Today we will consider a rather boring topic, without understanding which it is not possible to move on. This topic is called "rounding numbers" or in other words "approximate values of numbers".

Lesson content

Approximate values

Approximate (or approximate) values are used when the exact value of something cannot be found, or it does not matter that this value is accurate for the object under study.

For example, one can say in words that half a million people live in the city, but this statement will not be true, since the number of people in the city changes - people come and go, are born and die. Therefore, it would be more correct to say that the city is home to approximately half a million people.

Another example. Classes begin at nine in the morning. We left the house at 8:30. After a while, on the way, we met our friend, who asked us what time it was. When we left the house it was 8:30, we spent some unknown time on the road. We do not know what time it is, so we answer our comrade: “now approximately about nine o'clock. "

In mathematics, approximate values are indicated using a special sign. It looks like this:

Reads like "Approximately (approximately) equal to" .

To indicate an approximate (approximate) value, resort to such an action as rounding numbers.

Rounding numbers

To find an approximate value, such an action is applied as rounding numbers.

The word "rounding" speaks for itself. To round a number is to make it round. Round is a number that ends in zero. For example, the following numbers are round:

10, 20, 30, 100, 300, 700, 1000

Any number can be made round. The procedure for making a number round is called rounding the number.

We've already done "rounding" numbers when dividing large numbers. Recall that for this we left the digit forming the most significant digit unchanged, and replaced the remaining digits with zeros. But these were just sketches that we made to facilitate division. A kind of life hack. In fact, it wasn't even a rounding off of numbers. That is why at the beginning of this paragraph we took the word rounding in quotation marks.

In fact, the point of rounding is to find the closest value from the original. At the same time, the number can be rounded up to a certain digit - to the tens, hundreds, thousands.

Let's look at a simple rounding example. Given number 17. It is required to round it to the place of tens.

Without getting ahead of ourselves, let's try to understand what it means to “round up to the rank of tens”. When they say to round off the number 17, we must understand that we are required to find the nearest round number from the number 17. Moreover, during this search, it is possible that changes will also affect the number that is located in the tens place of the number 17 (ie the number 1).

Let's represent the numbers from 10 to 20 using the following figure:

The figure shows that for the number 17 the nearest round number is 20. So the answer to the problem will be as follows: "17 approximately equal twenty"

17 ≈ 20

We found an approximate value for 17, that is, we rounded it up to the place of tens. It can be seen that after rounding, a new digit 2 appears in the tens place.

Let's try to find an approximate number for the number 12. To do this, again represent the numbers from 10 to 20 using the figure:

The figure shows that the nearest round number for 12 is the number 10. So the answer to the problem will be as follows: 12 approximatelyequals 10

12 ≈ 10

We have found an approximate value for 12, that is, we rounded it up to the tenth place. This time, the number 1, which was in the tens place in the number 12, did not suffer from rounding. We will tell you why this happened later.

Let's try to find the closest number for the number 15. Let's again represent the numbers from 10 to 20 using the figure:

The figure shows that the number 15 is equally distant from the round numbers 10 and 20. The question arises: which of these round numbers will be an approximate value for the number 15? For such cases, we agreed to take the larger number as an approximate one. 20 is greater than 10, so the approximate value for 15 would be 20

15 ≈ 20

Large numbers can also be rounded. Naturally, it is not possible for them to make drawings and depict numbers. There is a way for them. For example, round up 1456 to the tenth place.

So we have to round 1456 to the tens. The tens rank starts at five:

Now we temporarily forget about the existence of the first digits 1 and 4. Number 56 remains

Now let's see which round number is closer to the number 56. Obviously, the nearest round number for 56 is 60. So we replace the number 56 with the number 60

So, when rounding the number 1456 to the place of tens, we get 1460

1456 ≈ 1460

It can be seen that after rounding off the number 1456 to the tens digit, the changes also affected the tens digit itself. In the new received number, the digit of tens is now located in the digit 6, not 5.

You can round off numbers not only to the tens place. You can round off the number to the place of hundreds, thousands, tens of thousands, and so on.

After it becomes clear that rounding is nothing more than a search for the nearest number, you can apply ready-made rules that greatly facilitate rounding of numbers.

First rounding rule

In the previous examples, we saw that when rounding a number to a certain digit, the least significant digits are replaced with zeros. The numbers that are replaced by zeros are called discarded figures .

The first rounding rule is as follows:

If, when rounding numbers, the first of the discarded digits is 0, 1, 2, 3 or 4, then the stored digit remains unchanged.

For example, let's round the number 123 to the tens place.

First of all, we find the stored digit. To do this, you need to read the task itself. The digit to be stored is located in the digit referred to in the task. The assignment says: round the number 123 to the rank of tens.

We see that there is a two in the tens place. So the stored digit is the number 2

Now we find the first of the discarded digits. The first digit to be discarded is the digit that follows the digit to be stored. We see that the first digit after two is digit 3. So digit 3 is first discarded digit.

Now we apply the rounding rule. It says that if, when rounding numbers, the first of the discarded digits is 0, 1, 2, 3 or 4, then the stored digit remains unchanged.

So we do it. We leave the stored digit unchanged, and replace all the lower digits with zeros. In other words, we replace everything that follows the number 2 with zeros (more precisely, zero):

123 ≈ 120

This means that when the number 123 is rounded to the place of tens, we get the approximate number 120.

Now let's try to round the same number 123, but already up to rank of hundreds.

We need to round the number 123 to the hundredth place. Look for the stored digit again. This time, the stored digit is 1 as we round the number to the hundredth place.

