To record numbers, people came up with ten characters called numbers. These are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
You can write any natural number using ten digits.
Its name depends on the number of characters (digits) in a number.
A number consisting of one sign (digit) is called single-digit. The smallest single-digit natural number is 1, the largest is 9.
A number consisting of two characters (digits) is called two-digit. 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 numbers. The smallest three-digit number is 100, the largest is 999.
Each digit in the notation of a multi-digit number occupies a certain place - position.
Discharge- this is the place (position) where the digit appears in the notation of a number.
The same digit in a number may have different meanings depending on what category it is in.
Places 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 the last place in the number (in the ones 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 place that comes before the tens place. The number 5 means 5 hundreds if it is in third place from the end of the number (in the hundreds place).
If a number is missing any digit, then the number will be written in its place with the number 0 (zero).
Example. The number 807 contains 8 hundreds, 0 tens and 7 units - this notation is called digit composition of the number.
807 = 8 hundreds 0 tens 7 units
Every 10 units of any rank form a new unit of a higher rank. For example, 10 ones make 1 ten, and 10 tens make 1 hundred.
Thus, the value of a digit from digit to digit (from units to tens, from tens to hundreds) increases 10 times. Therefore, the counting system we use is called the decimal number system.
Classes and ranks
In writing 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 formed by the first three digits (to the right of the end of the number): units place, tens place and hundreds place.
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Place numbers
Sum of bit terms
Any natural number can be written as a sum of digit terms.
How this is done can be seen from the following example: the number 999 consists of 9 hundreds, 9 tens and 9 units, therefore:
999 = 9 hundreds + 9 tens + 9 ones = 900 + 90 + 9
The numbers 900, 90 and 9 are digit terms. Bit term is simply the number of ones in a given digit.
The sum of the bit terms can also be written as follows:
999 = 9 100 + 9 10 + 9 1
The numbers by which multiplication is performed (1, 10, 100, 1000, etc.) are called bit units. So, 1 is the unit of the units place, 10 is the unit of the tens place, 100 is the unit of the hundreds place, etc. Numbers that are multiplied by the place units express number of digit units.
Write any number in the form:
12 = 1 10 + 2 1 or 12 = 10 + 2
called decomposition of a number into digit terms(or 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 digit terms
This calculator will help you represent a number as a sum of digit terms. Just enter the desired number and click the Expand button.
Place terms in mathematics
A number is a mathematical concept for a quantitative description of something or its part; it also serves to compare the whole and parts, and arrange them in order. The concept of number is represented by signs or numbers in various combinations. Currently, numbers from 1 to 9 and 0 are used almost everywhere. Numbers in the form of seven Latin letters have almost no application and will not be considered here.
Integers
When counting: “one, two, three... forty-four” or arranging in order: “first, second, third... forty-four,” natural numbers are used, which are called natural numbers. This entire set is called the “series of natural numbers” and is denoted Latin letter N has no end, because there is always an even larger number, and the largest one simply does not exist.
Places and classes of numbers
This shows that the digit of a number is its position in digital notation, and any value can be represented through digit 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 others are composite.
For ease of recording and transmission, categories are grouped into classes of three in each. It is allowed to put a space between classes for ease of reading.
First - units, contains up to 3 characters:
Two hundred and thirteen contains the following bit terms: two hundred, one ten and three prime ones.
Forty-five is made up of four tens and five prime ones.
Second - thousand, from 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 digit.
Third - millions, from 7 to 9 digits:
This number contains nine digit 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 in this number above the 7th digit.
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.
Class 4 bit terms 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;
- millions;
- hundreds of thousands;
- tens of thousands;
- thousand;
- simple hundreds;
- simple tens;
- simple units.
The digit of a number is numbered starting from the smallest, and reading - from the largest.
If there are no intermediate values in the number of terms, zeros are placed when writing; when pronouncing the name of the missing digits, as well as the class of units, the name is not pronounced:
Four hundred billion four. The following names of categories are not pronounced here due to absence: tenth and eleventh fourth grade; ninth, eighth and seventh third and most? third class; the names of the second class and its ranks, as well as hundreds and tens of units, are also not announced.
The fifth is trillions, 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 forty-one.
The sixth is 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 digits.
- 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 - sextillion, 22-24 digits.
- 842 527 342 458 752 468 359 173
Eight hundred and forty-two sextillion, five hundred and 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 classes by numbering, for example, the number of class 11 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, when performing operations with such quantities, the number of zeros is reduced by raising them to a power. After all, it is much easier to write 10 31 than to add thirty-one zeros to one.
education.guru
What are bit terms?
Answers and explanations
For example: 5679=5000+600+70+9
That is, the number of units in the category
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- Flag violation
the sum of the digit terms of the number 526 is 500+20+6
“sum of digit terms” is a representation of a two (or more) digit number as the sum of its digits.
Place terms are the addition of numbers with different bit depths. For example, we divide the number 17.890 into digit terms: 17.890=10.000+7.000+800+90+0
Rule for multiplying any number by zero
Even at school, teachers tried to hammer into our heads the simplest rule: “Any number multiplied by zero equals zero!”, – but still a lot of controversy constantly arises around him. Some people just remember the rule and don’t bother themselves with the question “why?” “You can’t and that’s it, because they said so at school, the rule is the rule!” Someone can fill half a notebook with formulas, proving this rule or, conversely, its illogicality.
Who's right in the end?
During these disputes, both people with opposing 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 their horns on each other. The only difference between them is that one is slightly less educated than the other. Most often, those who consider this rule to be incorrect try to appeal to logic in this way:
I have two apples on my table, if I put zero apples on them, that is, I don’t put a single one, then my two apples will not disappear! 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 = 2. So let’s discard this conclusion right away - it is illogical, although it has the opposite purpose - to call to logic.
This is interesting: How to find the difference between numbers in mathematics?
