There are several ways to round numbers in Excel. Using cell format and using functions. These two methods should be distinguished as follows: the first is only for displaying values ​​or printing, and the second method is also for calculations and calculations.

Using the functions, it is possible to accurately round up or down to a user-specified digit. And the values ​​obtained as a result of calculations can be used in other formulas and functions. At the same time, rounding using cell format will not give desired result, and the results of calculations with such values ​​will be erroneous. After all, the format of the cells, in fact, does not change the value, only its display method changes. To quickly and easily understand this and avoid making mistakes, we will give a few examples.

How to round a number using cell format

Let's enter the value 76.575 in cell A1. Right-click to bring up the “Format Cells” menu. You can do the same using the “Number” tool on the main page of the Book. Or press the hotkey combination CTRL+1.

Select the number format and set the number of decimal places to 0.

Rounding result:

You can assign the number of decimal places in “monetary”, “financial”, “percentage” formats.

As you can see, rounding occurs according to mathematical laws. The last digit to be stored is increased by one if it is followed by a digit greater than or equal to "5".

Peculiarity this option: how more numbers we leave after the comma, the more accurate the result will be.



How to properly round a number in Excel

Using the ROUND() function (rounds to the number of decimal places required by the user). To call the “Function Wizard” we use the fx button. The function you need is in the “Mathematical” category.

Arguments:

1. “Number” - a link to a cell with the desired value(A1).
2. “Number of digits” - the number of decimal places to which the number will be rounded (0 – to round to a whole number, 1 – one decimal place will be left, 2 – two, etc.).

Now let's round the whole number (not a decimal). Let's use the ROUND function:

• the first argument of the function is a cell reference;
• the second argument is with the “-” sign (up to tens – “-1”, up to hundreds – “-2”, to round the number to thousands – “-3”, etc.).

How to round a number to thousands in Excel?

An example of rounding a number to thousands:

Formula: =ROUND(A3,-3).

You can round not only a number, but also the value of an expression.

Let's say there is data on the price and quantity of a product. It is necessary to find the cost accurate to the nearest ruble (rounded to the nearest whole number).

The first argument of the function is numeric expression to find the cost.

How to round up and down in Excel

To round to big side– “ROUND UP” function.

We fill in the first argument according to the already familiar principle - a link to a cell with data.

Second argument: "0" - rounding decimal to the whole part, “1” - the function rounds, leaving one decimal place, etc.

Formula: =ROUNDUP(A1;0).

Result:

To round down in Excel, use the ROUNDDOWN function.

Example formula: =ROUNDBOTTOM(A1,1).

Result:

The “ROUND UP” and “ROUND DOWN” formulas are used to round the values ​​of expressions (product, sum, difference, etc.).

How to round to a whole number in Excel?

To round up to a whole number, use the “ROUND UP” function. To round down to a whole number, use the “ROUND DOWN” function. The “ROUND” function and cell format also allow you to round to a whole number by setting the number of digits to “0” (see above).

IN Excel program For rounding to a whole number, the “ROLL” function is also used. It simply discards the decimal places. Essentially, no rounding occurs. The formula cuts off the numbers to the designated digit.

Compare:

The second argument is “0” - the function cuts to an integer; “1” - up to a tenth; “2” - up to a hundredth, etc.

A special Excel function that will return only an integer is “INTEGER”. It has a single argument – ​​“Number”. You can specify a numeric value or a cell reference.

The disadvantage of using the "INTEGER" function is that it only rounds down.

You can round to the nearest integer in Excel using the “OKRUP” and “OKRVDOWN” functions. Rounding occurs up or down to the nearest whole number.

Example of using functions:

The second argument is an indication of the digit to which rounding should occur (10 to tens, 100 to hundreds, etc.).

Rounding to the nearest even integer is performed by the “EVEN” function, rounding to the nearest odd integer is performed by the “ODD” function.

An example of their use:

Why does Excel round large numbers?

If large numbers are entered into spreadsheet cells (for example, 78568435923100756), Excel automatically rounds them like this by default: 7.85684E+16 is a feature of the “General” cell format. To avoid such display of large numbers, you need to change the format of the cell with the data a large number on "Numerical" (the most quick way press the hotkey combination CTRL+SHIFT+1). Then the cell value will be displayed like this: 78,568,435,923,100,756.00. If desired, the number of digits can be reduced: “Home” - “Number” - “Reduce digits”.

