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 the digits from the first non-zero digit on the left to the last digit written 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 indicated must have the last significant figure 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 operational 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 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.

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). We determine the boundaries of the 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 caliper error 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 the 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.

Tsarkovskaya Nadezhda Ivanovna

Sakharov Yury Georgievich

General physics

Guidelines to fulfillment laboratory work"Introduction to the theory of measurement errors" for students of all specialties

Format 60*84 1/16 Volume 1 app.-ed. l. Circulation 50 copies.

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Let's say you want to round a number to the nearest whole number because you don't care about decimals, or you want to express a number as a power of 10 to make it easier to approximate. There are several ways to round numbers.

Changing the number of decimal places without changing the value

On the sheet

In built-in number format

Rounding up

Rounding a number to the nearest value

Rounding a number to the nearest fractional value

Rounding a number to the specified number of significant digits

Significant digits are digits that affect the precision of a number.

The examples in this section use the functions ROUND, ROUNDUP and ROUNDDOWN. They show ways to round positive, negative, integer, and fractional numbers, but the examples given cover only a small part of the possible situations.

The list below contains general rules, which must be taken into account when rounding numbers to the specified number of significant digits. You can experiment with the rounding functions and substitute your own numbers and parameters to get a number with the number of significant digits you want.

Rounded negative numbers are first converted to absolute values ​​(values ​​without a minus sign). After rounding, the minus sign is reapplied. Although it may seem counterintuitive, this is how rounding works. For example, when using the function ROUNDDOWN to round -889 to two significant digits, the result is -880. First, -889 is converted to an absolute value (889). This value is then rounded to two significant digits (880). The minus sign is then reapplied, resulting in -880.

When applied to positive number functions ROUNDDOWN it always rounds down, and when applying the function ROUNDUP- up.

Function ROUND rounds fractional numbers as follows: if the fractional part is greater than or equal to 0.5, the number is rounded up. If the fractional part is less than 0.5, the number is rounded down.

Function ROUND rounds integers up or down in the same way, using 5 instead of 0.5.

In general, when rounding a number without a fractional part (an integer), you need to subtract the length of the number from the right amount significant ranks. For example, to round 2345678 down to 3 significant digits, use the function ROUNDDOWN with -4 option: = ROUNDDOWN(2345678,-4). This rounds the number up to 2340000, where the "234" portion is significant digits.

Rounding a number to a given multiple

Sometimes you may want to round a value to a multiple of a given number. For example, let's say a company ships goods in boxes of 18 units. Using the ROUND function, you can determine how many boxes will be required to deliver 204 items. In this case, the answer is 12 because 204 when divided by 18 is 11.333, which needs to be rounded up. There will be only 6 items in the 12th box.

May also need to be rounded negative meaning up to a multiple of a negative or fractional - up to a multiple of a fractional. You can also use the function for this ROUND.

Methods

Can be used in different areas various methods rounding. In all these methods, the "extra" signs are set to zero (discarded), and the sign preceding them is corrected according to some rule.

• Rounding to nearest integer(English) rounding) - the most commonly used rounding, in which the number is rounded up to an integer, the modulus of the difference with which this number has a minimum. In general, when a number in the decimal system is rounded up to the Nth decimal place, the rule can be formulated as follows:
  • if N+1 characters< 5 , then the Nth sign is retained, and N+1 and all subsequent ones are set to zero;
  • if N+1 characters ≥ 5, then the N-th sign is increased by one, and N + 1 and all subsequent ones are set to zero;
  For example: 11.9 → 12; -0.9 → -1; −1,1 → −1; 2.5 → 3.
• Rounding down modulo(rounding towards zero, integer Eng. fix, truncate, integer) is the most “simple” rounding, since after zeroing the “extra” signs, the previous sign is retained. For example, 11.9 → 11; −0.9 → 0; −1,1 → −1).
• Rounding Up(round to +∞, round up, eng. ceiling) - if the nullable signs are not equal to zero, the preceding sign is increased by one if the number is positive, or kept if the number is negative. In economic jargon - rounding in favor of the seller, creditor(of the person receiving the money). In particular, 2.6 → 3, −2.6 → −2.
• Rounding Down(round to −∞, round down, engl. floor) - if the nullable signs are not equal to zero, the preceding sign is retained if the number is positive, or incremented by one if the number is negative. In economic jargon - rounding in favor of the buyer, debtor(the person giving the money). Here 2.6 → 2, −2.6 → −3.
• Rounding up modulo(round towards infinity, round away from zero) is a relatively rarely used form of rounding. If the nullable characters are not equal to zero, the preceding character is incremented by one.