Now we find the first of the discarded digits. The first digit to be discarded is the digit that follows the digit to be stored. We see that the first digit after one is digit 2. So digit 2 is the first discarded digit:

Now let's apply the rule. It says that if, when rounding numbers, the first of the discarded digits is 0, 1, 2, 3 or 4, then the stored digit remains unchanged.

So we do it. We leave the stored digit unchanged, and replace all the lower digits with zeros. In other words, replace everything that follows the number 1 with zeros:

123 ≈ 100

This means that when the number 123 is rounded to the place of hundreds, we get the approximate number 100.

Example 3. Round 1234 to the tens place.

Here, the stored digit is 3. And the first digit to discard is 4.

So we leave the stored digit 3 unchanged, and replace everything after it with zero:

1234 ≈ 1230

Example 4. Round 1234 to the hundredth place.

Here the stored digit is 2. And the first discarded digit is 3. According to the rule, if the first discarded digit is 0, 1, 2, 3 or 4 when rounding numbers, then the stored digit remains unchanged.

So we leave the stored digit 2 unchanged, and replace everything after it with zeros:

1234 ≈ 1200

Example 3. Round 1234 to the nearest thousand.

Here the stored digit is 1. And the first discarded digit is 2. According to the rule, if the first discarded digit is 0, 1, 2, 3 or 4 when rounding numbers, then the stored digit remains unchanged.

So we leave the stored digit 1 unchanged, and replace everything after it with zeros:

1234 ≈ 1000

Second rounding rule

The second rounding rule is as follows:

If, when rounding numbers, the first of the discarded digits is 5, 6, 7, 8 or 9, then the stored digit is increased by one.

For example, round off 675 to the tens.

We see that there is a seven in the tens place. So the stored digit is the number 7

Now we find the first of the discarded digits. The first digit to be discarded is the digit that follows the digit to be stored. We see that the first digit after the seven is the digit 5. So the digit 5 is first discarded digit.

Our first of the discarded digits is 5. So we must increase the stored digit 7 by one, and replace everything that follows after it with zero:

675 ≈ 680

This means that when the number 675 is rounded to the place of tens, we get the approximate number 680.

Now let's try to round the same number 675, but already up to rank of hundreds.

We need to round 675 to the hundredth place. Look for the stored digit again. This time, the stored digit is 6 as we round the number to the hundredth place:

Now we find the first of the discarded digits. The first digit to be discarded is the digit that follows the digit to be stored. We see that the first digit after the six is the number 7. So the number 7 is the first discarded digit:

Now we apply the second rounding rule. It says that if, when rounding numbers, the first of the discarded digits is 5, 6, 7, 8 or 9, then the stored digit is increased by one.

Our first of the discarded digits is 7. So we must increase the stored digit 6 by one, and replace everything after it with zeros:

675 ≈ 700

This means that when rounding the number 675 to the place of hundreds, we get the approximate number of 700.

Example 3. Round 9876 to the nearest tens.

Here, the stored digit is 7. And the first digit to discard is 6.

This means we increase the stored digit 7 by one, and replace everything after it with zero:

9876 ≈ 9880

Example 4. Round 9876 to the nearest hundred.

Here the stored digit is 8. And the first discarded digit is 7. According to the rule, if the first discarded digit is 5, 6, 7, 8 or 9 when rounding off numbers, then the stored digit is increased by one.

This means we increase the stored digit 8 by one, and replace everything after it with zeros:

9876 ≈ 9900

Example 5. Round 9876 to the nearest thousand.

Here the stored digit is 9. And the first discarded digit is 8. According to the rule, if the first discarded digit is 5, 6, 7, 8 or 9 when rounding numbers, then the stored digit is increased by one.

This means we increase the stored digit 9 by one, and replace everything after it with zeros:

9876 ≈ 10000

Example 6. Round the number 2971 to hundreds.

When rounding this number to hundreds, you should be careful, since the stored digit here is 9, and the first discarded digit is 7. So the digit 9 should increase by one. But the fact is that after increasing the nine by one, it will turn out to be 10, and this figure will not fit into the hundreds of the new number.

In this case, in the place of hundreds of the new number, it is necessary to write 0, and transfer the unit to the next place and add it with the digit that is there. Next, replace all digits after the stored one with zeros:

2971 ≈ 3000

Rounding decimals

You should be especially careful when rounding decimal fractions, since a decimal fraction consists of an integer and a fractional part. And each of these two parts has its own categories:

Integer bits:

category of units;
rank of tens;
rank of hundreds;
rank of thousands.

Fractional digits:

tenth rank;
hundredth place;
thousandth

Consider the decimal fraction 123.456 - one hundred twenty three point four hundred fifty six thousandths. Here the whole part is 123, and the fractional part is 456. Moreover, each of these parts has its own digits. It is very important not to confuse them:

For the integer part, the same rounding rules apply as for regular numbers. The difference is that after rounding the integer part and replacing all digits after the stored digit with zeros, the fractional part is completely discarded.

For example, round up 123.456 to the rank of tens. Precisely before rank of tens, but not tenths... It is very important not to confuse these digits. Discharge dozens is located in the whole part, and the discharge tenths in fractional.

So we have to round 123.456 to the tens place. The stored digit here is 2, and the first digit to be discarded is 3

According to the rule, if, when rounding numbers, the first of the discarded digits is 0, 1, 2, 3 or 4, then the stored digit remains unchanged.

This means that the stored digit will remain unchanged, and everything else will be replaced by zero. But what about the fractional part? It is simply discarded (removed):

123,456 ≈ 120

Now let's try to round the same fraction 123.456 to units discharge... The stored digit here will be 3, and the first digit to be discarded is 4, which is in the fractional part:

According to the rule, if, when rounding numbers, the first of the discarded digits is 0, 1, 2, 3 or 4, then the stored digit remains unchanged.