What is multiplication
Originally the multiplication rule was defined only for natural numbers: multiplication is a number added to itself a certain number of times, which implies that the number is natural. 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 it follows that 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. Ancient Eastern scientists thought differently - they approached the issue philosophically and drew some parallels between emptiness and infinity and saw deep meaning in this number. After all, zero, which has the meaning of emptiness, standing next to any natural number, multiplies it ten times. Hence all the controversy about multiplication - this number carries so much inconsistency that it becomes difficult not to get confused. In addition, zero is constantly used to define empty digits in decimals, this is done both before and after the decimal point.
Is it possible to multiply by emptiness?
You can multiply by zero, but it is useless, because, whatever one may say, even when multiplying negative numbers, you will still get zero. It’s enough just to remember this simple 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. Below we will give the most logical explanation that this multiplication is useless, because when you multiply a number by it, you will still get the same thing - zero.
Returning to the very beginning, to the argument about two apples, 2 times 0 looks like this:
- If you eat two apples five times, then you eat 2?5 = 2+2+2+2+2 = 10 apples
- If you eat two of them three times, then you eat 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 yourself to a small child. Whatever one may say, the result will be 0, two or three can be replaced with absolutely any number and the result will be absolutely the same. And to put it simply, then zero is nothing, and when do you have there is nothing, then no matter how much you multiply, it’s still the same will be zero. There is no such thing as 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, another important rule follows:
You can't divide by zero!
This rule has also been persistently hammered into our heads since childhood. We just know that it’s impossible and that’s all without bothering ourselves. unnecessary information. If you are unexpectedly asked the question why it is forbidden to divide by zero, then the majority will be confused and will not be able to clearly answer the question. simple question from school curriculum, because there is not so much controversy and controversy surrounding this rule.
Everyone simply memorized the rule and did not divide by zero, not suspecting that the answer was hidden on the surface. Addition, multiplication, division and subtraction are unequal; of the above, only multiplication and addition are valid, 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. This means that the entry 10: 0 is the same abbreviation for 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,
So as not to divide by 0!
Cut 1 as you want, lengthwise,
Just don't divide by 0!
education.guru
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Our first lesson was called numbers. We have covered only a small part of this topic. In fact, the topic of numbers is quite extensive. It has a lot of subtleties and nuances, a lot of tricks and interesting features.
Today we will continue the topic of numbers, but again we will not consider it all, so as not to complicate learning with unnecessary information, which at first is not really needed. We'll talk about discharges.
Lesson contentWhat is a discharge?
If we talk in simple language, then the digit is the position of the digit in the number or the place where the digit is located. Let's take the number 635 as an example. This number consists of three digits: 6, 3 and 5.
The position where the number 5 is located is called units digit
The position where the number 3 is located is called tens place
The position where the number 6 is located is called hundreds place
Each of us has heard from school such things as “units”, “tens”, “hundreds”. The digits, in addition to playing the role of the position of the digit in the number, tell us some information about the number itself. In particular, the digits tell us the weight of the number. They tell you how many units, how many tens, and how many hundreds there are in a number.
Let's return to our number 635. In the ones place there is a five. What does this mean? And this means that the ones digit contains five ones. It looks like this:
In the tens place there is a three. This tells us that the tens place contains three tens. It looks like this:
There is a six in the hundreds place. This means that there are six hundreds in the hundreds place. It looks like this:
If we add up the number of resulting units, the number of tens and the number of hundreds, we get our original number 635
There are also higher digits such as the thousand digit, the tens of thousands digit, the hundreds of thousands digit, the millions digit and so on. We will rarely consider such large numbers, but nevertheless it is also desirable to know about them.
For example, in the number 1645832, the units digit contains 2 ones, the tens digit contains 3 tens, the hundreds digit contains 8 hundreds, the thousands digit contains 5 thousand, the tens of thousands digit contains 4 tens of thousands, the hundreds of thousands digit contains 6 hundred thousand, and the millions digit contains 1 million. .
At the first stages of studying digits, it is advisable to understand how many units, tens, hundreds a particular number contains. For example, the number 9 contains 9 ones. The number 12 contains two ones and one ten. The number 123 contains three ones, two tens and one hundred.
Grouping items
After counting certain items, ranks can be used to group these items. For example, if we count 35 bricks in the yard, then we can use discharges to group these bricks. In the case of grouping objects, the ranks can be read from left to right. Thus, the number 3 in the number 35 will indicate that the number 35 contains three tens. This means that 35 bricks can be grouped three times in ten pieces.
So, let’s group the bricks three times ten pieces each:
It turned out to be thirty bricks. But there are still five units of bricks left. We will call them as "five units"
The result was three dozen and five units of bricks.
And if we did not group the bricks into tens and ones, then we could say that the number 35 contains thirty-five units. This grouping would also be acceptable:
The same can be said about other numbers. For example, about the number 123. Earlier we said that this number contains three units, two tens and one hundred. But we can also say that this number contains 123 units. Moreover, you can group this number in another way, saying that it contains 12 tens and 3 ones.
Words units, tens, hundreds, replace the multiplicands 1, 10 and 100. For example, in the units place of the number 123 there is a digit 3. Using the multiplicand 1, we can write that this unit is contained in the ones place three times:
100 × 1 = 100
If we add up the results of 3, 20 and 100, we get the number 123
3 + 20 + 100 = 123
The same thing will happen if we say that the number 123 contains 12 tens and 3 ones. In other words, the tens will be grouped 12 times:
10 × 12 = 120
And units three times:
1 × 3 = 3
This can be understood from the following example. If there are 123 apples, then you can group the first 120 apples 12 times, 10 each:
It turned out to be one hundred and twenty apples. But there are still three apples left. We will call them as "three units"
If we add the results of 120 and 3, we again get the number 123
120 + 3 = 123
You can also group 123 apples into one hundred, two tens and three ones.
