What does it mean to write numbers as a sum of bit terms. Representing a number as a sum of bit terms

2.8 Three-digit numbers

1. The Scarecrow wrote down some numbers as a sum. What groups can these expressions be divided into? What numbers are written as a sum of bit terms?

Expressions can be divided into two groups: "Sums of bit terms" and "Ordinary sums".

"Sums of discharge terms":

600 + 9

700 + 20 + 2

400 + 10

"Regular Amounts":

259 + 1

340 + 1

200 + 52

Write down the numbers as a sum of bit terms: 205, 360, 415.

205 = 200 + 5;

360 = 300 + 60;

415 = 400 + 10 + 5.

2. Read the numbers: 410, 700, 420, 267, 807, 268, 1000.

410 - four hundred and ten;

700 - seven hundred;

420 - four hundred and twenty;

267 - two hundred and sixty seven;

807 - eight hundred seven;

268 - two hundred sixty eight;

1000 is one thousand.

Write them down in descending order. Underline the number in the hundreds place yellow, in the tens place - green, in the ones place - blue.

10 0 0; 8 0 7; 7 0 0; 4 2 0; 4 1 0; 2 6 8; 2 6 7.

Name the neighboring numbers for the smallest of the numbers in this row.

The smallest number is 267. The neighboring numbers for it are 266 and 268.

3. Calculate.

260 + 5 = 265 784 — 80 = 704 500 + 99 — 1 = 598

382 — 2 = 380 805 + 90 = 895 640 — 600 + 1 =41

The Scarecrow said that among the meanings of these expressions there are numbers that are written like this: 7 s. 4 pts, 5 s 9 d. 8 units, 2 d. 6 s. Is he right? Explain how the numbers seven hundred four and seven hundred forty are written. Why are they recorded like this?

Scarecrow rights not until the end. The numbers 704 and 598 are there, but the numbers 620 are not.

704 - 7 s, 0 d, 4 u;

740 - 7 s, 4 d, 0 units.

Name a series of natural numbers from 598 to 610.

598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610.

4. Express

a) in millimeters: 5 dm, 7 dm 4 cm;

b) in meters: 800 cm, 600 cm;

c) in decimeters: 90 cm, 320 cm;

d) in cubic decimeters: 1 m³.

a) 5 dm = 500 mm; 7 dm = 700 mm; 4 cm = 40 mm.

b) 800 cm = 8 m; 600 cm = 6 m.

c) 90 cm = 9 dm, 320 cm = 32 dm.

d) 1 m³ = 1000 dm³.

3. Choose a scheme and solve problems.

a) Goodwin received 47 letters from the good sorceress Villina and 39 letters from the good sorceress Stella. How much news did Villina tell Goodwin if there are 16 more news stories in her letters than in Stella's letters, and in each letter of the sorceresses there is an equal number of news stories?

We solve according to scheme b).

47 + 39 = 8 (letters) - so much more from Villina.

16:8 = 2 (news) - in every letter.

2 47 \u003d 94 (news) - Villina informed Goodwin in total.

Answer: 94 news.

b) The long-bearded soldier Din Gior gets mail from three mailboxes every morning. There are 3 compartments in the first box, 6 in the second, and 9 in the third. All these boxes contain 90 parcels. How many parcels can be placed in each mailbox if there is an equal number of parcels in each compartment of the parcel box?

We solve according to scheme a).

3 + 6 + 9 = 18 (compartments) - in all boxes.

90: 18 = 5 (parcels) - in one compartment of the box.

5 3 \u003d 15 (parcels) - in the first box.

5 6 \u003d 30 (parcels) - in the second box.

5 9 \u003d 45 (parcels) - in the third box.

Answer: 15, 30, 45 parcels.

To write numbers, people came up with ten characters, which are called numbers. They are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

With ten digits, you can write any natural number.

Its name depends on the number of characters (digits) in the number.

A number consisting of one sign (digit) is called a single digit. The smallest single natural number is 1, the largest is 9.

A number consisting of two characters (digits) is called a two-digit number. The smallest two-digit number is 10, the largest is 99.

Numbers written with two, three, four or more digits are called two-digit, three-digit, four-digit or multi-digit. The smallest three-digit number is 100, the largest is 999.

Each digit in the record of a multi-digit number occupies a certain place - a position.