This means that the stored digit will remain unchanged, and everything else will be replaced by zero. The remaining fractional part will be discarded:

123,456 ≈ 123,0

The zero left after the decimal point can also be discarded. So the final answer will look like this:

123,456 ≈ 123,0 ≈ 123

Now let's start rounding the fractional parts. The rules for rounding fractional parts are the same as for rounding whole parts. Let's try to round the fraction 123.456 to digit of tenths. The digit 4 is in the tenth place, which means it is a stored digit, and the first discarded digit is 5, which is in the hundredth place:

According to the rule, if, when rounding numbers, the first of the discarded digits is 5, 6, 7, 8 or 9, then the stored digit is increased by one.

This means that the stored digit 4 will increase by one, and the rest will be replaced with zeros

123,456 ≈ 123,500

Let's try to round the same fraction 123.456 to the hundredth place. The stored digit here is 5, and the first discarded digit is 6, which is in the thousandths place:

According to the rule, if, when rounding numbers, the first of the discarded digits is 5, 6, 7, 8 or 9, then the stored digit is increased by one.

This means that the stored digit 5 will increase by one, and the rest will be replaced with zeros

123,456 ≈ 123,460

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Many people are wondering how to round numbers. This need often arises for people who associate their lives with accounting or other activities that require calculations. Rounding can be done to whole, tenths, and so on. And you need to know how to do it correctly so that the calculations are more or less accurate.

And what is a round number in general? This is the one that ends in 0 (for the most part). In everyday life, the ability to round numbers makes shopping much easier. Standing at the checkout, you can roughly estimate the total cost of purchases, compare how much a kilogram of a product of the same name costs in packages of different weight. With numbers reduced to a convenient form, it is easier to make oral calculations without resorting to a calculator.

Why are numbers rounded?

A person is inclined to round off any numbers in cases where more simplified operations need to be performed. For example, a melon weighs 3,150 kilograms. When a person tells his friends about how many grams the southern fruit has, he may be considered a not very interesting interlocutor. Phrases like "Here I bought a three-kilogram melon" sound much more laconic without delving into any unnecessary details.

Interestingly, even in science, there is no need to always deal with the most accurate numbers. And if we are talking about periodic infinite fractions, which have the form 3.33333333 ... 3, then this becomes impossible. Therefore, the most logical option would be to round them off as usual. As a rule, the result is then slightly distorted. So how do you round off numbers?

A few important rules when rounding numbers

So, if you wanted to round a number, is it important to understand the basic principles of rounding? This is a change operation aimed at reducing the number of decimal places. To carry out this action, you need to know several important rules:

If the number of the required digit is in the range of 5-9, rounding is carried out upward.
If the number of the required digit is in the range of 1-4, rounding down is performed.

For example, we have the number 59. We need to round it up. To do this, you need to take the number 9 and add one to it to get 60. This is the answer to the question of how to round numbers. Now let's look at some special cases. Actually, we figured out how to round a number to tens using this example. Now all that remains is to use this knowledge in practice.

How to round a number to integers

It often happens that there is a need to round off, for example, the number 5.9. This procedure is not difficult. First, we need to omit the comma, and when rounding, the already familiar number 60 appears before our eyes. And now we put the comma in its place, and we get 6.0. And since zeros in decimal fractions, as a rule, are omitted, we end up with the number 6.

A similar operation can be performed with more complex numbers. For example, how do you round numbers like 5.49 to integers? It all depends on what goals you set for yourself. In general, according to the rules of mathematics, 5.49 is still not 5.5. Therefore, it cannot be rounded up. But you can round it up to 5.5, after which it becomes legal to round up to 6. But this trick does not always work, so you need to be extremely careful.

In principle, an example of the correct rounding of a number to tenths has already been considered above, so now it is important to display only the main principle. In fact, everything happens in much the same way. If the digit that is in the second position after the decimal point is within 5-9, then it is removed altogether, and the digit in front of it is increased by one. If less than 5, then this figure is removed, and the previous one remains in its place.

For example, at 4.59 to 4.6, the number "9" leaves, and one is added to the five. But when rounding 4.41, the unit is omitted, and the four remains in an unnamed form.

How do marketers use the inability of the mass consumer to round numbers?

It turns out that most people in the world are not in the habit of assessing the real cost of a product, which is actively exploited by marketers. Everyone knows the slogans of stocks like "Buy for just 9.99". Yes, we consciously understand that this is essentially ten dollars. Nevertheless, our brain is designed in such a way that it only perceives the first number. So the simple operation of bringing a number into a convenient form should become a habit.

Very often, rounding off allows a better estimate of intermediate successes, expressed in numerical form. For example, a person began to earn $ 550 per month. An optimist will say that it is almost 600, a pessimist - that it is a little more than 500. It seems that there is a difference, but it is more pleasant for the brain to "see" that the object has achieved something more (or vice versa).

There are countless examples where the skill of rounding turns out to be incredibly useful. It is important to be creative and, if possible, not be loaded with unnecessary information. Then success will be immediate.

The numbers are also rounded to other digits - tenths, hundredths, tens, hundreds, etc.

If the number is rounded to a certain digit, then all digits following this digit are replaced with zeros, and if they are after the decimal point, then they are discarded.

Rule # 1. If the first of the discarded digits is greater than or equal to 5, then the last of the stored digits is amplified, that is, increased by one.

Example 1. Given the number 45.769, which must be rounded to tenths. The first discarded digit is 6 ˃ 5. Therefore, the last of the stored digits (7) is amplified, that is, increased by one. And thus, the rounded number would be 45.8.

Example 2. Given the number 5.165, which must be rounded to the nearest hundredth. The first discarded digit is 5 = 5. Therefore, the last of the stored digits (6) is amplified, that is, it is increased by one. And thus, the rounded number would be - 5.17.