Let's group a hundred:
Let's group two dozen:
Let's group three units:
If we add up the results of 100, 20 and 3, we again get the number 123
100 + 20 + 3 = 123
And finally, let's consider the last possible grouping, where the apples will not be distributed into tens and hundreds, but will be collected together. In this case, the number 123 will be read as "one hundred twenty-three units" . This grouping would also be acceptable:
1 × 123 = 123
The number 523 can be read as 3 units, 2 tens and 5 hundreds:
1 × 3 = 3 (three units)
10 × 2 = 20 (two tens)
100 × 5 = 500 (five hundred)
3 + 20 + 500 = 523
Another number 523 can be read as 3 ones 52 tens:
1 × 3 = 3 (three units)
10 × 52 = 520 (fifty two tens)
3 + 520 = 523
You can also read it as 523 units:
1 × 523 = 523 (five hundred twenty-three units)
Where to apply the discharges?
Bits make some calculations much easier. Imagine that you are at the board and solving a problem. You are almost finished with the task, all that remains is to evaluate the last expression and get the answer. The expression to be calculated looks like this:
I don’t have a calculator at hand, but I want to quickly write down the answer and surprise everyone with the speed of my calculations. Everything is simple if you add up the units separately, the tens separately and the hundreds separately. You need to start with the ones digit. First of all, after the equal sign (=) you need to mentally put three dots. These points will be replaced by a new number (our answer):
Now let's start folding. The ones place of the number 632 contains the number 2, and the ones place of the number 264 contains the number 4. This means the ones place of the number 632 contains two ones, and the ones place of the number 264 contains four ones. Add 2 and 4 units and get 6 units. We write the number 6 in the units place of the new number (our answer):
Next we add up the tens. The tens place of 632 contains the number 3, and the tens place of 264 contains the number 6. This means that the tens place of 632 contains three tens, and the tens place of 264 contains six tens. Add 3 and 6 tens and get 9 tens. We write the number 9 in the tens place of the new number (our answer):
And finally, we add up the hundreds separately. The hundreds place of 632 contains the number 6, and the hundreds place of 264 contains the number 2. This means that the hundreds place of 632 contains six hundreds, and the hundreds place of 264 contains two hundred. Add 6 and 2 hundreds to get 8 hundreds. We write the number 8 in the hundreds place of the new number (our answer):
Thus, if you add 264 to the number 632, you get 896. Of course, you will calculate such an expression faster and those around you will begin to be surprised at your abilities. They will think that you are quickly calculating large numbers, but you were actually calculating small ones. Agree that small numbers are easier to calculate than large ones.
Bit overflow
A digit is characterized by a single digit from 0 to 9. But sometimes, when calculating a numerical expression, a digit overflow may occur in the middle of the solution.
For example, when adding the numbers 32 and 14, no overflow occurs. Adding the units of these numbers will give 6 units in the new number. And adding tens of these numbers will give 4 tens in the new numbers. The answer is 46, or six ones and four tens.
But when adding the numbers 29 and 13, an overflow will occur. Adding the ones of these numbers gives 12 ones, and adding the tens gives 3 tens. If you write the resulting 12 units in the units place in a new number, and the resulting 3 tens in the tens place, you will get an error:
The value of the expression 29+13 is 42, not 312. What should you do if there is an overflow? In our case, the overflow occurred in the units digit of the new number. When we add nine and three units, we get 12 units. And in the units digit you can only write numbers in the range from 0 to 9.
The fact is that 12 units is not easy "twelve units" . Otherwise, this number can be read as "two ones and one ten" . The units digit is for ones only. There is no place for dozens there. This is where our mistake lies. By adding 9 units and 3 units we get 12 units, which can be called in another way two ones and one ten. By writing two ones and one ten in one place, we made a mistake, which ultimately led to an incorrect answer.
To correct the situation, two units need to be written in the ones place of the new number, and the remaining ten must be transferred to the next tens place. After adding two tens and one ten, we add to the result the ten that remained when adding the units.
So, out of 12 units, we write two ones in the ones place of the new number, and move one ten to the next place
As you can see in the figure, we represented 12 units as 1 ten and 2 ones. We wrote two ones in the ones place of the new number. And one ten was transferred to the tens ranks. We will add this ten to the result of adding the tens of the numbers 29 and 13. In order not to forget about it, we wrote it above the tens of the number 29.
So, let's add up the tens. Two tens plus one ten is three tens, plus one ten, which remains from the previous addition. As a result, in the tens place we get four tens:
Example 2. Add the numbers 862 and 372 by digits.
We start with the ones digit. In the ones place of the number 862 there is a digit 2, in the ones place of the number 372 there is also a digit 2. This means that the ones place of the number 862 contains two ones, and the ones place of the number 372 also contains two ones. Add 2 units plus 2 units - we get 4 units. We write the number 4 in the units place of the new number:
Next we add up the tens. The tens place of 862 contains the number 6, and the tens place of 372 contains the number 7. This means that the tens place of 862 contains six tens, and the tens place of 372 contains seven tens. Add 6 tens and 7 tens and get 13 tens. A discharge has overflowed. 13 tens is a ten repeated 13 times. And if you repeat the ten 13 times, you get the number 130
10 × 13 = 130
The number 130 is made up of three tens and one hundred. We will write three tens in the tens place of the new number, and send one hundred to the next place:
As you can see in the figure, we represented 13 tens (the number 130) as 1 hundred and 3 tens. We wrote three tens in the tens place of the new number. And one hundred was transferred to the ranks of hundreds. We will add this hundred to the result of adding the hundreds of numbers 862 and 372. In order not to forget about it, we inscribed it above the hundreds of the number 862.
