The simplest number is natural number... They are used in everyday life for counting items, i.e. to calculate their number and order.
What is a natural number: natural numbersare the numbers that are used for counting items or to indicate the serial number of any item from all homogeneousitems.
Integers are numbers starting from one. They form naturally during counting.For example, 1,2,3,4,5 ... -first natural numbers.
Smallest natural number - one. There is no greatest natural number. When counting the number zero is not used, so zero is a natural number.
Natural series of numbers is a sequence of all natural numbers. Natural numbers notation:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ...
In a natural row, each number is greater than the previous one by one.
How many numbers are in a natural row? The natural number is infinite, the largest natural number does not exist.
Decimal, since 10 units of any digit form 1 unit of the most significant digit. Positional so how the meaning of a digit depends on its place in the number, i.e. from the category where it is written.
Classes of natural numbers.
Any natural number can be written using 10 Arabic numerals:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
To read natural numbers, they are divided, starting from the right, into groups of 3 numbers each. 3 first the numbers on the right are the class of units, the next 3 are the class of thousands, then the classes of millions, billions andetc. Each of the numbers of the class is called itdischarge.
Comparison of natural numbers.
Of the 2 natural numbers, the less is the number that is called earlier when counting. for instance, number 7 less 11 (written like this:7 < 11 ). When one number is greater than the second, it is written like this:386 > 99 .
Table of categories and classes of numbers.
|
1st class unit |
1st digit of the unit 2nd rank tens 3rd rank hundreds |
|
2nd class thousand |
1st digit units of thousand 2nd rank tens of thousands 3rd rank hundreds of thousands |
|
3rd grade millions |
1st digit unit million 2nd rank tens of millions 3rd rank hundreds of millions |
|
4th grade billions |
1st digit unit billion 2nd rank tens of billions 3rd rank hundreds of billions |
|
Numbers 5th grade and above are considered large numbers. 5th grade units - trillions, 6th class - quadrillions, 7th grade - quintillions, 8th grade - sextillions, 9th grade -eptillions. Basic properties of natural numbers.
Actions on natural numbers. 4. Division of natural numbers is an operation opposite to multiplication. If a b ∙ c \u003d athen
Division Formulas: a: 1 \u003d a a: a \u003d 1, a ≠ 0 0: a \u003d 0, a ≠ 0 (and ∙ b): c \u003d (a: c) ∙ b (and ∙ b): c \u003d (b: c) ∙ a Numeric expressions and numeric equalities. The notation where numbers are connected by action signs is numerical expression. For example, 10 ∙ 3 + 4; (60-2 ∙ 5): 10. Records where 2 numeric expressions are combined with an equal sign is numerical equalities. Equality has left and right sides. The order of performing arithmetic operations. Addition and subtraction of numbers are first-degree actions, and multiplication and division are second-degree actions. When a numerical expression consists of actions of only one degree, then they are performed sequentiallyfrom left to right. When expressions consist of actions of only the first and second degrees, then the actions are performed first second degree, and then - actions of the first degree. When there are brackets in the expression, the actions in the brackets are performed first. For example, 36: (10-4) + 3 ∙ 5 \u003d 36: 6 + 15 \u003d 6 + 15 \u003d 21. |
Let's say Achilles runs ten times faster than a turtle and is a thousand steps behind it. During the time it takes Achilles to run this distance, the turtle will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the turtle crawls another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the turtle.
This reasoning came as a logical shock to all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... All of them, one way or another, considered Zeno's aporias. The shock was so strong that " ... the discussions continue at the present time, the scientific community has not yet managed to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them has become a generally accepted solution to the question ..."[Wikipedia, Zeno's Aporia"]. 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 magnitude to. This transition involves applying instead of constants. As far as I understand, the mathematical apparatus for using variable units of measurement either has not yet been developed, or it has not been applied to Zeno's aporia. Applying our usual logic leads us into a trap. We, by inertia of thinking, apply constant time units to the reciprocal. From a physical point of view, it looks like time dilation until it stops completely at the moment when Achilles is level with the turtle. If time stops, Achilles can no longer overtake the turtle.
If we turn over the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his 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 catch up with the turtle."
How can you avoid this logical trap? Stay in constant time units and do not go backwards. In Zeno's language, it looks like this:
During the time during which Achilles will run a thousand steps, the turtle will crawl a hundred steps in the same direction. Over the next interval of time, equal to the first, Achilles will run another thousand steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the turtle.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insuperability of the speed of light is very similar to the Zeno aporia "Achilles and the Turtle". 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 Zeno tells about a flying arrow:
A flying arrow is motionless, since at every 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 a flying arrow rests at different points in space, which, in fact, is motion. Another point should be noted here. It is impossible to determine either the fact of its movement or the distance to it from a single photograph of a car on the road. To determine the fact of the movement of the car, two photographs are needed, taken from the same point at different points in time, but the distance cannot be determined from them. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but it is impossible to determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is the fact that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
wednesday, 4 July 2018
The distinction between set and multiset is very well described in Wikipedia. We look.
As you can see, "there can be no two identical elements in a set", but if there are identical elements in a set, such a set is called a "multiset". Such logic of absurdity will never be understood by rational beings. This is the level of talking parrots and trained monkeys, who lack intelligence from the word "absolutely". Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the incompetent engineer died under the rubble of his creation. If the bridge could withstand the load, a talented engineer would build other bridges.