Rule # 2. If the first of the discarded digits is less than 5, then no amplification is done.

Example: You are given the number 45.749, which must be rounded to the nearest tenth. The first discarded digit is 4

Rule # 3. If the discarded digit is 5, and there are no significant digits behind it, then rounding is performed to the nearest even number. That is, the last digit remains unchanged if it is even and is amplified if it is odd.

Example 1: Rounding 0.0465 to the third decimal place, we write - 0.046. We do not amplify, since the last stored digit (6) is even.

Example 2. Rounding the number 0.0415 to the third decimal place, we write - 0.042. We make gains, since the last stored digit (1) is odd.

Rounding of a natural number is understood to mean replacing it with such a closest number, in which one or more of the last digits in its record are replaced with zeros.

Rounding rule:

To round off a natural number, you need to select the digit to which the rounding is performed in the number entry.

The digit written in the selected digit:

does not change if the next digit on the right is 0, 1, 2, 3 or 4;

All digits to the right of this digit are replaced with zeros.

Example: 14 3 ≈ 140 (rounding to tens);
56 71 ≈ 5700 (round to the nearest hundred).

If the digit 9 is in the digit to which the rounding is performed and it is necessary to increase it by one, then the digit 0 is written in this digit, and the digit in the adjacent most significant digit (on the left) is increased by 1.

Example: 79 6 ≈ 800 (rounding to tens);
9 70 ≈ 1000 (round to the nearest hundred).

Rounding decimals

To round a decimal fraction, you need to select the digit to which the rounding is performed in the number entry. The digit written in this digit:

increases by one if the next digit on the right is 5,6,7,8 or 9.

All digits to the right of this digit are replaced with zeros. If these zeros are in the fractional part of the number, then they are not written.

Example: 143,6 4 ≈ 143.6 (round to tenths);
5,68 7 ≈ 5.69 (round to the nearest hundredth);
27 , 945 ≈ 28 (rounding to integers).

If the digit to which the rounding is performed contains the number 9 and it is necessary to increase it by one, then the digit 0 is written in this digit, and the digit in the previous digit (on the left) is increased by 1.

Example: 8 9, 6 ≈ 90 (rounding to tens);
0,09 7 ≈ 0.10 (round to the nearest hundredth).

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Rounding numbers

1) Rules for rounding natural numbers. Natural numbers are rounded up to units of some kind. Rounding a natural number to the units of a certain digit means to establish how many units of this digit are contained in a given number. For example, we want to round 237,456 to thousands. This means finding out how many thousands there are in this number. Obviously, there are 237 thousand in it. How did we know this? To do this, we all digits of a given number up to the thousandth place hundreds, tens and ones, replaced with zeros and got the number 237000, which can be written shorter as follows: 237 thousand. But you can, knowing that 1000 = 10 3, write this rounded number and like this: 237 * 10 3.

So, 237,456? 237 thousand or 237 456? 237 * 10 3.

Please note: here we put not the usual equal sign, but approximate equal sign (?).

Why exactly such a sign? Yes, because the numbers 237,456 and 237 thousand are not equal, the second number is slightly less than the first, namely, 456 less, therefore, replacing the number 237 456 with 237 thousand, we thereby make an error equal to 456, which means, that the numbers 237 456 and 237 thousand are only approximately equal. Therefore, a sign of approximate equality is put. Note that the error when rounding the number 237 456 to thousands was 456 units, which is less than half of one thousand. Therefore, if we need to round up the number 237 873 to thousands, then it is more reasonable to take 237 thousand as the rounded value of the number 237 873, then we will make an error equal to 873, which is more than half a thousand, i.e. 500. If, however, the rounded value is 238 thousand, then the error will be only 127, which is significantly less than half a thousand from these examples, you can deduce the following the general rule for rounding natural numbers to units of a certain digit: replace all digits to the right of this digit with zeros. If the first digit on the left from those replaced by zeros is less than 5, then the rounding is over and the resulting rounded number can be written in an abbreviated form. If it is equal to or greater than 5, then the digit of the digit to which the rounding was performed is replaced by a larger one.

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Rounding off natural numbers.

We often use rounding in everyday life. If the distance from home to school is 503 meters. We can say, rounded up, that the distance from home to school is 500 meters. That is, we brought the number 503 closer to the more easily perceived number 500. For example, a loaf of bread weighs 498 grams, then we can say, rounding up the result, that a loaf of bread weighs 500 grams.

Rounding- this is the approximation of a number to a "lighter" number for human perception.

As a result of rounding it turns out approximate number. Rounding is denoted by the symbol ≈, such a symbol is read “approximately equal”.

You can write 503-500 or 498-500.

One reads such an entry as “five hundred and three is approximately equal to five hundred” or “four hundred ninety-eight is approximately equal to five hundred”.

Let's take another example:

4 4 71≈4000 4 5 71≈5000

4 3 71≈4000 4 6 71≈5000

4 2 71≈4000 4 7 71≈5000

4 1 71≈4000 4 8 71≈5000

4 0 71≈4000 4 9 71≈5000

In this example, numbers have been rounded to the thousandth place. If we look at the regularity of rounding, we will see that in one case the numbers are rounded down, and in the other - up. After rounding, all other numbers after the thousand place were replaced with zeros.

Rounding rules for numbers:

1) If the digit to be rounded is 0, 1, 2, 3, 4, then the digit of the digit to which the rounding goes does not change, and the rest of the numbers are replaced with zeros.

2) If the digit to be rounded is 5, 6, 7, 8, 9, then the digit of the digit to which the rounding is going to become 1 more, and the rest of the numbers are replaced with zeros.