So let's add up the hundreds. Eight hundred plus three hundred is eleven hundred plus one hundred, which remains from the previous addition. As a result, in the hundreds place we get twelve hundred:
There is also an overflow in the hundreds place here, but this does not result in an error since the solution is complete. If desired, with 12 hundreds you can carry out the same actions as we did with 13 tens.
12 hundred is a hundred repeated 12 times. And if you repeat a hundred 12 times, you get 1200
100 × 12 = 1200
Of the 1200 there are two hundred and one thousand. Two hundred are written into the hundreds place of the new number, and one thousand is moved to the thousand place.
Now let's look at examples of subtraction. First, let's remember what subtraction is. This is an operation that allows you to subtract another from one number. Subtraction consists of three parameters: minuend, subtrahend, and difference. You also need to subtract by digits.
Example 3. Subtract 12 from 65.
We start with the ones digit. The ones place of the number 65 contains the number 5, and the ones place of the number 12 contains the number 2. This means that the ones place of the number 65 contains five ones, and the ones place of the number 12 contains two ones. Subtract two units from five units and get three units. We write the number 3 in the units place of the new number:
Now let's subtract the tens. In the tens place of the number 65 there is a digit 6, in the tens place of the number 12 there is a digit 1. This means that the tens place of the number 65 contains six tens, and the tens place of the number 12 contains one ten. Subtract one ten from six tens, we get five tens. We write the number 5 in the tens place of the new number:
Example 4. Subtract 15 from 32
The ones digit of 32 contains two ones, and the ones digit of 15 contains five ones. You cannot subtract five units from two units, since two units are less than five units.
Let's group 32 apples so that the first group contains three dozen apples, and the second group contains the remaining two units of apples:
So, we need to subtract 15 apples from these 32 apples, that is, subtract five ones and one ten apples. And subtract by rank.
You cannot subtract five units of apples from two units of apples. To perform a subtraction, two units must take some apples from an adjacent group (the tens place). But you can’t take as much as you want, since the dozens are strictly ordered in sets of ten. The tens place can only give two ones a whole ten.
So, we take one ten from the tens place and give it to two units:
The two units of apples are now joined by one dozen apples. Makes 12 apples. And from twelve you can subtract five, you get seven. We write the number 7 in the units place of the new number:
Now let's subtract the tens. Since the tens place gave one ten to the units, now it has not three, but two tens. Therefore, we subtract one ten from two tens. There will be only one dozen left. Write the number 1 in the tens place of the new number:
In order not to forget that in some category one ten (or a hundred or a thousand) was taken, it is customary to put a dot above this category.
Example 5. Subtract 286 from 653
The ones digit of 653 contains three ones, and the ones digit of 286 contains six ones. You cannot subtract six ones from three units, so we take one ten from the tens place. We put a dot over the tens place to remember that we took one ten from there:
One ten and three ones taken together make thirteen ones. From thirteen units you can subtract six units to get seven units. We write the number 7 in the units place of the new number:
Now let's subtract the tens. Previously, the tens place of 653 contained five tens, but we took one ten from it, and now the tens place contains four tens. You cannot subtract eight tens from four tens, so we take one hundred from the hundreds place. We put a dot over the hundreds place to remember that we took one hundred from there:
One hundred and four tens taken together make fourteen tens. You can subtract eight tens from fourteen tens to get 6 tens. We write the number 6 in the tens place of the new number:
Now let's subtract hundreds. Previously, the hundreds place of 653 contained six hundreds, but we took one hundred from it, and now the hundreds place contains five hundred. From five hundred you can subtract two hundred to get three hundred. Write the number 3 in the hundreds place of the new number:
It is much more difficult to subtract from numbers like 100, 200, 300, 1000, 10000. That is, numbers with zeros at the end. To perform a subtraction, each digit has to borrow tens/hundreds/thousands from the next digit. Let's see how this happens.
Example 6
The ones digit of 200 contains zero ones, and the ones digit of 84 contains four ones. You cannot subtract four ones from zero, so we take one ten from the tens place. We put a dot over the tens place to remember that we took one ten from there:
But in the tens place there are no tens that we could take, since there is also a zero there. In order for the tens place to give us one ten, we must take one hundred from the hundreds place for it. We put a dot over the hundreds place to remember that we took one hundred from there for the tens place:
One hundred taken is ten tens. From these ten tens we take one ten and give it to the units. This one ten taken and the previous zero ones together form ten ones. From ten units you can subtract four units to get six units. We write the number 6 in the units place of the new number:
Now let's subtract the tens. To subtract units, we turned to the tens place after one ten, but at that moment this place was empty. So that the tens place can give us one ten, we take one hundred from the hundreds place. We called this one hundred "ten tens" . We gave one ten to a few. So on this moment The tens place contains not ten, but nine tens. From nine tens you can subtract eight tens to get one ten. Write the number 1 in the tens place of the new number:
Now let's subtract hundreds. For the tens place, we took one hundred from the hundreds place. This means that now the hundreds category contains not two hundred, but one. Since there is no hundreds place in the subtrahend, we move this one hundred to the hundreds place of the new number:
Naturally, perform subtraction like this traditional method quite difficult, especially at first. Having understood the principle of subtraction itself, you can use non-standard methods.
The first way is to reduce a number that has zeroes at the end by one. Next, subtract the subtrahend from the result obtained and add the unit that was originally subtracted from the minuend to the resulting difference. Let's solve the previous example this way:
The number being reduced here is 200. Let's reduce this number by one. If you subtract 1 from 200, you get 199. Now in the example 200 − 84, instead of the number 200, we write the number 199 and solve the example 199 − 84. And solving this example is not particularly difficult. Let's subtract units from units, tens from tens, and simply transfer a hundred to a new number, since there are no hundreds in the number 84
We received the answer 115. Now to this answer we add one, which we initially subtracted from the number 200
The final answer was 116.
