Rounding numbers in a period. Empirical rules of arithmetic with rounding

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."

Approximate values

Approximate (or approximate) values ​​apply when exact value it is impossible to find anything, or this value is not important for the object under study.

For example, one can verbally 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 go, 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. Some time later, 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 don’t know what time it is, so we answer a friend: “now approximately around nine o'clock."

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

It is 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 round number is a number that ends in zero. For example, next numbers are round,

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

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

We have already dealt with the "rounding" of numbers when dividing big 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 only sketches that we made to facilitate division. Kind of a hack. In fact, it wasn't even rounding numbers. That is why at the beginning of this paragraph we took the word rounding in quotation marks.

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

Consider a simple rounding example. The number 17 is given. It is required to round it up to the digit of tens.

Without looking ahead, let's try to understand what it means to "round to the digit of tens." When they say to round the number 17, we are required to find the nearest round number for the number 17. At the same time, during this search, the number that is in the tens place in the number 17 (i.e. units) may also be changed.

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 number 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 of 12, was not affected by rounding. Why this happened, we will consider later.

Let's try to find the nearest number to the number 15. Again, imagine 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 an approximate value for the number 15? For such cases, we agreed to take a larger number as an approximation. 20 is greater than 10, so the approximate value for 15 is the number 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 have to round 1456 to the tens place. The tens digit starts at five:

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

Now we look at which round number is closer to the number 56. Obviously, the nearest 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 digit, the changes also affected the tens digit itself. The new resulting number now has a 6 instead of a 5 in the tens place.

You can round numbers not only to the digit of tens. You can also round up to the discharge of hundreds, thousands, tens of thousands.

After it becomes clear that rounding is nothing more than a search for the nearest number, you can apply ready-made rules, which make it much easier to round numbers.

First rounding rule

From the previous examples, it became clear that when rounding a number to a certain digit, the lower digits are replaced by zeros. Digits that are replaced by zeros are called discarded figures.

The first rounding rule looks like this:

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. In the discharge, which is mentioned in the task, there is a stored figure. The task says: round the number 123 up to tens digit.

We see that there is a deuce 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 retained. We see that the first digit after the two is the number 3. So the number 3 is first discarded digit.

Now 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. We leave the stored digit unchanged, and replace all the lower digits with zeros. In other words, everything that follows after the number 2 is replaced by zeros (more precisely, zero):

123 ≈ 120

So when rounding the number 123 to the digit of tens, we get the approximate number 120.

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

We need to round the number 123 to the hundreds place. Again we are looking for a saved figure. This time, the stored digit 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 follows the digit to be retained. We see that the first digit after the unit is the number 2. So the number 2 is 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. We leave the stored digit unchanged, and replace all the lower digits with zeros. In other words, everything that follows after the number 1 is replaced with zeros:

123 ≈ 100

So when rounding the number 123 to the hundreds place, we get the approximate number 100.

Example 3 Round the number 1234 to the tens place.

Here the digit to be kept is 3. And the first digit to be discarded is 4.

So we leave the saved number 3 unchanged, and replace everything after it with zero:

1234 ≈ 1230

Example 4 Round the number 1234 to the hundreds place.

Here, the stored 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 stored digit remains unchanged.

So we leave the saved number 2 unchanged, and replace everything after it with zeros:

1234 ≈ 1200

Example 3 Round the number 1234 to the thousandth place.

Here, the stored 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 stored digit remains unchanged.

So we leave the saved number 1 unchanged, and replace everything after it with zeros:

1234 ≈ 1000

Second rounding rule

The second rounding rule looks like this:

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

First of all, we find the stored digit. To do this, you need to read the task itself. In the discharge, which is mentioned in the task, there is a stored figure. The task says: round the number 675 up to tens digit.

We see that in the category of tens there is a seven. 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 retained. We see that the first digit after the seven is the number 5. So the number 5 is first discarded digit.

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

675 ≈ 680

So when rounding the number 675 to the digit of tens, we get the approximate number 680.

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

We need to round the number 675 to the hundreds place. Again we are looking for a saved figure. This time, the stored digit is 6, because we're 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 follows the digit to be retained. We see that the first digit after the six is ​​​​the number 7. So the number 7 is first discarded digit:

Now 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 retained digit is increased by one.

We have the 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

So when rounding the number 675 to the hundreds place, we get the number 700 approximate to it.

Example 3 Round the number 9876 to the tens place.

Here the digit to be kept is 7. And the first digit to be discarded is 6.

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

9876 ≈ 9880

Example 4 Round the number 9876 to the hundreds place.

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

So we increase the saved number 8 by one, and replace everything that is located after it with zeros:

9876 ≈ 9900

Example 5 Round the number 9876 to the thousandth place.

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

So we increase the saved number 9 by one, and replace everything that is located after it with zeros:

9876 ≈ 10000

Example 6 Round the number 2971 to the nearest hundred.