1) Round to the tens place of 364.

The place of tens in this example is the number 6. After the six there is the number 4. According to the rounding rule, the number 4 does not change the place of tens. We write zero instead of 4. We get:

2) Round to the hundredth place of 4 781.

The place of hundreds in this example is the number 7. After the seven is the number 8, which affects whether the place of hundreds will change or not. According to the rounding rule, the digit 8 increases the hundreds place by 1, and replace the remaining digits with zeros. We get:

3) Round to the thousand places of 215,936.

The thousand place in this example is the number 5. After the five is the number 9, which affects whether the thousand place changes or not. According to the rounding rule, the digit 9 increases the thousand place by 1, and the rest of the digits are replaced with zeros. We get:

21 5 9 36≈21 6 000

4) Round to the tens of thousands of 1,302,894.

The thousand place in this example is the number 0. After the zero is the number 2, which affects whether the tens of thousands place changes or not. According to the rounding rule, the digit 2 does not change the digit of tens of thousands, we replace this digit and all the least significant digits with zero. We get:

13 0 2 894≈13 0 0000

If the exact value of the number is not important, then the value of the number is rounded and you can perform computational operations with approximate values... The result of the calculation is called an estimate of the result of actions.

For example: 598⋅23≈600⋅20≈12000 compare with 598⋅23 = 13754

An estimate of the result of actions is used in order to quickly calculate the answer.

Examples for assignments on the topic of rounding:

Example # 1:
Determine to what digit the rounding is done:
a) 3457987≈3500000 b) 4573426≈4573000 c) 16784≈17000
Let's remember what the digits are on the number 3457987.

7 - the place of ones,

8 - tens place,

9 - hundreds rank,

7 - place of thousands,

5 - tens of thousands,

4 - place of hundreds of thousands,
3 - millions place.
Answer: a) 3 4 57 987≈3 5 00 000 digit of hundreds of thousands b) 4 57 3 426≈4 57 3000 digit of thousands c) 1 6 7 841≈1 7 0 000 digit of tens of thousands.

Example # 2:
Round the number up to 5,999,994 digits: a) tens b) hundreds c) millions.
Answer: a) 5,999 99 4 ≈5,999,990 b) 5,999 9 9 4≈6,000,000 (since the digits of hundreds, thousands, tens of thousands, hundreds of thousands are digit 9, each digit has increased by 1) 5 9 99 994≈6,000,000.

Rounding rules for natural numbers

Rounding rules for natural numbers.
Rounding a number to a certain place.

From time to time, a population census is conducted in the country. Every day people are born, die, change their place of residence, so the number of residents is constantly changing. Let's say that there are 34,489 inhabitants in one city. Accordingly, with the movement of people in this number, the digits of the digits of units, tens and even hundreds will change. These numbers are replaced with zeros, and we get a simpler number. We can say that lives in the city approximately 34,000 inhabitants.

Number 34 489 rounded up to thousand 3 4 000.
If we want to round some number, then we apply the rule:
45 | 245 - the line shows to which digit we want to round.

If the first digit following the digit to which the number is rounded (to the right of the line) is 5, 6, 7, 8, 9, then the last remaining digit is increased by 1, and the rest of the digits after the line are replaced with zeros. In other cases, the last remaining digit is not changed.

The given number and the number obtained by rounding it approximately equal. This is written using the sign " » «.
45 | 245 "45 000, since the digit following the thousand place is 2.
124 7 | 89 "124 800, since the digit following the hundreds place is 8.

Round off the number 12 344; 12 343; 12342; 12 340; 12 341 to tens.
.

Rounding off natural numbers is used when calculating the price. Subtractions are made orally, and an estimate of the result is made. For example:
358 56 = 20 048

For simplified multiplication, let's round each number:
358 "400 and 56" 60 400 x 60 = 24,000

It can be seen that this answer is approximately equal to the first answer.

1. Give examples where you can use rounding of numbers ..
.
.

2. Explain to what place the numbers are rounded. The first column was rounded up to tens. The second column was rounded up to thousands

6789 "6800. 12,897 "10,000.
12 544 "12 500. 2,344,672 "2,340,000.
245 673 "245 700. 78 358 "78 360.
26 577 "30 000. 34 057 123 "34 100 000.

Rounding numbers

Numbers are rounded when full precision is unnecessary or impossible.

Round off the number to a certain digit (sign), then replace it with a number close in value with zeros at the end.

Natural numbers are rounded to tens, hundreds, thousands, etc. The names of numbers in the digits of a natural number can be recalled in the topic natural numbers.

Depending on to which digit the number needs to be rounded off, we replace the digit in the digits of ones, tens, etc. with zeros.

If the number is rounded up to tens, then we replace the digit in the one place with zeros.

If the number is rounded up to hundreds, then the digit zero must be in both the ones and tens places.

The number obtained by rounding off is called the approximate value of this number.

Record the rounding result after the special sign "≈". This sign reads “approximately equal”.

When rounding a natural number to any digit, you must use rounding rules.

Underline the digit of the digit to which the number should be rounded.
Separate all digits to the right of this digit with a vertical bar.
If there is a digit 0, 1, 2, 3 or 4 to the right of the underlined digit, then all digits that are separated to the right are replaced with zeros. The digit of the category to which we rounded off is left unchanged.
If there is a digit 5, 6, 7, 8 or 9 to the right of the underlined digit, then all the digits that are separated to the right are replaced with zeros, and 1 is added to the digit of the digit to which they were rounded.

Let us explain with an example. Let's round 57,861 to thousands. Let's execute the first two points of the rounding rules.

After the underlined digit is the number 8, which means that we add 1 to the digit of the thousand place (we have it 7), and replace all the numbers separated by a vertical line with zeros.