Example 7. Subtract 91899 from 100000
Subtract one from 100000, we get 99999
Now subtract 91899 from 99999
To the result 8100 we add one, which we subtracted from 100000
We received the final answer 8101.
The second way to subtract is to treat the digit in the digit as a number in its own right. Let's solve a few examples this way.
Example 8. Subtract 36 from 75
So, in the units place of the number 75 there is the number 5, and in the units place of the number 36 there is the number 6. You cannot subtract six from five, so we take one unit from the next number, which is in the tens place.
In the tens place there is the number 7. Take one unit from this number and mentally add it to the left of the number 5
And since one unit is taken from the number 7, this number will decrease by one unit and turn into the number 6
Now in the ones place of the number 75 there is the number 15, and in the ones place of the number 36 the number 6. From 15 you can subtract 6, you get 9. We write the number 9 in the ones place of the new number:
Let's move on to the next number, which is in the tens place. Previously, the number 7 was located there, but we took one unit from this number, so now the number 6 is located there. And in the tens place of the number 36 there is the number 3. From 6 you can subtract 3, you get 3. We write the number 3 in the tens place of the new number:
Example 9. Subtract 84 from 200
So, in the ones place of the number 200 there is a zero, and in the ones place of the number 84 there is a four. You cannot subtract four from zero, so we take one unit from the next number in the tens place. But in the tens place there is also a zero. Zero cannot give us one. In this case, we take 20 as the next number.
We take one unit from the number 20 and mentally add it to the left of the zero located in the ones place. And since one unit is taken from the number 20, this number will turn into the number 19
Now the number 10 is in the ones place. Ten minus four equals six. We write the number 6 in the units place of the new number:
Let's move on to the next number, which is in the tens place. Previously, there was a zero there, but this zero, together with the next digit 2, formed the number 20, from which we took one unit. As a result, the number 20 turned into the number 19. It turns out that now the number 9 is located in the tens place of the number 200, and the number 8 is located in the tens place of the number 84. Nine minus eight equals one. We write the number 1 in the tens place of our answer:
Let's move on to the next number, which is in the hundreds place. Previously, the number 2 was located there, but we took this number, together with the number 0, as the number 20, from which we took one unit. As a result, the number 20 turned into the number 19. It turns out that now in the hundreds place of the number 200 there is the number 1, and in the number 84 the hundreds place is empty, so we transfer this unit to the new number:
This method at first seems complicated and makes no sense, but in fact it is the easiest. We will mainly use it when adding and subtracting numbers in a column.
Column addition
Column addition is a school operation that many people remember, but it doesn’t hurt to remember it again. Column addition occurs by digits - units are added with units, tens with tens, hundreds with hundreds, thousands with thousands.
Let's look at a few examples.
Example 1. Add 61 and 23.
First, write down the first number, and below it the second number so that the units and tens of the second number are under the units and tens of the first number. We connect all this with an addition sign (+) vertically:
Now we add the units of the first number with the units of the second number, and the tens of the first number with the tens of the second number:
We got 61 + 23 = 84.
Example 2. Add 108 and 60
Now we add the units of the first number with the units of the second number, the tens of the first number with the tens of the second number, the hundreds of the first number with the hundreds of the second number. But only the first number 108 has a hundred. In this case, the digit 1 from the hundreds place is added to the new number (our answer). As they said at school, “it’s being demolished”:
It can be seen that we have added the number 1 to our answer.
When it comes to addition, it makes no difference in what order you write the numbers. Our example could easily be written like this:
The first entry, where the number 108 was at the top, is more convenient for calculation. A person has the right to choose any entry, but one must remember that units must be written strictly under units, tens under tens, hundreds under hundreds. In other words, the following entries will be incorrect:
If suddenly, when adding the corresponding digits, you get a number that does not fit into the digit of the new number, then you need to write down one digit from the low-order digit and move the remaining one to the next digit.
Speech in in this case This is about the overflow of the bit that we talked about earlier. For example, when you add 26 and 98, you get 124. Let's see how it turned out.
Write the numbers in a column. Units under units, tens under tens:
Add the units of the first number with the units of the second number: 6+8=14. We received the number 14, which does not fit into the units category of our answer. In such cases, we first take out the digit from 14 that is in the ones place and write it in the units place of our answer. In the units place of the number 14 there is the number 4. We write this number in the units place of our answer:
Where should I put the number 1 from the number 14? This is where the fun begins. We transfer this unit to the next category. It will be added to the dozens of our answer.
Adding tens with tens. 2 plus 9 equals 11, plus we add the unit that we got from the number 14. By adding our unit to 11, we get the number 12, which we write in the tens place of our answer. Since this is the end of the solution, there is no longer a question of whether the resulting answer will fit into the tens place. We write down 12 in its entirety, forming the final answer.
We received a response of 124.
Using the traditional addition method, adding 6 and 8 units together results in 14 units. 14 units is 4 units and 1 ten. We wrote down four ones in the ones place, and sent one ten to the next place (to the tens place). Then, adding 2 tens and 9 tens, we got 11 tens, plus we added 1 ten, which remained when adding ones. As a result, we got 12 tens. We wrote down these twelve tens in their entirety, forming the final answer 124.
This simple example demonstrates a school situation in which they say “we write four, one in mind” . If you solve examples and after adding the digits you still have a number that you need to keep in mind, write it down above the digit where it will be added later. This will allow you not to forget about it:
Example 2. Add the numbers 784 and 548
Write the numbers in a column. Units under units, tens under tens, hundreds under hundreds:
Add the units of the first number with the units of the second number: 4+8=12. The number 12 does not fit into the units category of our answer, so we take out the number 2 from 12 from the ones category and write it into the units category of our answer. And we move the number 1 to the next digit:
Now we add up the tens. We add 8 and 4 plus the unit that remained from the previous operation (the unit remained from 12, in the figure it is highlighted in blue). Add 8+4+1=13. The number 13 will not fit into the tens place of our answer, so we write the number 3 in the tens place, and move the unit to the next place:
Now we add up the hundreds. We add 7 and 5 plus the unit that remains from the previous operation: 7+5+1=13. Write the number 13 in the hundreds place:
Column subtraction
Example 1. Subtract the number 53 from the number 69.
