

Number	Masculine, feminine and neuter
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 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 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 Ninety	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 About ninety

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

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

Number	Nominative	Genitive	Dative	Accusative	Instrumental case	Prepositional
100 200 300 400 500 600 700 800 900	Hundred Two hundred Three hundred Four hundred Five hundred Six hundreds Seven hundred Eight hundred Nine hundreds	Sta two hundred three hundred four hundred five hundred six hundred Seven hundred eight hundred nine hundred	Sta two hundred Tremstam four hundred five hundred Six hundred seven hundred eight hundred Nine hundred	Hundred Two hundred Three hundred Four hundred Five hundred Six hundreds Seven hundred Eight hundred Nine hundreds	Sta two hundred Three hundred four hundred five hundred six hundred seven hundred eight hundred Nine hundred	About a hundred About two hundred About three hundred About four hundred About five hundred About six hundred About seven hundred About eight hundred About nine hundred

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

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

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

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

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

Class trillion	Class billion	Class million	Thousand class	Unit class
			134	678
		31	013	736
	23	476	009	434
2	533	467	001	222

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

Example 1

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

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

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

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

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

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

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 can also read the other given numbers:

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

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

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

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

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

Definition 7

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

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

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

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

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

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

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

Definition 8

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

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

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

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

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

Briefly about the decimal number system

Definition 9

Notation- a method of writing numbers using signs.

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

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

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

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

Integers

Natural numbers definition are positive integers. Natural numbers are used to count objects and for many other purposes. Here are the numbers:

This is a natural series of numbers.
Zero is a natural number? No, zero is not natural number.
How many natural numbers are there? There is an infinite set of natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It cannot be specified, because there is an infinite set of natural numbers.

The sum of natural numbers is a natural number. So, the addition of 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 natural numbers is a natural number, otherwise it is not.

The quotient of natural numbers There is not always a natural number. If for natural numbers a and b

where c is a natural number, it means that a is evenly 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 the natural number by which the first number is evenly divisible.

Every natural number is divisible by 1 and itself.

Simple natural numbers are only divisible by 1 and themselves. Here we mean divided completely. Example, numbers 2; 3; five; 7 is only divisible by 1 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 consists of 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 natural numbers, zero and the opposite of natural numbers.

Numbers opposite to 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 integers and fractions.

Any rational number can be represented as a periodic fraction. Examples:

1,(0); 3,(6); 0,(0);...

It can be seen from the examples that any integer is a periodic fraction with a period of zero.

Any rational number can be represented as a fraction m/n, where m integer,n natural number. Let's represent the number 3,(6) from the previous example as such a fraction.

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:

Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet been able to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.

If we turn the logic we are used to, everything falls into place. Achilles runs with constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells of a flying arrow:

A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow is at rest at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.

Wednesday, July 4, 2018

Very well the differences between set and multiset are described in Wikipedia. We look.

As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.

No matter how mathematicians hide behind the phrase "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.

We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.

First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will begin to convulsively recall physics: different coins there is a different amount of dirt, the crystal structure and the arrangement of atoms of each coin is unique...

And now I have the most interest Ask: where is the boundary beyond which elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science here is not even close.

Look here. We select football stadiums with the same field area. The area of ​​the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us about either a set or a multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."

Sunday, March 18, 2018

The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but they are shamans for that, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.

Do you need proof? Open Wikipedia and try to find the "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics, the task sounds like this: "Find the sum of graphic symbols representing any number." Mathematicians cannot solve this problem, but shamans can do it elementarily.

Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let's say we have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.

1. Write down the number on a piece of paper. What have we done? We have converted the number to a number graphic symbol. This is not a mathematical operation.

2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.

3. Convert individual graphic characters to numbers. This is not a mathematical operation.

4. Add up the resulting numbers. Now that's mathematics.

The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that is not all.

From the point of view of mathematics, it does not matter in which number system we write the number. So, in different systems reckoning, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With a large number of 12345, I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have already done that. Let's look at the result.

As you can see, in different number systems, the sum of the digits of the same number is different. This result has nothing to do with mathematics. It's like finding the area of ​​a rectangle in meters and centimeters would give you completely different results.

Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.

The result obtained should be considered as proof that number systems are units of measurement of numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.

What is real mathematics? This is when the result of a mathematical action does not depend on the value of the number, the unit of measure used, and on who performs this action.

Sign on the door Opens the door and says:

Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?

Female... A halo on top and an arrow down is male.

If you have such a work of design art flashing before your eyes several times a day,

Then it is not surprising that you suddenly find a strange icon in your car:

Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.

1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.

Where does the study of mathematics begin? Yes, that's right, from the study of natural numbers and actions with them.Integers (fromlat. naturalis- natural; natural numbers)numbers that arise naturally when counting (for example, 1, 2, 3, 4, 5, 6, 7, 8, 9 ...). The sequence of all natural numbers arranged in ascending order is called the natural number.

There are two approaches to the definition of natural numbers:

1. counting (numbering) items ( first, second, the third, fourth, fifth"…);
2. natural numbers are numbers that occur when quantity designation items ( 0 items, 1 item, 2 items, 3 items, 4 items, 5 items ).

In the first case, the series of natural numbers starts from one, in the second - from zero. There is no common opinion for most mathematicians on the preference of the first or second approach (that is, whether to consider zero as a natural number or not). The vast majority of Russian sources have traditionally adopted the first approach. The second approach, for example, is used in the worksNicolas Bourbaki , where natural numbers are defined aspower finite sets .

Negative and non-integer (rational , real ,…) numbers are not classified as natural.

The set of all natural numbers usually denoted by the symbol N (fromlat. naturalis- natural). The set of natural numbers is infinite, since for any natural number n there is a natural number greater than n.

The presence of zero facilitates the formulation and proof of many theorems in the arithmetic of natural numbers, so the first approach introduces the useful notion extended natural series , including zero. The extended row is denoted by N 0 or Z0 .

TOclosed operations (operations that do not output a result from the set of natural numbers) on natural numbers include the following arithmetic operations:

• addition: term + term = sum;
• multiplication: multiplier × multiplier = product;
• exponentiation: a b , where a is the base of the degree, b is the exponent. If a and b are natural numbers, then the result will also be a natural number.

Additionally, two more operations are considered (from a formal point of view, they are not operations on natural numbers, since they are not defined for allpairs of numbers (sometimes they exist, sometimes they don't)):

• subtraction: minuend - subtrahend = difference. In this case, the minuend must be greater than the subtrahend (or equal to it, if we consider zero as a natural number)
• division with remainder: dividend / divisor = (quotient, remainder). The quotient p and the remainder r from dividing a by b are defined as follows: a=p*r+b, and 0<=r

It should be noted that the operations of addition and multiplication are fundamental. In particular,