In some cases, the exact number when dividing a certain amount by a specific number cannot be determined in principle. For example, when dividing 10 by 3, we get 3.3333333333.....3, that is, this number cannot be used to count specific items in other situations. Then this number should be reduced to a certain digit, for example, to an integer or to a number with a decimal place. If we reduce 3.3333333333…..3 to an integer, we get 3, and if we reduce 3.3333333333…..3 to a number with a decimal place, we get 3.3.

Rounding rules

What is rounding? This is discarding a few digits that are the last in the series of an exact number. So, following our example, we discarded all the last digits to get the integer (3) and discarded the digits, leaving only the tens places (3,3). The number can be rounded to hundredths and thousandths, ten thousandths and other numbers. It all depends on how accurate the number needs to be. For example, in the manufacture medical supplies, the amount of each of the ingredients of the medicine is taken with the greatest accuracy, since even a thousandth of a gram can lead to fatal outcome. If it is necessary to calculate the progress of students at school, then most often a number with a decimal or hundredth place is used.

Let's look at another example where rounding rules apply. For example, there is a number 3.583333 that needs to be rounded to thousandths - after rounding, we should be left with three digits after the decimal point, that is, the result will be the number 3.583. If we round this number to tenths, then we get not 3.5, but 3.6, since after “5” there is the number “8”, which is already equal to “10” during rounding. Thus, following the rules of rounding numbers, you need to know that if the digits are greater than "5", then the last digit to be stored will be increased by 1. If there is a digit less than "5", the last digit to be stored remains unchanged. These rules for rounding numbers apply regardless of whether to a whole number or to tens, hundredths, etc. you need to round the number.

In most cases, when you need to round a number in which the last digit is “5,” this process is not performed correctly. But there is also a rounding rule that applies specifically to such cases. Let's look at an example. It is necessary to round the number 3.25 to the nearest tenth. Applying the rules for rounding numbers, we get the result 3.2. That is, if there is no digit after “five” or there is a zero, then the last digit remains unchanged, but only if it is even - in our case, “2” is an even digit. If we were to round 3.35, the result would be 3.4. Because, in accordance with the rules of rounding, if there is an odd digit before the “5” that must be removed, the odd digit is increased by 1. But only on the condition that there are no significant digits after the “5”. In many cases, simplified rules can be applied, according to which, if the last stored digit is followed by digits from 0 to 4, the stored digit does not change. If there are other digits, the last digit is increased by 1.

This CMEA standard establishes the rules for recording and rounding numbers expressed in the decimal number system.

The rules for recording and rounding numbers established in this CMEA standard are intended for use in regulatory, technical, design and technological documentation.

This CMEA standard does not apply to special rounding rules established in other CMEA standards.

1. RULES FOR RECORDING NUMBERS

1.1. Significant figures given number- these are all the digits from the first one on the left, which is not equal to zero, to the last recorded digit on the right. In this case, the zeros resulting from the factor 10 n are not taken into account.

	1. Number 12.0	has three significant figures;
	2. Number 30	has two significant figures;
	3. Number 120 10 3	has three significant figures;
	4. Number 0.514 10	has three significant figures;
	5. Number 0.0056	has two significant figures.

1.2. When it is necessary to indicate that a number is exact, the word "exact" must be written after the number or the last significant digit must be printed in bold.

Example. In printed text:

1 kWh = 3,600,000 J (exact), or = 3,600,000 J

1.3. Records of approximate numbers should be distinguished by the number of significant digits.

Examples:

1. It is necessary to distinguish between the numbers 2.4 and 2.40. The entry 2,4 means that only the whole and tenth digits are correct; true meaning the numbers can be for example 2.43 and 2.38. Writing 2.40 means that hundredths of the number are also correct; the true number may be 2.403 and 2.398, but not 2.421 or 2.382.

2. The entry 382 means that all numbers are correct; if you cannot vouch for the last digit, then the number should be written 3.8·10 2.

3. If in the number 4720 only the first two digits are correct, it should be written 47·10 2 or 4.7·10 3.

1.4. The number for which the permissible deviation is indicated must have the last significant digit of the same digit as the last significant digit of the deviation.

Examples:

1.5. It is advisable to write down the numerical values ​​of a quantity and its error (deviation) indicating the same unit of physical quantities.

Example. 80.555±0.002 kg

1.6. The intervals between numerical values ​​of quantities should be written down:

From 60 to 100 or from 60 to 100

Over 100 to 120 or over 100 to 120

Over 120 to 150 or over 120 to 150.