Now let's round 756,485 to hundreds.

Let's round 364 to tens.

3 6 | 4 ≈ 360 - there is 4 in the ones place, so we leave 6 in the tens place unchanged.

On the number axis, the number 364 is enclosed between the two "round" numbers 360 and 370. These two numbers are called approximate values of 364 with an accuracy of tens.

Number 360 - approximate downside value, and the number 370 is an approximate excess value.

In our case, having rounded 364 to tens, we got 360 - an approximate value with a disadvantage.

Rounded results are often written without zeros, adding the abbreviations "thousand" (thousand), "million" (million) and "billion" (billion).

8 659 000 = 8 659 thousand
3,000,000 = 3 million

Rounding is also used to roughly check the answer in calculations.

Before an accurate calculation, let's make an estimate of the answer, rounding the factors to the highest digit.

794 52 ≈ 800 50 ≈ 40,000

We conclude that the answer will be close to 40,000.

794 52 = 41 228

Similarly, you can perform an estimate by rounding and dividing numbers.

In practical human activity, there are two types of numbers: exact and approximate. Often, knowledge of only an approximate number is enough to understand the essence of the matter. Sometimes approximate numbers are used, since the exact number is not required, and sometimes the exact number cannot be found in principle.

Approximate values

Sometimes in calculations it is not necessary to use exact numeric values... To speed up or simplify calculations, it is often sufficient to obtain approximate result... To do this, round off the numbers that are involved in the calculations, as well as the final result of the calculations. Approximate values are used when the exact value of something cannot be found, or this value is not important for the object under study.

For example, we can say that it takes half an hour to get home. This is an approximate value, because to say exactly how long it will take to get home is either too difficult or in most cases not so important. The main thing is to indicate the order of the numbers, and this is quite enough.

In mathematics, approximate values are indicated using a special sign.

\ [\ LARGE \ approx \]

Rounding is used to indicate the approximate value of something.

Rounding numbers

The essence of rounding is to find the closest value from the original. At the same time, the number can be rounded up to a certain digit - to the tens, hundreds, thousands.

First rounding rule:

less 5 (0, 1, 2, 3, 4), then the last of the left digits remains unchanged (amplification or increase is not performed).

The number 47.271 is rounded off as 47.3. In this case, the number 2 will be amplified to 3, since the first cut-off number 7 is greater than 5.

Second rounding rule:

If, when rounding numbers, the first of the separated digits more 5 (5, 6, 7, 8, 9), then the last of the remaining digits is increased by one (amplification is made).

The number 64.28 is rounded off as 64. 64 is closest to the number to be rounded than 65.

Third rounding rule:

If the digit 5 is cut off, and there are no significant digits behind it, then rounding is performed to the nearest even number, in other words, the last digit left remains unchanged if it is even, and is amplified if it is odd.

The number 0.0465 is rounded off as - 0.046. In this case, no amplification is done, since the last digit 6 left is even. The number 0.935 is rounded off as - 0.94. The last digit 3 to be left is amplified as it is odd.

How to round a number to an integer

The rule for rounding a number to an integer

To round a number to an integer (or round a number to one), you need to drop the comma and all numbers after the comma.

If the first of the discarded digits is 0, 1, 2, 3 or 4, then the number will not change.

If the first of the discarded digits is 5, 6, 7, 8, or 9, the previous digit must be increased by one.

Examples of rounding a number to an integer:

\ [86, \ underline 2 4 \ approx 86 \]
To round a number to an integer, discard the comma and all numbers after it. Since the first discarded digit is 2, we do not change the previous digit. They read: "eighty six point twenty four hundredths is approximately equal to eighty six points."

\ [274, \ underline 8 39 \ approx 275 \]
Rounding off the number to the nearest integer, discard the comma and all following numbers. Since the first of the discarded digits is 8, we increase the previous one by one. They read: "Two hundred seventy-four point eight hundred thirty-nine thousandths is approximately equal to two hundred seventy-five points."

\ [0, \ underline 5 2 \ approx 1 \]
When rounding a number to an integer, discard all the numbers behind it. Since the first of the discarded digits is 5, we increase the previous one by one. They read: "Zero point fifty-two hundredths is approximately equal to one whole."

\ [0, \ underline 3 97 \ approx 0 \]
We discard the comma and all the numbers after it. The first of the discarded digits is 3, so we do not change the previous digit. They read: "Zero point three hundred ninety-seven thousandths is approximately equal to zero points."

\ [39, \ underline 7 04 \ approx 40 \]
The first of the discarded digits is 7, which means that the digit in front of it is increased by one. They read: "Thirty-nine point seven hundred and four thousandths is approximately equal to forty points." And a couple more examples for rounding a number to integers:

How to round to tenths

The rule for rounding numbers to tenths.

To round a decimal fraction to tenths, you need to leave only one digit after the decimal point, and discard all the other digits following it.

Examples of rounding to tenths:

\ [23.7 \ underline 5 \ approx 23.8 \]
To round the number to tenths, leave the first digit after the decimal point, and discard the rest. Since the first discarded digit is 5, we increase the previous digit by one. They read: "Twenty-three point seventy-five hundredths is approximately equal to twenty-three point eight tenths."

\ [348.3 \ underline 1 \ approx 348.3 \]
To round this number to tenths, leave only the first digit after the decimal point, discard the rest. The first discarded digit is 1, so we don't change the previous digit. They read: "Three hundred forty-eight point thirty-one hundredth is approximately equal to three hundred forty-one point three."

\ [49.9 \ underline 6 2 \ approx 50.0 \]
Rounding to tenths, leave one digit after the decimal point, and discard the rest. The first of the discarded digits is 6, which means that we increase the previous one by one. They read: "Forty-nine points, nine hundred sixty-two thousandths is approximately equal to fifty points, zero tenths."