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

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

2971 ≈ 3000

Rounding decimals

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

Bits of the integer part:

• unit digit
• tens place
• hundreds place
• thousand digit

Fractional digits:

• tenth place
• hundredth place
• thousandth place

Consider decimal 123.456 is one hundred and twenty-three point four hundred and 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 ordinary 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, let's round the fraction 123.456 to tens digit. Exactly up to tens place, but not tenth place. It is very important not to confuse these categories. Discharge dozens is located in the integer part, and the discharge tenths in fractional.

We have to round 123.456 to the tens place. The digit to be stored 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 retained digit remains unchanged.

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

123,456 ≈ 120

Now let's try to round the same fraction 123.456 up to unit digit. The digit to be stored 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 retained 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 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 do the 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. In the tenth place is the number 4, which means it is the 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 retained digit is increased by one.

So the stored number 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 stored here is 5, and the first digit to discard 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 retained digit is increased by one.

So the saved number 5 will increase by one, and the rest will be replaced by zeros

123,456 ≈ 123,460

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When rounding, leave only true signs, the rest are discarded.

Rule 1. Rounding is achieved by simply discarding digits if the first of the discarded digits is less than 5.

Rule 2. If the first of the discarded digits is greater than 5, then the last digit is increased by one. The last digit is also incremented when the first of the discarded digits is 5 followed by one or more non-zero digits. For example, various roundings of the number 35.856 would be 35.86; 35.9; 36.

Rule 3. If the discarded figure is 5, and there are no significant figures behind it, then rounding is performed to the nearest even number, i.e. the last digit stored remains unchanged if it is even and incremented by one if it is odd. For example, 0.435 is rounded up to 0.44; 0.465 is rounded up to 0.46.

8. EXAMPLE OF MEASUREMENT RESULTS PROCESSING

Determination of the density of solids. Suppose a rigid body has the shape of a cylinder. Then the density ρ can be determined by the formula:

where D is the diameter of the cylinder, h is its height, m ​​is the mass.

Let the following data be obtained as a result of measurements of m, D, and h:

Let us define the mean value D̃:

Find the errors of individual measurements and their squares

Let us determine the root-mean-square error of a series of measurements:

We set the reliability value α = 0.95 and find the Student's coefficient t α from the table. n=2.8 (for n=5). Defining boundaries confidence interval:

Since the calculated value ΔD = 0.07 mm significantly exceeds the absolute error of the micrometer, equal to 0.01 mm (measured with a micrometer), the resulting value can serve as an estimate of the confidence interval boundary:

D = D̃ ± Δ D; D= (12.61 ±0.07) mm.

Let us define the value of h̃:

Hence:

For α = 0.95 and n = 5 Student's coefficient t α , n = 2.8.

Determining the boundaries of the confidence interval

Since the obtained value Δh = 0.11 mm is of the same order as the error of the caliper equal to 0.1 mm (h is measured with a caliper), the boundaries of the confidence interval should be determined by the formula:

Hence:

Let us calculate the average value of the density ρ:

Let's find an expression for relative error:

where

7. GOST 16263-70 Metrology. Terms and Definitions.

8. GOST 8.207-76 Direct measurements with multiple observations. Methods for processing the results of observations.

9. GOST 11.002-73 (art. SEV 545-77) Rules for assessing the anomalous results of observations.

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This CMEA standard establishes rules for recording and rounding off 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. The significant digits of a given number are all digits from the first to the left, not zero, to the last recorded digit on the right. In this case, zeros following from the factor 10 n are not taken into account.

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

Example. In printed text:

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

1.3. It is necessary to distinguish between records of approximate numbers by the number of significant digits.

Examples:

1. One should distinguish between the numbers 2.4 and 2.40. The entry 2.4 means that only integers and tenths are correct; the true value of the number can be, for example, 2.43 and 2.38. Recording 2.40 means that the hundredths of the number are also true; the true number may be 2.403 and 2.398, but not 2.421 or 2.382.

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

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

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

Examples:

1.5. It is expedient to record the numerical values ​​of a quantity and its errors (deviations) with the indication of the same unit of physical quantities.

Example. 80.555±0.002 kg

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

60 to 100 or 60 to 100

Over 100 to 120 or over 100 to 120

Over 120 to 150 or over 120 to 150.

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

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

2. ROUNDING RULES

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

Example. Rounding 132.48 to four significant digits is 132.5.

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

Example. Rounding 12.23 to three significant digits gives 12.2.

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

Example. Rounding 0.145 to two significant figures gives 0.15.

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

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

Example. Rounding to one significant figure the number 0.15 (obtained after 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 the transition, if necessary, to the next digits).

Example. Rounding the number 0.25 (obtained 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 stored digit is increased by one.

Example. Rounding 0.156 to two significant digits gives 0.16.

2.5. Rounding should be performed immediately to the desired number of significant digits, and not in stages.

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

565.46 in stage I - to 565.5,

and in stage II - 566 (erroneously).

2.6. Whole numbers are rounded in the same way as fractional numbers.

Example. Rounding the number 12456 to two significant figures gives 12·10 3 .

Subject 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:

5. The term of the first check is 1981, the frequency of checks is 5 years.

Rounding numbers is the simplest mathematical operation. To be able to correctly round numbers, you need to know three rules.