1.7. Numerical values ​​of quantities must be indicated in standards with the same number of digits, which is necessary to ensure the required performance properties and product quality. The recording of numerical values ​​of quantities up to the first, second, third, etc. decimal place for different standard sizes, types of product brands of the same name, as a rule, should be the same. For example, if the thickness gradation of a hot-rolled steel strip is 0.25 mm, then the entire range of strip thicknesses must be indicated accurate to the second decimal place.

Depending on the technical characteristics and purpose of the product, the number of decimal places of numerical values ​​of the same parameter, size, indicator or norm may have several stages (groups) and should be the same only within this stage (group).

2. ROUNDING RULES

2.1. Rounding a number is the removal of significant digits on the right to a certain digit with a possible change in the digit of this digit.

Example. Rounding 132.48 to four significant figures becomes 132.5.

2.2. If the first of the discarded digits (counting from left to right) is less than 5, then the last saved digit does not change.

Example. Rounding 12.23 to three significant figures gives 12.2.

2.3. If the first of the discarded digits (counting from left to right) is 5, then the last retained digit is increased by one.

Example. Rounding the number 0.145 to two significant figures gives 0.15.

Note. In cases where the results of previous rounding must be taken into account, proceed as follows:

1) if the discarded digit was obtained as a result of the previous rounding up, then the last saved digit is retained;

Example. Rounding to one significant digit the number 0.15 (resulting from rounding the number 0.149) gives 0.1.

2) if the discarded digit was obtained as a result of the previous rounding down, then the last remaining digit is increased by one (with a transition to the next digits, if necessary).

Example. Rounding the number 0.25 (resulting from the previous rounding of the number 0.252) gives 0.3.

2.4. If the first of the discarded digits (counting from left to right) is greater than 5, then the last retained digit is increased by one.

Example. Rounding the number 0.156 to two significant figures gives 0.16.

2.5. Rounding should be done immediately to the desired number of significant figures, rather than in stages.

Example. Rounding the number 565.46 to three significant figures is done directly by 565. Rounding by stages would result in:

565.46 in stage I - to 565.5,

and in stage II - 566 (wrong).

2.6. Whole numbers are rounded according to the same rules as fractions.

Example. Rounding 12,456 to two significant figures gives 12·10 3 .

Topic 01.693.04-75.

3. The CMEA standard was approved at the 41st meeting of the PCC.

4. Dates for the start of application of the CMEA standard:

CMEA member countries	Deadline for the start of application of the CMEA standard in contractual legal relations on economic, scientific and technical cooperation	The start date for the application of the CMEA standard is national economy
NRB	December 1979	December 1979
VNR	December 1978	December 1978
GDR	December 1978	December 1978
Republic of Cuba
MPR
Poland
SRR
USSR	December 1979	December 1979
Czechoslovakia	December 1978	December 1978

5. The date of the first inspection is 1981, the frequency of inspection is 5 years.

Introduction........................................................ ........................................................ ..........
TASK No. 1. Series of preferred numbers.................................................... ....
TASK No. 2. Rounding measurement results..................................................
TASK No. 3. Processing of measurement results..................................................
TASK No. 4. Tolerances and fits of smooth cylindrical joints...
TASK No. 5. Tolerances of shape and location.................................................... .
TASK No. 6. Surface roughness.................................................. .....
TASK No. 7. Dimensional chains.................................................... ............................
Bibliography................................................ ............................................

Task No. 1. Rounding measurement results

When performing measurements, it is important to follow certain rules for rounding and recording their results in technical documentation, since if these rules are not followed, significant errors in the interpretation of measurement results are possible.

Rules for writing numbers

1. The significant digits of a given number are all digits from the first on the left, which is not equal to zero, to the last on the right. In this case, the zeros resulting from the multiplier of 10 are not taken into account.

Examples.

a) Number 12,0has three significant figures.

b) Number 30has two significant figures.

c) Number 12010 8 has three significant figures.

G) 0,51410 -3 has three significant figures.

d) 0,0056has two significant figures.

2. If it is necessary to indicate that a number is exact, the word “exactly” is indicated after the number or the last significant digit is printed in bold. For example: 1 kW/h = 3600 J (exactly) or 1 kW/h = 360 0 J .