No matter how mathematicians hide behind the phrase "chur, 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 the mathematicians themselves.
We studied mathematics very well and now we are sitting at the checkout, giving out salaries. A mathematician comes to us for his money. We count the entire amount to him and lay 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 hand the mathematician his "mathematical set of salary". We explain the mathematics that he will receive the rest of the bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: "You can apply it to others, you can not apply it to me!" Further, we will begin to assure us that there are different banknote numbers on bills of the same denomination, which means they cannot be considered the same elements. Okay, let's count the salary in coins - there are no numbers on the coins. Here the mathematician will start to frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms in each coin is unique ...
And now I have the most interesting question: where is the line beyond which the 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 did not lie anywhere near here.
Look here. We select football stadiums with the same pitch. The area of \u200b\u200bthe 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, one and the same set of elements is both a set and a multiset at the same time. How is it correct? And here the mathematician-shaman-shuller takes a trump ace out of his sleeve and begins to tell us either about the set or about the 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'll show you, without any "thinkable as not a single whole" or "not thinkable as a whole."
sunday, 18 March 2018
The sum of the digits of the 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 that is why they are shamans in order to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Need proof? Open Wikipedia and try to find the Sum of Digits of a Number page. It 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 the help of 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 - it is elementary.
Let's see what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What should be done in order to find the sum of the digits of this number? Let's go through all the steps in order.
1. We write down the number on a piece of paper. What have we done? We have converted the number to a graphic number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols to numbers. This is not a mathematical operation.
4. Add up the resulting numbers. Now that's mathematics.
The sum of the digits of 12345 is 15. These are the "courses of cutting and sewing" 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 number systems, 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 12345, I do not want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal notation systems. We will not look at every step under a microscope, we have already done that. Let's see 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 the same as if you would get completely different results when determining the area of \u200b\u200ba rectangle in meters and centimeters.
Zero in all number systems looks the same and has no sum of digits. This is another argument for the fact that. A question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists - no. Reality is not all about numbers.
The result should be considered as proof that number systems are units of measurement for 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 their comparison, 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 measurement used and on who performs this action.
Oh! Isn't this a women's toilet?
- Girl! This is a laboratory for the study of the indiscriminate holiness of souls during the ascension to heaven! Halo above and arrow pointing up. What other toilet?
Female ... The nimbus above and the down arrow is male.
If you have such a work of design art before your eyes several times a day,
Then it is not surprising that in your car you suddenly find a strange icon:
Personally, I make an effort on myself so that in a pooping person (one picture), I can see minus four degrees (a composition of several pictures: minus sign, number four, degrees designation). And I don't think this girl is a fool who doesn't know physics. She just has a stereotype of perception of graphic images. And mathematicians constantly teach us this. Here's an example.
1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive the number and the letter as one graphic symbol.
Natural numbers are familiar to a person and intuitive, because they surround us from childhood. In the article below, we will give a basic understanding 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 understanding of natural numbers
At a certain stage in the development of mankind, the task arose of counting certain objects and designating their number, which, in turn, required finding a tool to solve this problem. Natural numbers have become such a tool. The main purpose of natural numbers is also clear - to give an idea of \u200b\u200bthe 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 are natural ways of transmitting information.
Consider the basic skills of sounding (reading) and displaying (writing) natural numbers.
Decimal notation of a natural number
Let us recall how the following characters are depicted (we indicate them separated by commas): 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . We call these signs numbers.
Now let's take, as a rule, that when displaying (recording) any natural number, only the indicated numbers are used without the participation of any other symbols. Let the numbers when writing a natural number have the same height, they are written one after the other in a line and there is always a nonzero number on the left.
Let's give examples of the correct notation of natural numbers: 703, 881, 13, 333, 1 023, 7, 500 001. The indents between the numbers 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 recording a natural number, all 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, since on the left is the number 0.
The correct record of a natural number, made taking into account all the described requirements, is called decimal notation of a natural number.
The 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 records of which coincide with the records of numbers, i.e.: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 .
Imagine an object, for example, this: Ψ. You 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 seen as a whole. If there is a set, then any element of it can be designated by one. For example, out of a multitude of mice, any mouse is a unit; any flower of many flowers is a unit.
Now imagine: Ψ Ψ. We see one object and one more object, i.e. in the record it will be - 2 items. We read the natural number 2 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 recording of the number coincides with the recording of the digit 0, then such a number is called "zero". Zero is not a natural number, but consider it together with other natural numbers. Zero denotes absence, i.e. zero items means none.
Single-digit natural numbers
It is an obvious fact that, writing down each of the natural numbers, which were 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 character - 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, which are recorded using two characters - 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 numbers.
Consider the meaning of two-digit numbers. We will rely on the already known quantitative meaning of single-digit natural numbers.
Let's introduce such a concept as "ten".
Imagine a set of items, which consists of nine and one more. In this case, we can talk about 1 dozen ("one dozen") items. If we imagine one dozen and one more, then we will talk about 2 tens ("two tens"). Adding one more to two tens, we get three tens. And so on: continuing to add ten at a time, we will get four tens, five tens, six tens, seven tens, eight tens, and finally nine tens.
Let's look at a two-digit number as a set of single-digit numbers, one of which is written on the right, the other on the left. The number on the left will indicate the number of tens in 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 recorded 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-digit natural numbers, we introduce the concept "a hundred".