Discharge- this is the place (position) at which the digit stands in the notation of the number.

The same digit in a number entry can have different meanings depending on which digit it is in.

The digits are counted from the end of the number.

Units digit is the least significant digit that ends any number.

The number 5 - means 5 units, if the five is in last place in the number entry (in the units place).

Tens place is the digit that comes before the units digit.

The number 5 means 5 tens if it is in the penultimate place (in the tens place).

Hundreds place is the digit that comes before the tens digit. The number 5 means 5 hundreds if it is in the third place from the end of the number (in the hundreds place).

If there is no digit in the number, then the digit 0 (zero) will be in its place in the number entry.

Example. The number 807 contains 8 hundreds, 0 tens and 7 units - such an entry is called bit composition of the number.

807 = 8 hundreds 0 tens 7 ones

Every 10 units of any rank form a new unit of a higher rank. For example, 10 ones make 1 tens, and 10 tens make 1 hundred.

Thus, the value of a digit from digit to digit (from ones to tens, from tens to hundreds) increases 10 times. Therefore, the counting system (calculus) that we use is called decimal system reckoning.

Classes and ranks

In the notation of a number, the digits, starting from the right, are grouped into classes of three digits each.

Unit class or the first class is the class that the first three digits form (to the right of the end of the number): units place, tens place and hundreds place.

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Bit terms of a number

The sum of bit terms

Any natural number can be written as a sum of bit terms.

How this is done can be seen from the following example: the number 999 consists of 9 hundreds, 9 tens and 9 ones, so:

999 = 9 hundreds + 9 tens + 9 units = 900 + 90 + 9

The numbers 900, 90 and 9 are bit terms. Discharge term is simply the number of 1s in the given digit.

The sum of the bit terms can also be written as follows:

999 = 9 100 + 9 10 + 9 1

The numbers that are multiplied by (1, 10, 100, 1000, etc.) are called bit units. So, 1 is the unit of the digit of units, 10 is the unit of the digit of tens, 100 is the unit of the digit of hundreds, etc. Numbers that are multiplied by bit units express number of bit units.

Write any number in the form:

12 = 1 10 + 2 1 or 12 = 10 + 2

called decomposing a number into bit terms(or the sum of bit terms).

3278 = 3 1000 + 2 100 + 7 10 + 8 1 = 3000 + 200 + 70 + 8
5031 = 5 1000 + 0 100 + 3 10 + 1 1 = 5000 + 30 + 1
3700 = 3 1000 + 7 100 + 0 10 + 0 1 = 3000 + 700

Calculator for decomposing a number into bit terms

To represent a number as a sum of digit terms, this calculator will help you. Just enter the desired number and click the Decompose button.

Bit terms in mathematics

A number is a mathematical concept for a quantitative description of something or a part of it, it also serves to compare the whole and parts, arrange in order. The concept of number is represented by signs or numbers in various combinations. At present, numbers from 1 to 9 and 0 are used almost everywhere. Numbers in the form of seven Latin letters have almost no use and will not be considered here.

Integers

When counting: “one, two, three ... forty-four” or arranging in turn: “first, second, third ... forty-fourth”, natural numbers are used, which are called natural numbers. This whole set is called “a series of natural numbers” and is denoted by the Latin letter N and has no end, because there is always a number even more, and the largest simply does not exist.

Digits and classes of numbers

This shows that the bit of a number is its position in the digital notation, and any value can be represented through bit terms in the form nnn = n00 + n0 + n, where n is any digit from 0 to 9.

One ten is a unit of the second digit, and one hundred is a unit of the third. Units of the first category are called simple, all the rest are composite.

For the convenience of recording and transmission, a grouping of digits into classes of three in each is used. A space is allowed between classes for readability.

The first - units, contains up to 3 characters:

Two hundred and thirteen contains the following digit terms: two hundred, one ten and three simple units.

Forty-five is made up of four tens and five primes.

Second - thousand, 4 to 6 characters:

679 812 = 600 000 + 70 000 + 9 000 + 800 +10 + 2.

This sum consists of the following bit terms:

six hundred thousand;
seventy thousand;
nine thousand;
eight hundred;
ten;

3 456 = 3000 + 400 +50 +6.

There are no terms above the fourth category.