Rounding options 0.5 to nearest integer

A separate description is required by the rounding rules for the special case when (N+1)th digit = 5 and subsequent digits are zero. If in all other cases, rounding to the nearest integer provides a smaller rounding error, then this special case It is characteristic that for a single rounding it is formally indifferent whether to produce it “up” or “down” - in both cases an error is introduced exactly in 1/2 of the least significant digit. There are the following variants of the rounding rule to the nearest integer for this case:

• Mathematical rounding- rounding is always up (the previous digit is always increased by one).
• Bank rounding(English) banker's rounding) - rounding for this case occurs to the nearest even number, i.e. 2.5 → 2, 3.5 → 4.
• Random rounding- rounding up or down random order, but with equal probability (can be used in statistics).
• Alternate rounding- Rounding occurs up or down alternately.

In all cases, when the (N + 1)th sign is not equal to 5 or subsequent signs are not equal to zero, rounding occurs according to the usual rules: 2.49 → 2; 2.51 → 3.

Mathematical rounding simply formally corresponds to the general rounding rule (see above). Its disadvantage is that when rounding a large number of values, accumulation can occur. rounding errors. Typical example: rounding to whole rubles of monetary amounts. So, if in the register of 10,000 lines there are 100 lines with amounts containing the value of 50 in terms of kopecks (and this is a very realistic estimate), then when all such lines are rounded “up”, the sum of the “total” according to the rounded register will be 50 rubles more than the exact .

The other three options are just invented in order to reduce the total error of the sum during rounding. a large number values. Rounding "to the nearest even number" is based on the assumption that when large numbers rounded values ​​that have 0.5 in the rounded remainder, on average, half will be to the left and half to the right of the nearest even number, so rounding errors cancel each other out. Strictly speaking, this assumption is true only when the set of numbers being rounded has the properties of a random series, which is usually true in accounting applications where we are talking about prices, amounts in accounts, and so on. If the assumption is violated, then rounding “to even” can lead to systematic errors. For such cases, the following two methods work best.

The last two rounding options ensure that approximately half special values will be rounded one way, half rounded the other way. But the implementation of such methods in practice requires additional efforts to organize the computational process.

Applications

Rounding is used to work with numbers within the number of digits that corresponds to the actual accuracy of the calculation parameters (if these values ​​are real values ​​​​measured in one way or another), the realistically achievable calculation accuracy, or the desired accuracy of the result. In the past, the rounding of intermediate values ​​and the result was of practical importance (because when calculating on paper or using primitive devices such as the abacus, taking into account extra decimal places can seriously increase the amount of work). Now it remains an element of scientific and engineering culture. In accounting applications, in addition, the use of rounding, including intermediate ones, may be required to protect against computational errors associated with the finite bit capacity of computing devices.

Using rounding when working with numbers of limited precision

Real physical quantities are always measured with some finite accuracy, which depends on the instruments and methods of measurement and is estimated by the maximum relative or absolute deviation of the unknown actual value from the measured one, which in decimal representation of the value corresponds either to a certain number of significant digits or to a certain position in the number notation, all digits after ( to the right) which are insignificant (lie within the measurement error). The measured parameters themselves are recorded with such a number of characters that all figures are reliable, perhaps the last one is doubtful. The error in mathematical operations with numbers of limited accuracy is preserved and changes according to known mathematical laws, so when intermediate values ​​and results with a large number of digits appear in further calculations, only a part of these digits are significant. The remaining figures, being present in the values, do not actually reflect any physical reality and only take time for calculations. As a result, intermediate values ​​and results in calculations with limited accuracy are rounded to the number of decimal places that reflects the actual accuracy of the values ​​obtained. In practice, it is usually recommended to store one more digit in intermediate values ​​for long "chained" manual calculations. When using a computer, intermediate roundings in scientific and technical applications most often lose their meaning, and only the result is rounded.