Definition 5
One hundred (1 hundred) Is a set of ten dozen. One hundred and one more hundred will amount to two hundred. Add one more hundred and get 3 hundred. Adding gradually one hundred, we get: four hundred, five hundred, six hundred, seven hundred, eight hundred, nine hundred.
Consider the very notation of a three-digit number: the single-digit natural numbers included in it are written one after another from left to right. The one-digit number on the right indicates the number of units; the next single-digit number to the left - by the number of tens; the leftmost single-digit number - by the number of hundreds. If the number 0 participates in the recording, 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.
Multidigit natural numbers
From all of the above, it is now possible to move on to the definition of multi-valued natural numbers.
Definition 6
Multiple natural numbers - natural numbers, which are recorded using two or more characters. Multi-digit natural numbers are two-digit, three-digit, and so on numbers.
One thousand is a multitude of ten hundred; one million is one thousand thousand; one billion - one thousand million; one trillion - one thousand billion. Even larger sets also have names, but they are rarely used.
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 hundred, 2 thousand, 1 ten thousand, 9 hundred thousand and 4 million.
Summing up, we examined 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 the designation of the number of units in each of such sets.
Reading natural numbers, classes
In theory, above we have designated 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 notation:
| Number | Masculine gender | Feminine gender | 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 |
Of 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 |
One Two Three Four Five Six Family Eight Nine |
About one About two About three About four Oh five About six About seven About eight About nine |
To correctly read and write 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 Forty 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 Forty 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 Forty 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 About forty About fifty About sixty About seventy About eighty About ninety |
To read other natural two-digit numbers, we will use the data of both tables, consider this with an example. Let's say we need to read the natural two-digit number 21. This number contains 1 unit and 2 tens, i.e. 20 and 1. Referring to the tables, we will read the indicated number as "twenty-one", while the union "and" between 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 items in the genitive case: "there are no 21 apples." In this case, the pronunciation will sound as follows: "there are no twenty one apples."
Let's give for clarity another example: the number 76, which will be 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 |
One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Semist Eight hundred Nine hundred |
One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
One hundred 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 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, read: "three hundred and five" or in declension by case, for example, like this: "three hundred and five meters."
Let's read another number: 543. According to the rules of the tables, the specified number will sound like this: "five hundred forty three" or in declension by cases, for example, "there is no five hundred forty three rubles."
Let's move on to the general principle of reading multi-digit natural numbers: to read a multi-digit number, it is necessary to split it from right to left into groups of three digits, and the leftmost group can contain 1, 2 or 3 digits. Such groups are called classes.
The far right class is the class of units; then the next class, to the left is 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 signs (16, 17 and more) are rarely used in reading, it is rather difficult to perceive them by ear.
For ease of reading, 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 class name). The name of the class of units is not pronounced, and those classes that make up three digits 0 are not pronounced. If in the composition of one class on the left there are one or two digits 0, then they are not used in any way when reading. For example, 054 reads fifty-four or 001 reads one.
Example 1
Let us analyze in detail the reading of the number 2 533 467 001 222:
We read the number 2 as a constituent of the trillion class - "two";
By adding the class name, 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 and 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, let's read 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 skill of breaking 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 at which the digit in the number is recorded. That is, for example, the number 3 in the natural number 314 denotes the number of hundreds, namely, 3 hundreds. Number 2 is the number of tens (1 dozen), and 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 a given number. The number 1 stands in the tens place and serves as the tens place value. The number 3 is in the hundreds place and is the value in the hundreds place.
Definition 7
Discharge - this is the position of the digit in the record of a natural number, as well as the value of this digit, which is determined by its position in the 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 memorization, you can use the following table (we indicate 15 digits):
Let's clarify this detail: the number of digits in a given multi-digit number is the same as the number of characters in the number. For example, this table contains the names of all the digits for a number with 15 characters. Subsequent discharges also have names, but they 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 into the table so that the rightmost digit is written in the one digit and then in each digit by digit. For example, let's write a poly-valued natural number 56 402 513 674 as follows:
Pay attention to the number 0 in the tens of millions place - it means the absence of units of this category.
Let us also introduce the concepts of the lowest and highest digits of a multi-digit number.
Definition 8
The lowest (least significant) digit any multi-digit natural number - the place of ones.
Highest (senior) category any multi-digit natural number - the position corresponding to the leftmost digit in the record of the given number.
So, for example, in the number 41 781: the lowest rank - the rank of ones; the highest rank is the rank of tens of thousands.
It logically follows that it is possible to talk about the seniority of the categories 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 subsequent digit is higher (higher) than the previous one. For example, the thousands rank is older than the hundreds but less than the millions.
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 9Notation - a method of writing numbers using signs.
Positional number systems - those in which the value of the digit in the number depends on its position in the number record.
According to this definition, we can say that, studying above natural numbers and the way they are written, we used the positional number system. The number 10 plays a special role here. We keep counting in tens: ten units make ten, ten tens will combine 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 her, there are other number systems. For example, computer science uses a binary system. When we keep track of time, we use the sixagesimal number system.
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1.1 Definition
The numbers used by people when counting are called natural (for example, one, two, three, ..., one hundred, one hundred and one, ..., three thousand two hundred twenty one, ...) To write natural numbers, special signs (symbols) are used, called figures.