Third - million, 7 to 9 digits:

This number contains nine bit terms:

800 million;
80 million;
7 million;
200 thousand;
10 thousand;
3 thousand;
6 hundreds;
4 tens;
4 units;

7 891 234.

There are no terms higher than 7 digits in this number.

The fourth is billions, from 10 to 12 digits:

Five hundred sixty-seven billion eight hundred ninety-two million two hundred thirty-four thousand nine hundred seventy-six.

Bit terms of class 4 are read from left to right:

units of hundreds of billions;
units of tens of billions;
units of billions;
hundreds of millions;
tens of millions;
million;
hundreds of thousands;
tens of thousands;
thousand;
simple hundreds;
simple tens;
simple units.

The numbering of the digit of the number is made starting from the smallest, and reading - from the largest.

If there are no intermediate values in the number of terms, zeros are put during recording, when pronouncing the name of the missing bits, as well as the class of units, it is not pronounced:

Four hundred billion four. Here, due to lack, the following names of ranks are not pronounced: tenth and eleventh fourth grade; ninth, eighth and seventh third and most? third class; the names of the second class and its categories, as well as hundreds and tens of units, are also not voiced.

Fifth - trillion, from 13 to 15 characters.

Four hundred eighty-seven trillion seven hundred eighty-nine billion six hundred fifty-four million four hundred twenty-seven two hundred and forty-one.

Sixth - quadrillion, 16-18 digits.

321 546 818 492 395 953;

Three hundred twenty one quadrillion five hundred forty six trillion eight hundred eighteen billion four hundred ninety two million three hundred ninety five thousand nine hundred fifty three.

Seventh - quintillion, 19-21 signs.

771 642 962 921 398 634 389.

Seven hundred seventy one quintillion six hundred forty two quadrillion nine hundred sixty two trillion nine hundred twenty one billion three hundred ninety eight million six hundred thirty four thousand three hundred eighty nine.

Eighth - sextillions, 22-24 digits.

842 527 342 458 752 468 359 173

Eight hundred and forty-two sextillion five hundred twenty-seven quintillion three hundred and forty-two quadrillion four hundred and fifty-eight trillion seven hundred and fifty-two billion four hundred and sixty-eight million three hundred and fifty-nine thousand one hundred and seventy-three.

You can simply distinguish between classes by numbering, for example, the number 11 of the class contains from 31 to 33 characters when written.

But in practice, writing such a number of characters is inconvenient and most often leads to errors. Therefore, during operations with such values, the number of zeros is reduced by raising to a power. After all, it is much easier to write 10 31 than to attribute thirty-one zeros to one.

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What are bit terms

Answers and explanations

For example: 5679=5000+600+70+9
That is, the number of units in the discharge

Comments (1)
Flag Violation

the sum of the bit terms of the number 526 is 500+20+6

The "sum of bit terms" is the representation of a two (or more) digit number as the sum of its bits.

Bit terms are the addition of numbers with different bit depths. For example, the number 17.890 is divided into bit terms: 17.890=10.000+7.000+800+90+0

Rule for multiplying any number by zero

Even at school, teachers tried to hammer the simplest rule into our heads: "Any number multiplied by zero equals zero!", - but still a lot of controversy constantly arises around him. Someone just memorized the rule and does not bother with the question “why?”. “You can’t do everything here, because at school they said so, the rule is the rule!” Someone can fill half a notebook with formulas, proving this rule or, conversely, its illogicality.

Who is right in the end

During these disputes, both people, having opposite points of view, look at each other like a ram, and prove with all their might that they are right. Although, if you look at them from the side, you can see not one, but two rams resting against each other with their horns. The only difference between them is that one is slightly less educated than the other. Most often, those who consider this rule to be wrong try to call for logic in this way:

I have two apples on my table, if I put zero apples to them, that is, I don’t put a single one, then my two apples will not disappear from this! The rule is illogical!

Indeed, apples will not disappear anywhere, but not because the rule is illogical, but because a slightly different equation is used here: 2 + 0 \u003d 2. So we will immediately discard such a conclusion - it is illogical, although it has the opposite goal - to call to logic.

This is interesting: How to find the difference of numbers in mathematics?