3. Records of approximate numbers are distinguished by the number of significant digits. For example, there are numbers 2.4 and 2.40. Writing 2.4 means that only whole and tenths are correct; the true value of the number could be, for example, 2.43 and 2.38. Writing 2.40 means that hundredths are also true: the true value of the number can be 2.403 and 2.398, but not 2.41 and not 2.382. Writing 382 means that all the numbers are correct: if you cannot vouch for the last digit, then the number should be written 3.810 2. If only the first two digits of the number 4720 are correct, it should be written as: 4710 2 or 4.710 3.

4. The number for which the permissible deviation is indicated must have the last significant digit of the same digit as the last significant digit of the deviation.

Examples.

a) Correct: 17,0 + 0,2. Wrong: 17 + 0,2or 17,00 + 0,2.

b) Correct: 12,13+ 0,17. Wrong: 12,13+ 0,2.

c) Correct: 46,40+ 0,15. Wrong: 46,4+ 0,15or 46,402+ 0,15.

5. It is advisable to write down the numerical values ​​of a quantity and its error (deviation) indicating the same unit of quantity. For example: (80.555 + 0.002) kg.

6. It is sometimes advisable to write the intervals between numerical values ​​of quantities in text form, then the preposition “from” means “”, the preposition “to” – “”, the preposition “over” – “>”, the preposition “less” – “<":

"d takes values ​​from 60 to 100" means "60 d100",

"d takes values ​​greater than 120 less than 150" means "120<d< 150",

"d takes values ​​over 30 to 50" means "30<d50".

Rules for rounding numbers

1. Rounding a number is the removal of significant digits to the right to a certain digit with a possible change in the digit of this digit.

2. If the first of the discarded digits (counting from left to right) is less than 5, then the last saved digit is not changed.

Example: Rounding a number 12,23gives up to three significant figures 12,2.

3. If the first of the discarded digits (counting from left to right) is equal to 5, then the last saved digit is increased by one.

Example: Rounding a number 0,145gives up to two digits 0,15.

Note . In cases where the results of previous rounding should be taken into account, proceed as follows.

4. If the discarded digit is obtained as a result of rounding down, then the last remaining digit is increased by one (with a transition to the next digits, if necessary), otherwise - vice versa. This applies to both fractions and integers.

Example: Rounding a number 0,25(obtained as a result of the previous rounding of the number 0,252) gives 0,3.

4. If the first of the discarded digits (counting from left to right) is more than 5, then the last saved digit is increased by one.

Example: Rounding a number 0,156gives to two significant figures 0,16.

5. Rounding is performed immediately to the desired number of significant figures, and not in stages.

Example: Rounding a number 565,46gives up to three significant figures 565.

6. Whole numbers are rounded according to the same rules as fractions.

Example: Rounding a number 23456gives to two significant figures 2310 3

The numerical value of the measurement result must end with a digit of the same digit as the error value.

Example:Number 235,732 + 0,15should be rounded to 235,73 + 0,15, but not until 235,7 + 0,15.

7. If the first of the discarded digits (counting from left to right) is less than five, then the remaining digits do not change.

Example: 442,749+ 0,4rounded up to 442,7+ 0,4.

8. If the first digit to be discarded is greater than or equal to five, then the last digit to be retained is increased by one.

Example: 37,268 + 0,5rounded up to 37,3 + 0,5; 37,253 + 0,5 must be roundedbefore 37,3 + 0,5.

9. Rounding should be done immediately to the desired number of significant figures; rounding incrementally may lead to errors.

Example: Step by step rounding of a measurement result 220,46+ 4gives at the first stage 220,5+ 4and on the second 221+ 4, while the correct rounding result is 220+ 4.

10. If the error of a measuring instrument is indicated with only one or two significant digits, and the calculated error value is obtained with a large number of digits, only the first one or two significant digits should be left in the final value of the calculated error, respectively. Moreover, if the resulting number begins with the digits 1 or 2, then discarding the second character leads to a very large error (up to 3050%), which is unacceptable. If the resulting number begins with the number 3 or more, for example, with the number 9, then preserving the second character, i.e. indicating an error, for example, 0.94 instead of 0.9, is misinformation, since the original data does not provide such accuracy.

Based on this, the following rule has been established in practice: if the resulting number begins with a significant digit equal to or greater than 3, then only one is retained in it; if it begins with significant figures less than 3, i.e. from numbers 1 and 2, then two significant figures are stored in it. In accordance with this rule, the standardized values ​​of errors of measuring instruments are established: two significant figures are indicated in the numbers 1.5 and 2.5%, but in numbers 0.5; 4; 6% only one significant figure is indicated.

Example:On an accuracy class voltmeter 2,5with measurement limit x TO = 300 In a reading of the measured voltage x = 267,5Q. In what form should the measurement result be recorded in the report?