Let's write the numbers in a column. Units under units, tens under tens. Then we subtract by digits. From the units of the first number, subtract the units of the second number. From the tens of the first number, subtract the tens of the second number:
We received a response of 16.
Example 2. Find the value of the expression 95 − 26
The ones place of the number 95 contains 5 ones, and the ones place of the number 26 contains 6 ones. You cannot subtract six ones from five units, so we take one ten from the tens place. This ten and the existing five ones together make 15 units. From 15 units you can subtract 6 units to get 9 units. We write the number 9 in the units place of our answer:
Now let's subtract the tens. The tens place of 95 used to contain 9 tens, but we took one ten from that place, and now it contains 8 tens. And the tens place of the number 26 contains 2 tens. You can subtract two tens from eight tens to get six tens. We write the number 6 in the tens place of our answer:
Let's use it in which each digit included in a number is considered as a separate number. When subtracting large numbers in a column this method is very convenient.
In the units place of the minuend is the number 5. And in the units place of the subtrahend is the number 6. You cannot subtract a six from a five. Therefore, we take one unit from the number 9. The taken unit is mentally added to the left of the five. And since we took one unit from the number 9, this number will decrease by one unit:
As a result, the five turns into the number 15. Now we can subtract 6 from 15. We get 9. We write the number 9 in the units place of our answer:
Let's move on to the tens category. Previously, the number 9 was located there, but since we took one unit from it, it turned into the number 8. In the tens place of the second number there is the number 2. Eight minus two is six. We write the number 6 in the tens place of our answer:
Example 3. Let's find the value of the expression 2412 − 2317
We write this expression in the column:
In the ones place of the number 2412 there is the number 2, and in the ones place of the number 2317 there is the number 7. You cannot subtract seven from two, so we take one from the next number 1. We mentally add the taken one to the left of the two:
As a result, two turns into the number 12. Now we can subtract 7 from 12. We get 5. We write the number 5 in the units place of our answer:
Let's move on to tens. In the tens place of the number 2412 there used to be the number 1, but since we took one unit from it, it turned into 0. And in the tens place of the number 2317 there is the number 1. You cannot subtract one from zero. Therefore, we take one unit from the next number 4. We mentally add the taken unit to the left of zero. And since we took one unit from the number 4, this number will decrease by one unit:
As a result, zero turns into the number 10. Now you can subtract 1 from 10. You get 9. We write the number 9 in the tens place of our answer:
In the hundreds place of the number 2412 there used to be a number 4, but now there is a number 3. In the hundreds place of the number 2317 there is also a number 3. Three minus three equals zero. The same goes for the thousand places in both numbers. Two minus two equals zero. And if the difference between the most significant digits is zero, then this zero is not written down. Therefore, the final answer will be the number 95.
Example 4. Find the value of the expression 600 − 8
In the units place of the number 600 there is a zero, and in the units place of the number 8 this number itself is located. You can’t subtract eight from zero, so we take one from the next number. But the next number is also zero. Then we take the number 60 as the next number. We take one unit from this number and mentally add it to the left of zero. And since we took one unit from the number 60, this number will decrease by one unit:
Now the number 10 is in the ones place. From 10 you can subtract 8, you get 2. Write the number 2 in the units place of the new number:
Let's move on to the next number, which is in the tens place. There used to be a zero in the tens place, but now there is a number 9 there, and in the second number there is no tens place. Therefore, the number 9 is transferred, as it is, to the new number:
Let's move on to the next number, which is in the hundreds place. There used to be a number 6 in the hundreds place, but now there is a number 5 there, and in the second number there is no hundreds place. Therefore, the number 5 is transferred, as it is, to the new number:
Example 5. Find the value of the expression 10000 − 999
Let's write this expression in a column:
In the units place of the number 10000 there is a 0, and in the units place of the number 999 there is a number 9. You cannot subtract nine from zero, so we take one unit from the next number, which is in the tens place. But the next digit is also zero. Then we take 1000 as the next number and take one from this number:
The next number in this case was 1000. Taking one from it, we turned it into the number 999. And we added the taken unit to the left of zero.
Further calculations were not difficult. Ten minus nine equals one. Subtracting the numbers in the tens place of both numbers gave zero. Subtracting the numbers in the hundreds place of both numbers also gave zero. And the nine from the thousands place was moved to a new number:
Example 6. Find the value of the expression 12301 − 9046
Let's write this expression in a column:
In the units place of the number 12301 there is the number 1, and in the units place of the number 9046 there is the number 6. You cannot subtract six from one, so we take one unit from the next number, which is in the tens place. But in the next digit there is a zero. Zero can't give us anything. Then we take 1230 as the next number and take one from this number:
PURPOSE: to create conditions for introducing the concept of “bit terms”.
- Learn to represent numbers as a sum of digit terms.
- Systematize and deepen students’ knowledge about natural numbers.
- To develop students' computing skills and the ability to recognize geometric shapes.
1. Organizational moment.
Teacher: Guys, let's check your readiness for the lesson. Solve the problem:
There were 8 ears sticking out from behind the bush. These are the bunnies hiding. How many are there?
Teacher: How did you reason?
Timur: I counted 2 - 2, and even 2 would be 4 ears. These are 2 bunnies. 2 more, and 2 more, 2 more bunnies. Only 4 bunnies.
Teacher: How many legs do they have?
Artem: 16. I thought like this - 4+4 =8, 8+4=12, 12+4=16.
Teacher: How many tails do they have?