In our time, adopted decimal notation... The decimal system (or method) of writing numbers uses Arabic numerals. These are ten different character numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 .
The fewest natural number is a number one, it written using a decimal digit - 1. The next natural number is obtained from the previous one (except for one) by adding 1 (one). This addition can be done many times (an infinite number of times). It means that no the greatest a natural number. Therefore, they say that the series of natural numbers is unlimited or infinite, since it has no end. Natural numbers are written using decimal digits.
1.2. Number "zero"
To indicate the absence of something, use the number " zero" or " zero".
It is written using numbers 0 (zero).
For example, all balls in the box are red. How many of them are green? - Answer: zero .
So there are no green balls in the box! The number 0 can mean that something is over. For example, Masha had 3 apples. She shared two with friends, ate one herself. So she has left 0
(zero) apples, i.e. not one was left. The number 0 may mean that something has not happened. For example, the hockey match Russian National Team - Canada National Team ended with the score 3:0
(we read "three - zero") in favor of the Russian national team. This means that the Russian team scored 3 goals, and the Canadian national team 0 goals, could not score a single goal. We must remember that the number zero is not natural.
1.3. Natural numbers notation
In decimal notation of a natural number, each digit can mean a different number. It depends on the place of this digit in the number recording. A certain place in the notation of a natural number is called position.Therefore, the decimal notation system for numbers is called positional. Consider the decimal notation 7777 of the number seven thousand seven hundred seventy seven. There are seven thousand, seven hundred, seven tens, and seven units in this record.
Each of the places (positions) in the decimal notation of the number is called discharge... Every three digits are combined into class. This union is performed from right to left (from the end of the number). The various categories and classes have their own names. The range of natural numbers is unlimited. Therefore, the number of digits and classes is also not limited ( infinitely). Consider the names of the digits and classes using the example of a decimal number
38 001 102 987 000 128 425:
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Classes and ranks |
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quintillions |
hundreds of quintillion |
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tens of quintillion |
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quintillions |
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quadrillion |
hundreds of quadrillion |
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tens of quadrillion |
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quadrillion |
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trillions |
hundreds of trillion |
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tens of trillions |
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trillions |
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billions |
hundreds of billions |
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tens of billions |
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billions |
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millions |
hundreds of millions |
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tens of millions |
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millions |
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hundreds of thousands |
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tens of thousands |
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So, the classes, starting with the youngest, have names: units, thousands, millions, billions, trillions, quadrillions, quintillions.
1.4. Bit units
Each of the classes in the representation of natural numbers consists of three digits. Each rank has bit units... The following numbers are called bit units:
1 - bit unit of the units,
10 - digit unit of the tens digit,
100 - bit unit of the category of hundreds,
1,000 is a thousand-bit unit,
10,000 - a bit unit of the rank of tens of thousands,
100,000 - a bit unit of the category of hundreds of thousands,
1 000 000 - the bit unit of the million, and so on.
A digit in any of the digits indicates the number of units of this digit. So, the number 9, in the place of hundreds of billions, means that the number 38 001 102 987 000 128 425 includes nine billion (ie 9 times 1,000,000,000 or 9 digit units of the billions category). An empty place of hundreds of quintillions means that there are no hundreds of quintillions in this number, or their number is zero. The number 38 001 102 987 000 128 425 can be written like this: 038 001 102 987 000 128 425.
You can write it differently: 000 038 001 102 987 000 128 425. Zeros at the beginning of a number indicate empty most significant bits. Usually they are not written, unlike the zeros inside the decimal notation, which must be used to mark empty digits. So, three zeros in the class of millions means that the digits of hundreds of millions, tens of millions and units of millions are empty.
1.5. Abbreviations in notation of numbers
Abbreviations are used when writing natural numbers. Here are some examples:
1,000 \u003d 1,000 (one thousand)
23,000,000 \u003d 23 million (twenty-three million)
5,000,000,000 \u003d 5 billion (five billion)
203,000,000,000,000 \u003d 203 trillion (two hundred three trillion)
107,000,000,000,000,000 \u003d 107 kvdr. (one hundred seven quadrillion)
1,000,000,000,000,000,000 \u003d 1 kw. (one quintillion)
Box 1.1. Dictionary
Compile a glossary of new terms and definitions from §1. To do this, write words from the list of terms below in the blank cells. In the table (at the end of the block), indicate for each definition the term number from the list.
Box 1.2. Self-preparation
In a world of big numbers
Economy .
- The budget of Russia for the next year will be: 6328251684128 rubles.
- Expenditures planned for this year: 5124983252134 rubles.
- The country's revenues exceeded spending by 1203268431094 rubles.
Questions and tasks
- Read all three numbers
- Write down the numbers in the million class of each of the three numbers
- Which section in each of the numbers belongs to the number in the seventh position from the end of the number recording?
- What number of bit units does the digit 2 in the first number represent? ... in the second and third numbers?
- What is the digit unit for the eighth position from the end in the three numbers.
Geography (length)
- Equatorial radius of the Earth: 6378245 m
- Equatorial circumference: 40075696 m
- The greatest depth of the world ocean (Mariana Trench in the Pacific Ocean) 11,500 m
Questions and tasks
- Convert all three values \u200b\u200bto centimeters and read the resulting numbers.