What is multiplication

The original multiplication rule was defined only for natural numbers: multiplication is a number added to itself a certain number of times, which implies the naturalness of the number. Thus, any number with multiplication can be reduced to this equation:

25?3 = 75
25 + 25 + 25 = 75
25?3 = 25 + 25 + 25

From this equation follows the conclusion, that multiplication is a simplified addition.

What is zero

Any person knows from childhood: zero is emptiness. Despite the fact that this emptiness has a designation, it does not carry anything at all. The ancient Eastern scientists thought differently - they approached the issue philosophically and drew some parallels between emptiness and infinity and saw a deep meaning in this number. After all, zero, which has the value of emptiness, standing next to any natural number, multiplies it ten times. Hence all the controversy over multiplication - this number carries so much inconsistency that it becomes difficult not to get confused. In addition, zero is constantly used to determine empty digits in decimal fractions, this is done both before and after the decimal point.

Is it possible to multiply by emptiness

It is possible to multiply by zero, but it is useless, because, whatever one may say, but even when multiplying negative numbers, zero will still be obtained. It is enough just to remember this simplest rule and never ask this question again. In fact, everything is simpler than it seems at first glance. There are no hidden meanings and secrets, as ancient scientists believed. The most logical explanation will be given below that this multiplication is useless, because when multiplying a number by it, the same thing will still be obtained - zero.

Going back to the very beginning, the argument about two apples, 2 times 0 looks like this:

If you eat two apples five times, then eaten 2 × 5 = 2+2+2+2+2 = 10 apples
If you eat two of them three times, then eaten 2? 3 = 2 + 2 + 2 = 6 apples
If you eat two apples zero times, then nothing will be eaten - 2?0 = 0?2 = 0+0 = 0

After all, eating an apple 0 times means not eating a single one. It will be clear even to a small child. Like it or not, 0 will come out, two or three can be replaced with absolutely any number and absolutely the same thing will come out. And to put it simply, zero is nothing and when you have there is nothing, then no matter how much you multiply - it's all the same will be zero. There is no magic, and nothing will make an apple, even if you multiply 0 by a million. This is the simplest, most understandable and logical explanation of the rule of multiplication by zero. For a person who is far from all formulas and mathematics, such an explanation will be enough for the dissonance in the head to resolve and everything to fall into place.

From all of the above follows another important rule:

You can't divide by zero!

This rule, too, has been stubbornly hammered into our heads since childhood. We just know that it is impossible and that's it, without stuffing our heads with unnecessary information. If you are suddenly asked the question, for what reason it is forbidden to divide by zero, then the majority will be confused and will not be able to clearly answer the simplest question from the school curriculum, because there are not so many disputes and contradictions around this rule.

Everyone just memorized the rule and does not divide by zero, not suspecting that the answer lies on the surface. Addition, multiplication, division and subtraction are unequal, only multiplication and addition are full of the above, and all other manipulations with numbers are built from them. That is, the entry 10: 2 is an abbreviation of the equation 2 * x = 10. Hence, the entry 10: 0 is the same abbreviation of 0 * x = 10. It turns out that division by zero is a task to find a number, multiplying by 0, you get 10 And we have already figured out that such a number does not exist, which means that this equation has no solution, and it will be a priori incorrect.

Let me tell you

To not divide by 0!

Cut 1 as you like, along,

Just don't divide by 0!

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Integers

Digits and classes of numbers

Discharges

dozens

10…90;

100…900.

This shows that the bit of a number is its position in the digital notation, and any value can be represented through bit terms in the form nnn = n00 + n0 + n, where n is any digit from 0 to 9.

One ten is a unit of the second digit, and one hundred is a unit of the third. Units of the first category are called simple, all the rest are composite.

For the convenience of recording and transmission, a grouping of digits into classes of three in each is used. A space is allowed between classes for readability.

Classes

The first - units, contains up to 3 characters:

200 + 10 +3 = 213.

Two hundred and thirteen contains the following digit terms: two hundred, one ten and three simple units.

40 + 5 = 45;

Forty-five is made up of four tens and five primes.

Second - thousand, 4 to 6 characters:

679 812 = 600 000 + 70 000 + 9 000 + 800 +10 + 2.

This sum consists of the following bit terms:

six hundred thousand;
seventy thousand;
nine thousand;
eight hundred;
ten;

3 456 = 3000 + 400 +50 +6.

There are no terms above the fourth category.