Teacher: How did you reason?
Children: There were 4 bunnies in total, which means they had 4 tails.
Teacher: Who hunts bunnies?
Children: Fox.
2. Updating knowledge. Working with numbers.
Teacher: Today a fox came to our lesson, but an unusual one.<Рисунок 1 >She will help us make a discovery today. Look, she's holding some secret in her paws. She has prepared a task for you. Read the numbers: 4,1,6,3.
Teacher: What can these numbers in the picture mean?
Children: 4 - circles.
3 - daisies on the fox's dress.
1 - pentagon, 1 flower in the fox's paw.
6 - triangles, both small and large...
Artem: 1- octagon.
Teacher: Where in the picture, Artem, did you find such a figure? Can you show me? (Artem goes to the board, starts counting... Counts 9 sides.)
Teacher: What is the name of such a figure?
Artem: Ninegon.
Ksyusha: 1 - oval. This is the mouth of a fox.
Polina: 1 - triangle.
Teacher: Which one?
Polina: The fox has a nose on its face.
Teacher: Did I understand you correctly....Did you talk about the brown triangle?
Polina: Yes.
Teacher: Or maybe some other numbers can be found in the picture?
Children: 2 - yellow circles, 2 - orange...
Teacher: What can you say about these numbers?
Children: Natural numbers. The numbers are single digits. The numbers are not in order. Numbers are missing…..If the numbers are inserted, you get a natural series.
Teacher: Children, do you agree with Artem? What are the numbers and in what order will they go?
(Write 1,2,3,4,5,6 on the board)
Teacher: Is this entry a natural series of numbers?
Alina: This is a segment of a natural series of numbers.
Teacher: How can we make this record become a natural series of numbers?
Nastya: We need to put points.
Teacher: Why?
Alina: This will mean that the numbers will go further.
Teacher: What feature of the natural series were you talking about?
Nastya: About infinity.
Teacher: Guys, was it easy to complete the assignments? Do you want a more difficult task?
Teacher: Using these numbers, compose and write in your notebook double figures, in which there are more tens than ones. How did you understand?
Artem: I will make up numbers in which there are more tens than ones.
Teacher: Go ahead. (Children complete the task in notebooks and on the board.)
As a result of the check, the entry appears: 65, 64, 61, 54, 51, 41.
Teacher: Are there other options for completing the task?
Dasha: Yes. I wrote down the numbers 66, 11,44, 33.
Teacher: Guys, what can you say about Dasha’s work?
Children: Dasha, you used the same numbers in the recording, but the task was different.
Teacher: How are these numbers different from these?
Children: They have tens and ones. There are two numbers in the entry.
Teacher: Underline the numbers in the tens place with one line, and in the ones place with two lines. (A card is attached to the board - tens place, units place)
Teacher: Do you think this is all we know about two-digit numbers? Do you want to know? Why do you need this?
Children: - We will learn to add two-digit numbers. This will be useful to us.
My brother solves such examples in which……. must be multiplied by ………. . First you need to find out everything about such numbers.
Teacher: How are we going to do this?
Children: You have prepared a task for us.
3. Studying new material. Introduction to the concept of bit terms.
Teacher: Try to guess which number is missing. I distribute sheets only to the first desks, and there are only 6 of them.)
Oh guys, what should I do? I only have 6 sheets, but there are a lot of you. What should I do?
Children: let's work in groups... (On the sheets there are equalities with in which terms are missing. In several equalities, the terms are digit terms. For one group, in which the weaker students are, all equalities are written as the sum of digit terms).
| 54+…=61 | 60 +…=61 |
| 60 + …=64 | 60 +…=64 |
| 59 +…=63 | 60 +…=63 |
| 40 + …= 43 | 40 +…= 41 |
| 37 + ….=41 | 40 +…=43 |
| 27 +…=31 | 30 +…= 31 |
Teacher: Check that you did it correctly.
Teacher: Who noticed which group completed the task first? (I finished the work before everyone else, just the group in which I studied weaker.)
Teacher: Why do you think?
Children: Their equality is easier.
Teacher: How is this?
Children: There are tens and ones, so it was easier to look for the missing numbers.
Teacher: Did I understand you correctly that the first term is tens, and the second is units? What does the I term mean? And the second term? Try to come up with a name using this term...
Children confer in groups.
Teacher: What options did you get?
Children: -We just named tens and units.
We couldn't come up with one.
We called the bit terms.
Teacher: What do you think, how can you check the correctness of your answers? Open the textbook on page 25, find on the page the name of such terms.... (Children read with buzz reading).
Teacher: Let's check, what did the fox bring us... (The card is turned over, and there is a note on it - BITS.)
Teacher: Who guessed what topic we are working on today?
Teacher: Using cards, show the place value terms of the numbers 39 and 93.
4. Physical exercise. The attention exercise “Desk” is carried out (If the teacher calls the word DESK before the movement, then the students perform the action, and if the word is not named or some other word is named, then the students do not perform the movement.)
5. Reinforcing the concept of bit terms.
Teacher: Maybe it’s the numbers - they are easy for you, and you completed the task easily? Can you handle other numbers? Complete step 4 of task No. 60.
Teacher: What will you do?
Teacher: I also want to work, I will complete the task with you on the board. (On the board I make a note in which the “trap” is made)
20 +9 =29
72+4=76
60+5=65
52+3=56
10+7=17
Teacher: Check your work with the model.
Teacher: Our fox seems sad. Maybe because of the assignment? What do you think needs to be done? (To the left and right of the fox there are cards with expressions. For example: 80+12, 32+4, 50+8, 42+10, 60+6, 50+ 14, 70+5, 80+7)
Children: Find the sums of the bit terms.
Teacher: Go ahead.
MUTUAL CHECK. After completing the task, the cards with the sums of the bit terms are removed.