\ [7.0 \ underline 2 8 \ approx 7.0 \]
We round to tenths, so after the decimal point we leave only the first of the numbers, and discard the rest. The first of the discarded digits is 4, which means we leave the previous digit unchanged. They read: "Seven point twenty eight thousandths is approximately equal to seven point zero tenths."

\ [56.8 \ underline 7 06 \ approx 56.9 \]
To round this number to tenths, leave one digit after the decimal point, and discard all following it. Since the first discarded digit is 7, therefore, we add one to the previous one. They read: "Fifty six point eight thousand seven hundred six ten thousandth is approximately equal to fifty six point nine tenths."

How to round a number to the nearest hundredth

The rule for rounding a number to hundredths

To round a number to hundredths, leave two digits after the decimal point, and discard the rest.

If the first of the discarded digits is 0, 1, 2, 3 or 4, then the previous digit is not changed.

If the first of the discarded digits is 5, 6, 7, 8 or 9, then we increase the previous digit by one.

An example of rounding a number to hundredths:

\ [32.78 \ underline 6 \ approx 32.79 \]
To round the number to hundredths, leave two digits after the decimal point, and discard the next digit. Since this figure is 9, we increase the previous figure by one. They read: "Thirty-two point seven hundred eighty-six thousandths is approximately equal to thirty-two point seventy-nine hundredths."

\ [6.96 \ underline 1 \ approx 6.96 \]
Rounding this number to hundredths, we leave two digits after the decimal point, and discard the third. Since the discarded digit is 1, we leave the previous digit unchanged. They read: "Six point nine hundred sixty one thousandth is approximately equal to six point ninety six hundredths."

\ [17.48 \ underline 3 9 \ approx 17.48 \]
When rounding to hundredths, leave two digits after the decimal point, discard the rest. The first of the discarded digits is 3, so we do not change the previous digit. They read: "Seventeen point four thousand thirty nine ten thousandth is approximately equal to seventeen point forty eight hundredths."

\ [0.12 \ underline 5 4 \ approx 0.13 \]
To round this number to hundredths, we will leave only two digits after the decimal point, and discard the rest. The first of the discarded digits is 5, so we increase the previous digit by one. They read: "Zero point one thousand two hundred fifty-four thousandths is approximately equal to zero point thirteen hundredths."

\ [549.30 \ underline 7 3 \ approx 549.31 \]
When rounding a number to hundredths, leave two digits after the decimal point, discard the rest. Since the first of the discarded digits is 7, we increase the previous digit by one. We read: "Five hundred and forty-nine points, three thousand seventy-three ten-thousandths is approximately equal to five hundred and forty-nine points, thirty-one hundredth."

How to round a number to thousandths

Rounding rule to thousandths

To round a decimal fraction to thousandths, you need to leave only three digits after the decimal point, and discard the remaining digits following it.

If the first of the discarded digits is 0, 1, 2, 3 or 4, then the previous digit is not changed.

If the first of the discarded digits is 5, 6, 7, 8 or 9, then we increase the previous digit by one.

An example of rounding a number to thousandths:

\ [3.785 \ underline 4 \ approx 3.785 \]
To round a number to thousandths, you only need to leave three digits after the decimal point, and discard the fourth. Since the discarded digit is 4, we leave the previous digit unchanged. They read: "Three whole, seven thousand eight hundred fifty four ten thousandths is approximately equal to three whole, seven hundred eighty five thousandths."

\ [37.207 \ underline 6 \ approx 37.208 \]
To round this number to thousandths, leave three digits after the decimal point, and discard the fourth. The discarded digit is 6, which means we increase the previous digit by one. They read: "Thirty-seven point two thousand seventy-six ten-thousandths is approximately equal to thirty-seven point two hundred and eight thousandths."

\ [69,999 \ underline 8 1 \ approx 70,000 \]
Rounding the number to thousandths, we leave three digits after the decimal point, and discard all the rest. Since the first of the discarded digits is 8, we add one to the previous one. They read: "Sixty-nine point ninety-nine thousand nine hundred and eighty-one hundred thousandth is approximately equal to seventy point zero thousandths."

\ [863,124 \ underline 2 3 \ approx 863,124 \]
We round the number to thousandths, so we leave the first three digits after the decimal point, and discard the following ones. Since the first of the discarded digits is 2, we do not change the previous digit. They read: "Eight hundred sixty three point twelve thousand four hundred twenty three hundred thousandth is approximately equal to eight hundred sixty three point one hundred twenty four thousandths."

\ [0.003 \ underline 5 9 \ approx 0.004 \]
To round this number to thousandths, we leave the first three digits after the decimal point, and discard all the rest. The first of the discarded digits is 5, which means that the previous digit should be increased by one. They read: "Zero point three hundred and fifty-nine hundred thousandths is approximately equal to zero point four thousandths."

How to round a number to tens

The rule for rounding numbers to tens

To round a number to tens, you need to replace the digit in the ones place with zero, and if there are digits after the decimal point in the number record, then they should be discarded.

If the first of the discarded digits is 0, 1, 2, 3 or 4, then the previous digit is not changed.

If the first of the discarded digits is 5, 6, 7, 8 or 9, then we increase the previous digit by one.

Examples of rounding a number to tens:

\ [58 \ underline 3 \ approx 580 \]
To round the number to tens, replace the digit in the ones place (that is, the last digit in the record of a natural number) with zero. Since this figure is 3, we do not change the previous figure. They read: "Five hundred and eighty-three is approximately equal to five hundred and eighty."