- For the first number (in cm), write down the numbers standing in the sections:
hundreds of thousands _______
tens of millions _______
thousand _______
billion _______
hundreds of millions _______
- For the second number (in cm), write down the digit units corresponding to the numbers 4, 7, 5, 9 in the number
- Convert the third value to millimeters, read the resulting number.
- For all positions in the record of the third number (in mm), indicate the digits and bit units in the table:
Geography (square)
- The entire surface area of \u200b\u200bthe Earth is 510,083 thousand square kilometers.
- The surface area of \u200b\u200bthe sums on Earth is 148,628 thousand square kilometers.
- The area of \u200b\u200bthe Earth's water surface is 361,455 thousand square kilometers.
Questions and tasks
- Convert all three values \u200b\u200bto square meters and read the resulting numbers.
- Name the classes and digits corresponding to nonzero digits in the representation of these numbers (in sq. M).
- In the record of the third number (in square meters), name the digit units corresponding to the numbers 1, 3, 4, 6.
- In two records of the second quantity (in sq. Km. And sq. M), indicate which digits the number 2 belongs to.
- Write down the digit units for the number 2 in the second value entries.
Box 1.3. Dialogue with the computer.
It is known that large numbers are often used in astronomy. Here are some examples. The average distance of the Moon from the Earth is 384 thousand km. The distance of the Earth from the Sun (average) is 149504 thousand km, the Earth from Mars 55 million km. On a computer, using the text editor Word, create tables so that each digit in the recording of the indicated numbers is in a separate cell (cell). To do this, execute the commands on the toolbar: table → add a table → number of rows (use the cursor to put "1") → number of columns (count yourself). Create tables for other numbers (block "Self-study").
Box 1.4. Relay of large numbers
The first line of the table contains a large number. Read it. Then complete the tasks: by moving the numbers in the number to the right or left, get the following numbers and read them. (Do not move the zeros at the end of the number!). In the classroom, the baton can be carried out by passing it on to each other.
Line 2 . Move all digits of the number in the first line to the left after two cells. Replace the digits 5 with the next digit. Fill in empty cells with zeros. Read the number.
Line 3 . Move all digits of the number in the second line to the right through three cells. Replace the digits 3 and 4 in the number with the following digits. Fill in empty cells with zeros. Read the number.
Line 4. Move all digits of the number in line 3 one cell to the left. Replace the number 6 in the trillion class with the previous figure, and in the billion class with the next figure. Fill in empty cells with zeros. Read the resulting number.
Line 5 . Move all the digits of the number in line 4 one cell to the right. Replace the number 7 in the tens of thousands category with the previous one, and in the tens of millions category with the next one. Read the resulting number.
Line 6 . Move all digits of the number in line 5 to the left after 3 cells. Replace the digit 8 in the hundreds of billions with the previous digit and the 6 in the hundreds of millions with the next digit. Fill in empty cells with zeros. Calculate the resulting number.
Line 7 . Move all the digits of the number in line 6 to the right one cell. Swap the tens of quadrillion and tens of billions. Read the resulting number.
Line 8 . Move all the digits of the number in line 7 to the left through one cell. Swap the quintillion and quadrillion digits. Fill in empty cells with zeros. Read the resulting number.
Line 9 . Move all digits of the number in line 8 to the right through three cells. Swap two adjacent numbers in a number row from the classes of millions and trillions. Read the resulting number.
Line 10 . Move all digits of the number in line 9 one cell to the right. Read the resulting number. Highlight the numbers indicating the year of the Moscow Olympiad.
Box 1.5. let's play
Light the fire
The playing field is a drawing of a New Year tree. It has 24 bulbs. But only 12 of them are connected to the mains. To choose the connected lamps, you must correctly answer the questions with the words "Yes" or "No". The same game can be played on a computer. The correct answer "lights" the light bulb.
- Is it true that numbers are special characters for writing natural numbers? (1 - yes, 2 - no)
- Is it true that the number 0 is the smallest natural number? (3 - yes, 4 - no)
- Is it true that in the positional number system, the same number can mean different numbers? (5 - yes, 6 - no)
- Is it true that a certain place in the decimal notation of numbers is called a place? (7 - yes, 8 - no)
- Given the number 543 384. Is it true that the number of the most significant bit units is 543, and the least significant ones are 384? (9 - yes, 10 - no)
- Is it true that in the class of billions, the oldest of the bit units is one hundred billion, and the lowest is one billion? (11 - yes, 12 - no)
- Given the number 458 121. Is it true that the sum of the number of the most significant bit units and the number of the least significant ones is 5? (13 - yes, 14 - no)
- Is it true that the most senior of the trillion class is a million times the highest of the millions? (15 - yes, 16 - no)
- You are given two numbers 637 508 and 831. Is it true that the most significant bit unit of the first number is 1000 times the most significant bit unit of the second number? (17 - yes, 18 - no)
- Given the number 432. Is it true that the most significant bit unit of this number is 2 times the least significant one? (19 - yes, 20 - no)
- The number given is 100,000,000. Is it true that the number of bit units in 10,000 is 1,000? (21 - yes, 22 - no)
- Is it true that before the trillion class is the quadrillion class, and before this class the quintillion class? (23 - yes, 24 - no)
1.6. From the history of numbers
Since ancient times, a person was faced with the need to count the number of things, to compare the number of objects (for example, five apples, seven arrows ...; there are 20 men and thirty women in the tribe, ...). There was also a need to establish order within a number of objects. For example, on a hunt, the leader of the tribe goes first, the most powerful warrior of the tribe comes second, etc. For these purposes, numbers were used. Special names were invented for them. In speech, they are called numerals: one, two, three, etc. are cardinal numbers, and the first, second, third are ordinal numbers. Numbers were recorded using special characters - numbers.