It is more convenient to calculate the error in the following order: first you need to find the absolute error, and then the relative one. Absolute error  X =  0 X TO/100, for the reduced voltmeter error  0 = 2.5% and the measurement limits (measurement range) of the device X TO= 300 V:  X= 2.5300/100 = 7.5 V ~ 8 V; relative error  =  X100/X = 7,5100/267,5 = 2,81 % ~ 2,8 % .

Since the first significant digit of the absolute error value (7.5 V) is greater than three, this value should be rounded according to the usual rounding rules to 8 V, but in the relative error value (2.81%) the first significant digit is less than 3, so here two decimal places must be retained in the answer and  = 2.8% must be indicated. Received value X= 267.5 V must be rounded to the same decimal place as the rounded absolute error value, i.e. up to whole units of volts.

Thus, the final answer should state: “The measurement was made with a relative error of = 2.8%. The measured voltage X= (268+ 8) B".

In this case, it is more clear to indicate the limits of the uncertainty interval of the measured value in the form X= (260276) V or 260 VX276 V.

Today we will look at 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 the value is not important to the item being examined.

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

Another example. Classes start at nine in the morning. We left the house at 8:30. After some time on the road, we met a 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 don’t know what time it is, so we answer our friend: “now approximately about nine o'clock."

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

Read as "approximately equal."

To indicate the approximate value of something, they resort to such an operation as rounding numbers.

Rounding numbers

To find an approximate value, an operation such as rounding numbers.

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

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

Any number can be made round. The procedure by which a number is made round is called rounding the number.

We have already dealt with “rounding” numbers when we divided large numbers. Let us 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 make division easier. A kind of life hack. In fact, this was not even a rounding of numbers. That is why at the beginning of this paragraph we put the word rounding in quotation marks.

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

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

Without getting ahead of ourselves, let’s try to understand what “round to the tens place” means. When they say to round the number 17, we are required to find the nearest round number for the number 17. Moreover, during this search, changes may also affect the number that is in the tens place in the number 17 (i.e., ones).

Let's imagine that all numbers from 10 to 20 lie on a straight line:

The figure shows that for the number 17 the nearest round number is 20. So the answer to the problem will be like this: 17 is approximately equal to 20

17 ≈ 20

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

Let's try to find an approximate number for the number 12. To do this, imagine again that all numbers from 10 to 20 lie on a straight line:

The figure shows that the nearest round number for 12 is the number 10. So the answer to the problem will be like this: 12 is approximately equal to 10

12 ≈ 10

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

Let's try to find the closest number for the number 15. Let's imagine again that all numbers from 10 to 20 lie on a straight line:

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 the 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 approximation for 15 is 20

15 ≈ 20

Large numbers can also be rounded. Naturally, it is not possible for them to draw a straight line and depict numbers. There is a way for them. For example, let's round the number 1456 to the tens place.

We must round 1456 to the tens place. The tens place begins at five:

Now we temporarily forget about the existence of the first numbers 1 and 4. The number remaining is 56

Now we look at which round number is closer to the number 56. Obviously, the closest round number for 56 is the number 60. So we replace the number 56 with the number 60

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

1456 ≈ 1460

It can be seen that after rounding the number 1456 to the tens place, the changes affected the tens place itself. The new number obtained now has a 6 in the tens place, not a 5.

You can round numbers not only to the tens place. You can also round to the hundreds, thousands, or tens of thousands place.

Once it becomes clear that rounding is nothing more than searching for the nearest number, you can apply ready-made rules that make rounding numbers much easier.

First rounding rule

From the previous examples it became clear that when rounding a number to a certain digit, the low-order digits are replaced by zeros. Numbers that are replaced by zeros are called discarded digits.

The first rounding rule is as follows:

If, when rounding numbers, the first digit to be discarded is 0, 1, 2, 3 or 4, then the retained digit remains unchanged.

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

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

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

Now we find the first of the discarded digits. The first digit to be discarded is the digit that comes after the digit to be stored. We see that the first digit after the two is the number 3. This means the number 3 is first digit to be discarded.

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

That's what we do. We leave the stored digit unchanged, and replace all low-order 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 rounding the number 123 to the tens place, we get the number 120 approximating it.

Now let's try to round the same number 123, but to hundreds place.

We need to round the number 123 to the hundreds place. Again we are looking for the number to be saved. This time the digit being stored is 1 because we are rounding the number to the hundreds place.