Rule 1

When we round a number to a certain digit, we must get rid of all the digits to the right of that digit.

For example, we need to round the number 7531 to the nearest hundred. This number is five hundred. To the right of this category are the numbers 3 and 1. We turn them into zeros and get the number 7500. That is, rounding the number 7531 to hundreds, we got 7500.

When rounding fractional numbers, everything happens in the same way, only the extra digits can simply be discarded. Let's say we need to round the number 12.325 to tenths. To do this, after the decimal point, we must leave one digit - 3, and discard all the numbers to the right. The result of rounding the number 12.325 to tenths is 12.3.

Rule 2

If to the right of the remaining digit the discarded digit is 0, 1, 2, 3 or 4, then the digit we leave does not change.

This rule worked in the previous two examples.

So, when rounding the number 7531 to hundreds, the closest figure to the discarded figure was a three. Therefore, the number we left - 5 - has not changed. The rounding result is 7500.

Similarly, when 12.325 was rounded to tenths, the digit we dropped after the three was a two. Therefore, the rightmost of the remaining digits (three) did not change during rounding. It turned out 12.3.

Rule 3

If the leftmost of the discarded digits is 5, 6, 7, 8, or 9, then the digit to which we round is increased by one.

For example, you need to round the number 156 to tens. There are 5 tens in this number. The units place we are going to get rid of is the number 6. This means that we should increase the tens place by one. Therefore, when rounding the number 156 to tens, we get 160.

Consider an example with a fractional number. For example, we are going to round 0.238 to the nearest hundredth. By rule 1, we must discard the eight, which is to the right of the hundredth place. And according to rule 3, we have to increase the three in the hundredth place by one. As a result, rounding the number 0.238 to hundredths, we get 0.24.

Task No. 1. Rounding off measurement results

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

Rules for writing numbers

1. Significant digits of a given number - all digits from the first on the left, not equal to zero, to the last on the right. In this case, the zeros following from the factor 10 are not taken into account.

Examples.

a) Number 12,0has three significant digits.

b) Number 30has two significant digits.

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

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

e) 0,0056has two significant digits.

2. If it is necessary to indicate that the 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. Distinguish records of approximate numbers by the number of significant digits. For example, the numbers 2.4 and 2.40 are distinguished. The entry 2.4 means that only integers and tenths are correct, the true value of the number can be, for example, 2.43 and 2.38. Writing 2.40 means that the hundredths are also correct: the true value of the number can be 2.403 and 2.398, but not 2.41 and not 2.382. Recording 382 means that all digits are correct: if the last digit cannot be vouched for, then the number should be written 3.810 2 . If only the first two digits are correct in the number 4720, it should be written as: 4710 2 or 4.710 3 .

4. The number for which they indicate tolerance, 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. Not right: 17 + 0,2or 17,00 + 0,2.

b) Correct: 12,13+ 0,17. Not right: 12,13+ 0,2.

c) Correct: 46,40+ 0,15. Not right: 46,4+ 0,15or 46,402+ 0,15.

5. The numerical values ​​of the quantity and its errors (deviations) should be recorded with the indication of the same unit of quantity. For example: (80,555 + 0.002) kg.

6. The intervals between the numerical values ​​​​of quantities are sometimes advisable to write in text form, then the preposition "from" means "", the preposition "to" - "", the preposition "above" - ​​">", the preposition "less" - "<":

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

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

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

Number Rounding Rules

1. Rounding a number is the rejection of significant digits on 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 stored digit is not changed.

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

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

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

Note . In those cases where it is necessary to take into account the results of previous roundings, 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 the transition, if necessary, to the next digits), otherwise, vice versa. This applies to both fractional and integer numbers.

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 stored digit is increased by one.

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

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

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

6. Whole numbers are rounded off according to the same rules as fractional ones.

Example: Rounding a number 23456up to two significant figures gives 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 + 0,15must be rounded up to 235,73 + 0,15but not before 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 of the discarded digits is greater than or equal to five, then the last retained digit 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 digits, incremental rounding may lead to errors.

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

10. If the error of measuring instruments 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, respectively, should be left in the final value of the calculated error. In this case, if the resulting number begins with the digits 1 or 2, then discarding the second sign 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 the preservation of 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 figure equal to or greater than 3, then only it is stored in it; if it starts with significant digits less than 3, i.e. with the numbers 1 and 2, then two significant digits are stored in it. In accordance with this rule, the normalized values ​​of the errors of measuring instruments are also established: in the numbers 1.5 and 2.5% two significant figures are indicated, but in the numbers 0.5; 4; 6% indicate only one significant figure.

Example:On a voltmeter of accuracy class 2,5with measurement limit x To = 300 In the readout 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 error of the voltmeter  0 \u003d 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 must be rounded to 8 V according to the usual rounding rules, but in the relative error value (2.81%) the first significant digit is less than 3, so here two decimal places must be stored in the answer and  = 2.8% indicated. Received value X= 267.5 V must be rounded to the same decimal place that ends the rounded absolute error value, i.e. to whole units of volts.

Thus, in the final answer it should be reported: "The measurement was made with a relative error  = 2.8% . 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.