Over time appeared number system. These are systems that include ways of writing numbers and various actions on them. The oldest known number systems are the Egyptian, Babylonian, Roman number systems. In Russia, in the old days, letters of the alphabet with a special sign ~ (titlo) were used to write numbers. Currently, the most widespread is the decimal number system. Binary, octal and hexadecimal number systems are widely used, especially in the computer world.
So, to write the same number, you can use different signs - numbers. So, the number four hundred and twenty-five can be written in Egyptian numbers - hieroglyphs:
This is the Egyptian way of writing numbers. The same number in Roman numerals: CDXXV (Roman way of writing numbers) or decimal digits 425 (decimal notation of numbers). In binary notation, it looks like this: 110101001 (binary or binary system of notation of numbers), and in octal - 651 (octal notation of numbers). In hexadecimal notation, it will be written: 1A9 (hexadecimal notation of numbers). You can do it quite simply: make, like Robinson Crusoe, four hundred and twenty-five notches (or strokes) on a wooden post - IIIIIIIII…... IIII. These are the very first images of natural numbers.
So, in the decimal notation of numbers (in the decimal notation of numbers), Arabic numerals are used. These are ten different symbols - numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ... In binary - two binary digits: 0, 1; in octal - eight octal digits: 0, 1, 2, 3, 4, 5, 6, 7; in hexadecimal - sixteen different hexadecimal digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F; in sexagesimal (Babylonian) - sixty different symbols - numbers, etc.)
Decimal digits came to European countries from the Middle East, Arab countries. Hence the name - arabic numerals... But they came to the Arabs from India, where they were invented around the middle of the first millennium.
1.7. Roman numeral system
One of the ancient number systems in use today is the Roman system. We give in the table the main digits of the Roman numeral system and the corresponding numbers of the decimal system.
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Roman numeral |
C |
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50 fifty |
500 five hundred |
1000 thousand |
Roman numeral system is addition system.In it, unlike positional systems (for example, decimal), each digit denotes the same number. So, the entry II - denotes the number two (1 + 1 \u003d 2), record III - number three (1 + 1 + 1 \u003d 3), record XXX - number thirty (10 + 10 + 10 \u003d 30), etc. The following rules apply to writing numbers.
- If the smaller figure is after larger, then it is added to the larger: Vii - number seven (5 + 2 \u003d 5 + 1 + 1 \u003d 7), XVII - number seventeen (10 + 7 \u003d 10 + 5 + 1 + 1 \u003d 17), MCL - number one thousand one hundred fifty (1000 + 100 + 50 \u003d 1150).
- If the smaller figure is front larger, then it is subtracted from the larger: IX - number nine (9 \u003d 10 - 1), LM - number nine hundred and fifty (1000 - 50 \u003d 950).
To write large numbers, you have to use (invent) new symbols - numbers. In this case, the recording of numbers turns out to be cumbersome, it is very difficult to perform calculations with Roman numerals. So the year of the launch of the first artificial Earth satellite (1957) in the Roman record has the form MCMLVII .
Block 1. 8. Punch card
Reading natural numbers
These tasks are checked using a map with circles. Let us explain its application. After completing all the tasks and finding the correct answers (they are indicated by the letters A, B, C, etc.), put a sheet of transparent paper on the map. Use X to mark the correct answers and the + alignment mark. Then place the transparent sheet over the page so that the registration marks line up. If all the "X" signs are in the gray circles on this page, then the tasks were completed correctly.
1.9. Reading order of natural numbers
When reading a natural number, proceed as follows.
- Mentally divide the number into triples (classes) from right to left, from the end of the number recording.
- Starting from the junior grade, from right to left (from the end of the number recording), the names of the classes are written: units, thousands, millions, billions, trillions, quadrillions, quintillions.
- Read the number starting in high school. In this case, the number of bit units and the name of the class are called.
- If the digit contains zero (the digit is empty), then it is not called. If all three digits of the named class are zeros (the digits are empty), then this class is not called.
Let's read (name) the number written in the table (see §1), according to steps 1 - 4. Mentally divide the number 38001102987000128425 into classes from right to left: 038 001 102 987 000 128 425. Indicate the names of classes in this number, starting from the end his records: units, thousands, millions, billions, trillions, quadrillions, quintillions. Now you can read the number, starting with high school. We name three-digit, two-digit and single-digit numbers, adding the name of the corresponding class. We do not name empty classes. We get the following number:
- 038 - thirty-eight quintillion
- 001 - one quadrillion
- 102 - one hundred two trillion
- 987 - nine hundred eighty seven billion
- 000 - do not name (do not read)
- 128 - one hundred twenty eight thousand
- 425 - four hundred twenty five
As a result, we read the natural number 38 001 102 987 000 128 425 as follows: "thirty-eight quintillion one quadrillion one hundred two trillion nine hundred and eighty-seven billion one hundred twenty-eight thousand four hundred twenty-five."
1.9. The order of writing natural numbers
Natural numbers are written in the following order.
- Three digits of each grade are recorded, starting with the senior grade to the one grade. Moreover, for the senior class, there can be two or one digits.