So, for example, if a force of 5815 gf is given with an accuracy of a gram of force and a shoulder length of 1.4 m with an accuracy of a centimeter, then the moment of force in kgf according to the formula, in the case of a formal calculation with all signs, will be equal to: 5.815 kgf 1.4 m = 8.141 kgf m. However, if we take into account the measurement error, then we get that the limiting relative error of the first value is 1/5815 ≈ 1,7 10 −4 , second - 1/140 ≈ 7,1 10 −3 , the relative error of the result according to the error rule of the multiplication operation (when multiplying approximate values, the relative errors add up) will be 7,3 10 −3 , which corresponds to the maximum absolute error of the result ±0.059 kgf m! That is, in reality, taking into account the error, the result can be from 8.082 to 8.200 kgf m, thus, in the calculated value of 8.141 kgf m, only the first digit is completely reliable, even the second is already doubtful! It will be correct to round the calculation result to the first doubtful digit, that is, to tenths: 8.1 kgf m, or, if necessary, a more accurate indication of the margin of error, present it in a form rounded to one or two decimal places with an indication of the error: 8.14 ± 0.06 kgf m.

Empirical rules of arithmetic with rounding

In cases where there is no need to accurately take into account computational errors, but only an approximate estimate of the number of exact numbers as a result of the calculation by the formula, you can use the set simple rules rounded calculations :

1. All raw values ​​are rounded to the actual measurement accuracy and recorded with the appropriate number of significant digits, so that all digits in the decimal notation are reliable (it is allowed that the last digit is doubtful). If necessary, values ​​are recorded with significant right-hand zeros so that the actual number of reliable characters is indicated in the record (for example, if a length of 1 m is actually measured to the nearest centimeter, “1.00 m” is written so that it can be seen that two characters are reliable in the record after the decimal point), or the accuracy is explicitly indicated (for example, 2500 ± 5 m - here only tens are reliable, and should be rounded up to them).
2. Intermediate values ​​are rounded off with one "spare" digit.
3. When adding and subtracting, the result is rounded to the last decimal place of the least accurate of the parameters (for example, when calculating a value of 1.00 m + 1.5 m + 0.075 m, the result is rounded to tenths of a meter, that is, to 2.6 m). At the same time, it is recommended to perform calculations in such an order as to avoid subtracting numbers that are close in magnitude and to perform operations on numbers, if possible, in ascending order of their modules.
4. When multiplying and dividing, the result is rounded up to the smallest number significant digits that the parameters have (for example, when calculating the speed of a uniform movement of a body at a distance of 2.5 10 2 m, for 600 s, the result should be rounded up to 4.2 m / s, since it is two digits that have distance, and time - three , assuming all digits in the entry are significant).
5. When calculating the function value f(x) it is required to estimate the value of the modulus of the derivative of this function in the vicinity of the calculation point. If a (|f"(x)| ≤ 1), then the result of the function is exact to the same decimal place as the argument. Otherwise, the result contains fewer exact decimal places by the amount log 10 (|f"(x)|), rounded to the nearest integer.

Despite the non-strictness, the above rules work quite well in practice, in particular, because of the rather high probability of mutual cancellation of errors, which is usually not taken into account when errors are accurately taken into account.

Mistakes

Quite often there are abuses of non-round numbers. For example:

• Write down numbers that have low accuracy, in unrounded form. In statistics: if 4 people out of 17 answered “yes”, then they write “23.5%” (while “24%” is correct).
• Pointer users sometimes think like this: “the pointer stopped between 5.5 and 6 closer to 6, let it be 5.8” - this is also prohibited (the graduation of the device usually corresponds to its actual accuracy). In this case, you need to say "5.5" or "6".

see also

• Observation Processing
• Rounding errors

Notes

Literature

• Henry S. Warren, Jr. Chapter 3// Algorithmic tricks for programmers = Hacker's Delight. - M .: Williams, 2007. - S. 288. - ISBN 0-201-91465-4

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 the tolerance 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. 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 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 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 figures, and the calculated value of the error is obtained with a large number characters, in the final value of the calculated error, respectively, only the first one or two significant digits should be left. 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.