Third - million, 7 to 9 digits:

887 213 644;

This number contains nine bit terms:

800 million;
80 million;
7 million;
200 thousand;
10 thousand;
3 thousand;
6 hundreds;
4 tens;
4 units;

7 891 234.

There are no terms higher than 7 digits in this number.

The fourth is billions, from 10 to 12 digits:

567 892 234 976;

Five hundred sixty-seven billion eight hundred ninety-two million two hundred thirty-four thousand nine hundred seventy-six.

Bit terms of class 4 are read from left to right:

units of hundreds of billions;
units of tens of billions;
units of billions;
hundreds of millions;
tens of millions;
million;
hundreds of thousands;
tens of thousands;
thousand;
simple hundreds;
simple tens;
simple units.

The numbering of the digit of the number is made starting from the smallest, and reading - from the largest.

If there are no intermediate values in the number of terms, zeros are put during recording, when pronouncing the name of the missing bits, as well as the class of units, it is not pronounced:

400 000 000 004;

Four hundred billion four. Here, due to lack, the following names of ranks are not pronounced: tenth and eleventh fourth grade; ninth, eighth and seventh third and third class itself; the names of the second class and its categories, as well as hundreds and tens of units, are also not voiced.

Fifth - trillion, from 13 to 15 characters.

487 789 654 427 241.

Reading on the left:

Four hundred eighty-seven trillion seven hundred eighty-nine billion six hundred fifty-four million four hundred twenty-seven two hundred and forty-one.

Sixth - quadrillion, 16-18 digits.

321 546 818 492 395 953;

Three hundred twenty one quadrillion five hundred forty six trillion eight hundred eighteen billion four hundred ninety two million three hundred ninety five thousand nine hundred fifty three.

Seventh - quintillion, 19-21 signs.

771 642 962 921 398 634 389.

Eighth - sextillions, 22-24 digits.

842 527 342 458 752 468 359 173

You can simply distinguish between classes by numbering, for example, the number 11 of the class contains from 31 to 33 characters when written.

Guys, open your textbook to page 24. Read the title of today's topic at the top.

Today we will learn what bit terms mean, and we will also learn to represent a number as a sum of bit terms. We carry out task number 1. I read the task, you listen carefully. Write down the numbers 18, 15, 19, 14 in your notebook.

The teacher writes these numbers on the blackboard.

For each number, underline the tens digit in red. What numbers will you underline?

The teacher on the board underlines the number 1 in red in each number.

In the same numbers, underline the units digits in blue. What numbers will you underline?

The teacher on the blackboard underlines the number 8, 5, 9, 4 in blue in each number.

How are these numbers similar?

How are these numbers different?

Write each of these two-digit numbers as a sum whose first term is 10.

In what form can the number 18 be added if this number consists of 1 ten and 8 units?

Now I will read how Masha presented the number 18. So, Masha presented the number 18 as a sum 10+8. This representation of numbers is called So we correctly represented the number 18 as a sum of 10 + 8?

Expand the rest of the numbers, 15, 19, 14, into bit terms. In the form of what amount will you present these numbers.

That's right guys, this representation of a number is called EXPANSION INTO DIFFERENT TERMS. Record these amounts in your notebook.

Task number 2. Write down the numbers 15, 16, 11, 10 in your notebook. Write these numbers in your notebook.

The teacher writes the numbers on the blackboard.

How many tens are in each of these numbers?

How many units are in each number?

Represent each number as a sum of bit terms.

The teacher writes the sums on the blackboard.

Task number 3. Look at the pictures and write down the numbers. What number will we write in the first picture?

The second picture, what number will we write down?

The teacher writes the number on the board.

The third picture, what number will we write down?

The teacher writes the number on the board.

The fourth picture, what number will we write down?

The teacher writes the number on the board.

Fifth figure, what number will we write down?

The teacher writes the number on the board.

How many tens and how many ones are in each of these numbers?

Write down a number that has 2 tens and 0 ones. What is this number?

The teacher writes the number 20 on the blackboard.

That's right, this number TWENTY.

- How is the number 20 represented in the last picture?

Write down all the numbers from 11 to 20 in order.

The teacher writes the numbers from 11 to 20 on the blackboard.

So guys, all the numbers from 11 to 20 - These are numbers in the second ten.

And now we're going to have a physical exercise.