- If the class or category is not named, then zeros are written in the corresponding bits.
For example, the number twenty five million three hundred two written in the form: 25 000 302 (the class of thousands is not named, therefore, zeros are written in all bits of the class of thousands).
1.10. Representation of natural numbers as a sum of bit terms
Here's an example: 7 563 429 is the decimal notation of a number seven million five hundred sixty three thousand four hundred twenty nine. This number contains seven million, five hundred thousand, six tens of thousands, three thousand, four hundred, two tens and nine units. It can be represented as the sum: 7,563,429 \u003d 7,000,000 + 500,000 + 60,000 + + 3,000 + 400 + 20 + 9. This is called the representation of a natural number as a sum of bit terms.
Box 1.11. let's play
Dungeon treasures
On the playing field is a drawing for Kipling's fairy tale "Mowgli". There are padlocks on five chests. To open them, you need to solve problems. In this case, by opening a wooden chest, you get one point. Opening a pewter chest gives you two points, a copper one three points, a silver one four, and a gold one five. The winner is the one who opens all the chests faster. The same game can be played on a computer.
- Wooden chest
Find how much money (in thousand rubles) is in this chest. To do this, you need to find the total number of the least significant bit units of the million class for the number: 125308453231.
- Pewter chest
Find how much money (in thousand rubles) is in this chest. To do this, in the number 12530845323 find the number of the least significant bit units of the class of ones and the number of the least significant bit units of the class of millions. Then find the sum of these numbers and on the right, add the number in the tens of millions place.
- Copper chest
To find the money of this chest (in thousand rubles), in the number 751305432198203, find the number of the lowest digit units in the trillion class and the number of the lowest units in the billion class. Then find the sum of these numbers and on the right, write down the natural numbers of the class of units of this number in the order of their arrangement.
- Silver Chest
The money of this chest (in million rubles) will be shown by the sum of two numbers: the number of the lowest bit units of the class of thousands and the middle bit units of the class of billions for the number 481534185491502.
- Golden chest
Given the number 800123456789123456789. If we multiply the numbers in the highest digits of all classes of this number, we get the money of this chest in a million rubles.
Box 1.12. Set correspondence
Recording natural numbers. Representation of natural numbers as a sum of bit terms
For each task in the left column, select a solution from the right column. Write the answer in the form: 1a; 2d; 3b ...
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Write down the number in digits: five million twenty five thousand |
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Write down the number in digits: five billion twenty five million |
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Write down the number in digits: five trillion twenty five |
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Write down the number in digits: seventy seven million seventy seven thousand seven hundred seventy seven |
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Write down the number in digits: seventy seven trillion seven hundred seventy seven thousand seven |
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Write down the number in digits: seventy seven million seven hundred seventy seven thousand seven |
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Write down the number in digits:one hundred twenty three billion four hundred fifty six million seven hundred eighty nine thousand |
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Write down the number in digits:one hundred twenty three million four hundred fifty six thousand seven hundred eighty nine |
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Write down the number in digits:three billion eleven |
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Write down the number in digits:three billion eleven million |
Option 2
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thirty two billion one hundred seventy five million two hundred ninety eight thousand three hundred forty one |
100000000 + 1000000 + 10000 + 100 + 1 |
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Imagine the number as a sum of bit terms:three hundred twenty one million forty one |
30000000000 + 2000000000 + 100000000 + 70000000 + 5000000 + 200000 + 90000 + 8000 + 300 + 40 + 1 |
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Imagine the number as a sum of bit terms: 321000175298341 |
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Imagine the number as a sum of bit terms: 101010101 |
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Imagine the number as a sum of bit terms: 11111 |
300000000 + 20000000 + 1000000 + |
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5000000 + 300000 + 20000 + 1000 |
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Write down in decimal notation the number represented as the sum of the bit terms:5000000 + 300 + 20 + 1 |
30000000000000 + 2000000000000 + 1000000000000 + 100000000 + 70000000 + 5000000 + 200000 + 90000 + 8000 + 300 + 40 + 1 |
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Write down in decimal notation the number represented as the sum of the bit terms: 10000000000 + 2000000000 + 100000 + 10 + 9 |
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Write down in decimal notation the number represented as the sum of the bit terms: 10000000000 + 2000000000 + 100000000 + 10000000 + 9000000 |
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Write down in decimal notation the number represented as the sum of the bit terms:9000000000000 + 9000000000 + 9000000 + 9000 + 9 |
10000 + 1000 + 100 + 10 + 1 |
Box 1.13. Facet test
The name of the test comes from the word "insect faceted eye". It is a complex eye, consisting of separate "eyes". Facet test items are formed from individual items indicated by numbers. Facet tests usually contain a large number of items. But in this test there are only four problems, but they are composed of a large number of elements. This is to teach you how to "collect" the problems of the test. If you can write them, you can easily handle other facet tests.
We will explain how the tasks are composed using the example of the third task. It is composed of test items numbered: 1, 4, 7, 11, 1, 5, 7, 9, 10, 16, 17, 22, 21, 25
« If a» 1) take numbers (figure) from the table; 4) 7; 7) put it in the category; 11) billion; 1) take a figure from the table; 5) 8; 7) put it in the digits; 9) tens of millions; 10) hundreds of millions; 16) hundreds of thousands; 17) tens of thousands; 22) place the numbers 9 and 6 in the thousands and hundreds. 21) fill the remaining digits with zeros; " TO» 26) we get a number equal to the time (period) of the revolution of the planet Pluto around the Sun in seconds (s); " This number is": 7880889600 p. In the answers, it is indicated by the letter "in".