Now we find the first of the discarded digits. The first digit to be discarded is the digit that comes after the digit to be stored. We see that the first digit after one is the number 2. This means that the number 2 is first digit to be discarded:

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

That's what we do. We leave the stored digit unchanged, and replace all low-order digits with zeros. In other words, we replace everything that follows the number 1 with zeros:

123 ≈ 100

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

Example 3. Round 1234 to the tens place.

Here the retained digit is 3. And the first discarded digit is 4.

This means we leave the saved number 3 unchanged, and replace everything that is located after it with zero:

1234 ≈ 1230

Example 4. Round 1234 to the hundreds place.

Here, the retained digit is 2. And the first discarded digit 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 retained digit remains unchanged.

This means we leave the saved number 2 unchanged, and replace everything that is located after it with zeros:

1234 ≈ 1200

Example 3. Round 1234 to the thousands place.

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

This means we leave the saved digit 1 unchanged, and replace everything that is located after it with zeros:

1234 ≈ 1000

Second rounding rule

The second rounding rule is as follows:

When rounding numbers, if the first digit to be discarded is 5, 6, 7, 8, or 9, then the retained digit is increased by one.

For example, let's round the number 675 to the tens place.

First of all, we find the digit to be stored. To do this, you need to read the task itself. The digit being stored is located in the digit referred to in the task. The assignment says: round the number 675 to tens place.

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

Now we find the first of the discarded digits. The first digit to be discarded is the digit that comes after the digit to be stored. We see that the first digit after seven is the number 5. This means that the number 5 is first digit to be discarded.

Our first discarded digit is 5. This means we must increase the retained digit 7 by one, and replace everything after it with zero:

675 ≈ 680

This means that when rounding the number 675 to the tens place, we obtain the approximate number 680.

Now let's try to round the same number 675, but to hundreds place.

We need to round the number 675 to the hundreds place. Again we are looking for the number to be saved. This time the digit being stored is 6, since we are rounding the number to the hundreds place:

Now we find the first of the discarded digits. The first digit to be discarded is the digit that comes after the digit to be stored. We see that the first digit after six is ​​the number 7. This means that the number 7 is first digit to be discarded:

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

Our first discarded digit is 7. This means we must increase the retained digit 6 by one, and replace everything after it with zeros:

675 ≈ 700

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

Example 3. Round the number 9876 to the tens place.

Here the retained digit is 7. And the first discarded digit is 6.

This means we increase the stored number 7 by one, and replace everything that is located after it with zero:

9876 ≈ 9880

Example 4. Round 9876 to the hundreds place.

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

This means we increase the stored number 8 by one, and replace everything that is located after it with zeros:

9876 ≈ 9900

Example 5. Round 9876 to the thousands place.

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

This means we increase the stored number 9 by one, and replace everything that is located after it with zeros:

9876 ≈ 10000

Example 6. Round 2971 to the nearest hundred.

When rounding this number to the nearest hundred, you should be careful because the digit being retained here is 9, and the first digit to be discarded is 7. This means that the digit 9 must be increased by one. But the fact is that after increasing nine by one, the result is 10, and this figure will not fit into the hundreds digit of the new number.

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

2971 ≈ 3000

Rounding decimals

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

Integer digits:

• units digit
• tens place
• hundreds place
• thousand digit

Fractional digits:

• tenth place
• hundredths place
• thousandth place

Consider the decimal fraction 123.456 - one hundred twenty-three point four hundred fifty-six thousandths. Here the integer 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 the fraction 123.456 to tens place. Exactly until tens place, but not tenth place. It is very important not to confuse these categories. Discharge dozens is located in the whole part, and the digit tenths in fractional

We must round 123.456 to the tens place. The digit retained here is 2, and the first digit discarded is 3

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

This means that the saved digit will remain unchanged, and everything else will be replaced by zero. What to do with the fractional part? It is simply discarded (removed):

123,456 ≈ 120

Now let's try to round the same fraction 123.456 to units digit. The digit to be retained 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 digit to be discarded is 0, 1, 2, 3 or 4, then the retained digit remains unchanged.

This means that the saved 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 that remains 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 fractional parts. The same rules apply for rounding fractional parts as for rounding whole parts. Let's try to round the fraction 123.456 to tenth place. The number 4 is in the tenths place, which means it is the retained digit, and the first digit to be discarded is 5, which is in the hundredths place:

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

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

123,456 ≈ 123,500

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

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

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

123,456 ≈ 123,460

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