\ [103 \ underline 7 \ approx 1040 \]
We round up to tens, so the number in the ones place is replaced by zero. Since this figure is 7, we increase the previous one by one. They read: "One thousand thirty-seven is approximately equal to one thousand and forty."

\ [35 \ underline 2, 78 \ approx 350 \]
Rounding off the decimal fraction to tens, replace the digit in the ones place (that is, the last digit before the comma) with zero, and discard the comma and all the digits after it. The digit replaced by zero is 2, which means that the previous digit does not need to be changed. They read: "Three hundred and fifty-two point seventy-eight hundredths are approximately equal to three hundred and fifty."

\ [247 \ underline 6.05 \ approx 2480 \]
To round this decimal fraction to tens, replace the digit in the ones place with zero, and discard the digits after the decimal point. Since the digit replaced by zero is 6, add one to the previous digit. They read: "Two thousand four hundred seventy six point five hundredths is approximately equal to two thousand four hundred and eighty."

\ [79 \ underline 9, 1 \ approx 800 \]
Rounding the decimal fraction to tens, replace the digit with zero in the ones place, and discard the comma and everything after the comma. Since 9 was replaced by zero, we increase the previous digit by one. They read: "Seven hundred and ninety-nine points, one tenth is approximately equal to eight hundred."

How to round a number to hundreds

Rounding Rule to Hundreds

To round a number to hundreds, you need to replace the digits in the ones and tens place with zeros. When rounding to hundreds of a decimal fraction, the comma and all the digits after it are discarded.

If the first of the discarded digits is 0, 1, 2, 3 or 4, then the previous digit is not changed.

If the first of the discarded digits is 5, 6, 7, 8 or 9, then we increase the previous digit by one.

Examples of rounding a number to hundreds:

\ [23 \ underline 1 7 \ approx 2300 \]
To round this number to hundreds, we replace the digits in the ones and tens (that is, the last two digits in the record) with zeros. Since the first digit replaced by zero is 1, we do not change the previous digit. They read: "Two thousand three hundred seventeen is approximately equal to two thousand three hundred."

\ [45 \ underline 8 1 \ approx 4600 \]
Rounding this number to hundreds, replace the last two digits in its entry with zeros. Since the first digit replaced by zero is 8, we increase the previous digit by one. They read: "Four thousand five hundred and eighty-one is approximately equal to four thousand six hundred."

\ [785 \ underline 0 9 \ approx 78500 \]
We round the number to hundreds, which means that the last two digits in the number record - tens and ones - are replaced with zeros. The first of the digits replaced by zero is equal to zero, so we rewrite the previous one without changes. They read: "Seventy-eight thousand five hundred and nine is approximately equal to seventy-eight thousand five hundred."

\ [939 \ underline 5 2 \ approx 94000 \]
To round this number to hundreds, we replace the digits in the digits of tens and units with zeros. Since the first of the digits replaced by zero is 9, we increase the previous one by one. They read: "Ninety-three thousand nine hundred and fifty-two is approximately equal to ninety-four thousand."

\ [14 \ underline 7 3.12 \ approx 1500 \]
To round the decimal fraction to hundreds, the comma and all the digits after the decimal point must be discarded, and the last two digits of the integer part (ones and tens) must be replaced with zeros. The first digit replaced by zero is 7, so we add one to the previous digit. They read: "One thousand four hundred seventy three point twelve hundredths is approximately equal to one thousand five hundred."

How to round a number to the nearest thousand

Rounding Rule to Thousands

To round a number to thousands, you need to replace the digits in the digits of hundreds, tens and ones with zeros. When rounding to thousands of decimal fractions, the comma and all the numbers after it must be discarded.

If the first of the discarded digits is 0, 1, 2, 3 or 4, then the previous digit is not changed.

If the first of the discarded digits is 5, 6, 7, 8 or 9, then we increase the previous digit by one.

Examples of rounding a number to thousands:

\ [82 \ underline 3 71 \ approx 82000 \]
To round this number to thousands, you need to replace the digits in the digits of hundreds, tens and ones with zeros (thousands have three zeros at the end of the record, the same number of zeros at the end of the number should be obtained when rounding to thousands). Since the first of the digits, which we replaced with zero, is equal to 3, we leave the previous digit unchanged. They read: "Eighty-two thousand three hundred seventy-one is approximately equal to eighty-two thousand."

\ [40 \ underline 6 28 \ approx 41000 \]
When rounding to thousands, the last three digits - in the digits of hundreds, tens and ones - are replaced with zeros. Since the first of the digits replaced by zero is 6, we increase the previous digit by one. They read: "Forty thousand six hundred twenty eight is approximately equal to forty one thousand."

\ [159 \ underline 7 32 \ approx 160000 \]
Rounding the given number to thousands, we replace the digits in the digits of hundreds, tens and ones with zeros. The first digit replaced by zero is 7, so we add one to the previous digit. They read: "One hundred fifty-nine thousand seven hundred thirty-two is approximately equal to one hundred and sixty thousand."

\ [238 \ underline 1 97 \ approx 238000 \]
We round the number to thousands, so we replace the digits in the digits of hundreds, tens and ones with zeros. Since the first of the digits, which we replaced with zero, is equal to 1, we rewrite the previous digit without changes. They read: "Two hundred thirty-eight thousand one hundred ninety-seven is approximately equal to two hundred and thirty-eight thousand."

\ [457 \ underline 2 49.83 \ approx 457000 \]
To round the decimal fraction to thousands, we discard the comma and all the digits after the decimal point, and replace the digits in the digits of hundreds, tens and ones with zeros. Since the first of the digits replaced by zero is 2, we do not change the previous digit. They read: "Four hundred and fifty-seven thousand two hundred and forty-nine points, eighty-three hundredths are approximately equal to four hundred and fifty thousand."

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