When solving problems, write numbers in the cells of the table with a pencil.
Facet test. Make up the number
The table contains numbers:
If a
1) take the number (s) from the table:
2) 4; 3) 5; 4) 7; 5) 8; 6) 9;
7) put this digit (s) in the category (s);
8) hundreds of quadrillion and tens of quadrillion;
9) tens of millions;
10) hundreds of millions;
11) billion;
12) quintillion;
13) tens of quintillion;
14) hundreds of quintillions;
15) trillions;
16) hundreds of thousands;
17) tens of thousands;
18) fill with it (them) the class (classes);
19) quintillion;
20) billion;
21) fill the remaining bits with zeros;
22) place numbers 9 and 6 in the digits of thousands and hundreds;
23) we get a number equal to the mass of the Earth in tens of tons;
24) we will receive a number approximately equal to the volume of the Earth in cubic meters;
25) we get a number equal to the distance (in meters) from the Sun to the farthest planet of the solar system, Pluto;
26) we get a number equal to the time (period) of the revolution of the planet Pluto around the Sun in seconds (s);
This number is equal to:
a) 5929000000000
b) 999990000000000000000
d) 598000000000000000000
Solve tasks:
1, 3, 6, 5, 18, 19, 21, 23
1, 6, 7, 14, 13, 12, 8, 21, 24
1, 4, 7, 11, 1, 5, 7, 10, 9, 16, 17, 22, 21, 26
1, 3, 7, 15, 1, 6, 2, 6, 18, 20, 21, 25
Answers
1, 3, 6, 5, 18, 19, 21, 23 - d
1, 6, 7, 14, 13, 12, 8, 21, 24 - b
1, 4, 7, 11, 1, 5, 7, 10, 9, 16, 17, 22, 21, 26 - c
1, 3, 7, 15, 1, 6, 2, 6, 18, 20, 21, 25 - a
Mathematics emerged from general philosophy around the sixth century BC. e., and from that moment began its 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 confusing, and the moment came when “the most complex mathematics began - all numbers disappeared from it”. But what was behind it?
The beginning of time
Natural numbers appeared on a par with the first mathematical operations. One spine, two spine, three spine ... They appeared thanks to Indian scientists who brought out the first positional
The word "positionality" means that the location of each digit in the 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 hundred, while the second includes only 4. The innovation of the Indians was picked up by the Arabs, who brought the numbers to the form that we know now.
In ancient times, numbers were given a mystical meaning, Pythagoras believed that number underlies the creation of the world along with the main 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 integers and positive numbers: 1, 2, 3,… + ∞. Zero is excluded. Used primarily for counting items and indicating order.
What is Mathematics? Peano's axioms
The N field is the base on which elementary mathematics is based. Over time, the fields of wholes, rational,
The works 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 of N.
What is a natural number, it was clarified earlier in simple language, below we will consider a mathematical definition based on Peano's axioms.
- The unit is considered a natural number.
- The number that follows the natural number is natural.
- There is no natural number in front of the unit.
- If the number b follows both the number c and the number d, then c \u003d d.
- The induction axiom, 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 works for a number n from the field of natural numbers N. Then the statement is true for n \u003d 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 \u200b\u200bof a number of operations below belong to it. They are closed and not. The main difference is that closed operations are guaranteed to keep the result within the set N regardless of which numbers are involved. It is enough that they are natural. The outcome of the rest of the numerical interactions is no longer so unambiguous and directly depends on what numbers are involved in the expression, since it may contradict the basic definition. So, closed operations:
- addition - x + y \u003d z, where x, y, z are included in the N field;
- multiplication - x * y \u003d z, where x, y, z are included in the field N;
- exponentiation - x y, where x, y are included in field N.
Other operations, the result of which may not exist in the context of the definition of "what is a natural number", are as follows:
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 movable property of addition is x + y \u003d y + x, where the numbers x, y are included in the field N. Or the well-known "the sum does not change from the change of places of the terms".
- The movable property of multiplication is x * y \u003d y * x, where the numbers x, y are included in the field N.
- Combination property of addition - (x + y) + z \u003d x + (y + z), where x, y, z are included in the field N.
- Combination property of multiplication - (x * y) * z \u003d x * (y * z), where numbers x, y, z are included in the field N.
- distribution property - x (y + z) \u003d x * y + x * z, where numbers x, y, z are included in field N.
Pythagoras table
One of the first steps in the knowledge of the whole structure of elementary mathematics by schoolchildren after they have figured out for themselves which numbers are called natural is the Pythagorean table. It can be viewed 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 was removed from it, and the numbers from 1 to 10 denote themselves, without taking into account the orders (hundreds, thousands ...). It is a table in which the titles of rows and columns are numbers, and the contents of the cells of their intersection are equal to their product.
In the practice of teaching in recent decades, there was a need to memorize the Pythagorean table "in order", that is, first there was memorization. Multiplication by 1 was excluded because the result was 1 or more. 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 in order to get the desired product. This system is by far more convenient than the one that was 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
At the 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. A natural number is the first thing that a child learns when studying himself and the world around him. 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, preparing the ground for great discoveries.