Teacher: What can you do with the remaining expressions?
Expected answers from children: You can find the values of the sum, or you can change the terms so that they become digits. The check is carried out according to the sample.
6. Summing up the lesson.
Teacher: What topic did you work on in class?
Which task was the most interesting?
The most difficult?
Teacher: Since there were difficulties, I suggest you complete the task at home (it was written down in advance, but covered with a sheet):
Choose the task that will be more interesting for you to work with.
Explanation of new material
to CEO you need to be smart. Today in the lesson we will talk about how to represent a multi-digit number as a sum of digit terms.
You have already done this kind of work with three digit numbers. Represent the number one hundred twenty-eight as a sum of digit terms~4~
That's right, the number one hundred twenty-eight consists of the sum of the digit terms one hundred, twenty and eight.
Multi-digit numbers are replaced by the sum of digit terms in the same way. Look at the following entry. The number four hundred twenty-seven thousand nine hundred and forty can be represented as a sum of digit terms: four hundred thousand, twenty thousand, seven thousand, nine hundred and forty. When decomposing numbers, remember that each class has three digits. Each class is written using three numbers.
To represent a number as a sum of digit terms you need:
Determine the number of bit terms (by the number of digits other than zero).
Stage of assimilation of new knowledge
Exercise
If you have good ingenuity, you can easily replace the following numbers with the sum of the digit terms.
Test yourself.
725 368 = 700 000+ 20 000 + 5 000 + 300 + 60 + 8
45 200 = 40 000 + 5 000 + 200
390 020= 300 000 + 90 000 + 20
500 068 = 500 000 + 60 + 8
610 707= 600 000 + 10 000 + 700 + 7
Exercise
Your company has competitors. They really don't like the fact that you are lucky and you are a leader among other companies. They decided to harm you and erased the numbers in the report. Will you be able to recover the document?
Fill in the missing numbers:
408 690 = 400 000 + … + 600 + 90
200 097 = 200 000 + … + 7
560 448 = … + 60 000 + … + 40 + 8
384 794 = 300 000 + 80 000 + … + 700 + 90 + …
62 058= … + 2 000 + … + 8
Test yourself.
408 690 = 400 000 + 8 000 + 600 + 90
200 097 = 200 000 + 90 + 7
560 448 = 500 000 + 60 000 + 400 + 40 + 8
384 794 = 300 000 + 80 000 + 4 000 + 700 + 90 + 4
62 058= 60 000 + 2 000 + 50 + 8
In the first expression we insert the number 8,000.
The number 90 is missing in the second expression
The numbers 500,000 and 400 are missing in the third expression.
In the fourth numerically the numbers 4,000 and 4 are missing.
The numbers 60,000 and 50 are missing in the fifth numerical expression.
Well done guys, you dealt with this quickly challenging task
Stage of assimilation of new knowledge
The president of the company needs to have a good understanding of financial statements. Let's see if you can handle the next task.
Write which numbers are represented as a sum of digit terms.
700 000 + 50 000 + 2 =
80 000 + 6 000 + 30 + 7 =
900 000 + 4 000 + 800 + 90 +3=
200 000 + 2 000 + 8 =
Test yourself.
Well done boys! Well done.
Exercise
Next task. The accountant made errors in calculations. Your task is to find and correct errors.
450 680 = 400 000 + 500 000 + 600 + 80
950 200 = 90 000 + 50 000 + 200
38 405 = 30 000 + 800 + 40 + 5
603 010 = 60 000 + 3 000 + 100
84 811 = 800 000 + 4 000 + 800 + 10 + 1
Test yourself.
450 680 = 400 000 + 50 000 + 600 + 80
950 200 = 900 000 + 50 000 + 200
38 405 = 30 000 + 8 000 + 400 + 5
603 010 = 600 000 + 3 000 + 10
84 811 = 80 000 + 4 000 + 800 + 10 + 1
Exercise
Now calculate the revenue from different branches. I think you know that a branch is your company located in another location and carrying out the same activities. Branch employees submitted reports containing errors. Find and fix errors.
800 000 + 30 000 + 400 + 50 + 2 =
50 000 + 7 000 + 800 + 10 = 507 810
600 000 + 40 000 + 900 + 1 = 640 091
30 000 + 4 000 + 20 = 34 200
4 000 + 600 + 30 + 7 = 40 637
Test yourself.
Let's remember once again what qualities a company director should have.
He must own competent speech.
Exercise
Read multi-digit numbers.
Six hundred eighty-nine thousand eight hundred, fifty-two thousand four hundred ten, seven hundred thousand four, three hundred one thousand two hundred forty-seven, eight hundred thousand sixty.
Exercise
The director of the company must be able to compare his profits with the profits of competitors.
Compare the numbers.
a+ 3150 a+ 3,015
Test yourself.
a+ 3150 a+ 3,015
Exercise
The director of the company must be able to distribute salaries among employees. To do this, complete the following task. Present numbers as a sum of digit terms.
Test yourself.
602 420 = 600 000 + 2 000 + 400 + 20
700 043 =700 000 + 40 + 3
86 480 = 80 000 + 6 000 + 400 + 80
301 071= 300 000 + 1 000 + 70 + 1
And of course, the director of the company must be able to count well. Find the sum of the bit terms.
400 000 + 50 000 + 300 + 8 =
80 000 + 2 000 + 100 +6 =
500 000 + 7 000 + 80 + 3 =
90 000 + 9 000 + 900 + 9 =
70 000 + 4 000 + 1 =
Test yourself.
If you completed all the tasks without errors, then when you grow up, you can become directors of companies.
Lesson summary
The owl speaks
Guys, let's remember how to correctly represent a number as a sum of digit terms.
To do this, you need to determine the number of bit terms (by the number of digits other than zero).
Then determine the number of zeros in each bit term.
Write down the sum of the bit terms.
