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

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

Why are numbers rounded?

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

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

A few important rules when rounding numbers

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

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

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

How to round a number to integers

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

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

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

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

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

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

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

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

When rounding, only correct signs are left, the rest are discarded.

Rule 1. Rounding is achieved by simply dropping 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 increased in the case when the first of the discarded digits is 5, followed by one or more digits other than zero. For example, different roundings of 35.856 would be 35.86; 35.9; 36.

Rule 3. If the discarded digit is 5, and there are no significant digits behind it, then rounding is performed to the nearest even number, ie. the last digit stored remains unchanged if it is even and incremented if it is odd. For example, round 0.435 to 0.44; Round 0.465 to 0.46.

8. EXAMPLE OF PROCESSING THE MEASUREMENT RESULTS

Determination of the density of solids. Suppose a solid is in 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:

P / p No.	m, g	Δm, g	D, mm	ΔD, mm	h, mm	Δh, mm	, g / cm 3	Δ, g / cm 3
	51,2	0,1	12,68	0,07	80,3	0,15	5,11	0,07	0,013
	12,63	80,2
	12,52	80,3
	12,59	80,2
	12,61	80,1
the average			12,61		80,2		5,11

Let us determine the average value of D̃:

Find the errors of individual measurements and their squares

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

We set the value of reliability α = 0.95 and from the table we find the Student's coefficient t α. n = 2.8 (for n = 5). 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 (the measurement is performed with a micrometer), the resulting value can serve as an estimate of the confidence interval:

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:

We 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 observation results.

9. GOST 11.002-73 (Art. CMEA 545-77) Rules for assessing the abnormality of observation results.

Tsarkovskaya Nadezhda Ivanovna

Sakharov Yuri Georgievich

General physics

Methodical instructions for laboratory work "Introduction to the theory of measurement errors" for students of all specialties

Format 60 * 84 1/16 Volume 1 book. l. Circulation 50 copies.

Order ______ Free

Bryansk State Engineering and Technological Academy

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Editorial and publishing department

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Methods

Different areas may apply different rounding methods. In all these methods, the "extra" signs are set to zero (discarded), and the preceding sign is corrected according to some rule.

• Round to the nearest integer(eng. rounding) - the most frequently used rounding, in which the number is rounded to the nearest integer, the modulus of the difference with which this number is minimal. In general, when a number in the decimal system is rounded to the Nth decimal place, the rule can be formulated as follows:
  • if N + 1 digit< 5 , then the N-th sign is preserved, and N + 1 and all subsequent ones are set to zero;
  • if N + 1 digit ≥ 5, then the N-th sign is increased by one, and N + 1 and all subsequent ones are zeroed;
  For example: 11.9 → 12; −0.9 → −1; −1.1 → −1; 2.5 → 3.
• Round down in absolute value(rounding towards zero, whole eng. fix, truncate, integer) is the "simplest" rounding, because after zeroing out the "extra" characters, the previous character is retained. For example, 11.9 → 11; −0.9 → 0; −1,1 → −1).
• Round up(rounding to + ∞, rounding 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 retained if the number is negative. In economic jargon - rounding in favor of the seller, creditor(the person receiving the money). In particular, 2.6 → 3, −2.6 → −2.
• Round down(rounding to −∞, rounding down, eng. floor) - if the nullable signs are not equal to zero, the preceding sign is retained if the number is positive, or increased 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(rounding towards infinity, rounding away from zero) is a relatively rarely used form of rounding. If the nullable characters are not zero, the preceding character is incremented by one.

Rounding options 0.5 to nearest integer

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

• Mathematical rounding- rounding is always upward in absolute value (the previous digit is always increased by one).
• Banking rounding(eng. banker "s rounding) - rounding for this case occurs to the nearest even, that is, 2.5 → 2, 3.5 → 4.
• Random rounding- rounding occurs in a random order, but with equal probability (can be used in statistics).
• Alternating rounding- rounding up or down one by one.

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

Mathematical rounding just formally follows 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 monetary amounts to whole rubles. So, if in the registry of 10,000 lines there are 100 lines with amounts containing the value of 50 in a part of kopecks (and this is a very real estimate), then when all such lines are rounded "up", the sum of the "total" in the rounded register will be 50 rubles more accurate ...

The other three options are just invented in order to reduce the total error of the sum when rounding a large number of values. Rounding to the nearest even is based on the assumption that for a large number of values ​​to be rounded that have 0.5 in the remainder to be rounded, on average, half will be to the left and half to the right of the nearest even, thus, rounding errors are canceled out. Strictly speaking, this assumption is true only when the set of numbers to be rounded has the properties of a random series, which is usually true in accounting applications, where we are talking about prices, amounts on 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 about half of the special values ​​are rounded one way and half the other. But the implementation of such methods in practice requires additional efforts to organize the computational process.

Applications

Rounding is used in order to work with numbers within the number of digits that corresponds to the real accuracy of the calculation parameters (if these values ​​are real values ​​measured in one way or another), the actually achievable accuracy of the calculations, or the desired accuracy of the result. In the past, the rounding of intermediate values ​​and the result was of practical importance (since when calculating on paper or using primitive devices such as an 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 roundings, including intermediate ones, may be required to protect against computational errors associated with the finite bit width of computing devices.

Using Rounding with Limited Precision Numbers

Real physical quantities are always measured with some finite accuracy, which depends on instruments and measurement methods 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 record, all the numbers after (to the right) of which are insignificant (within the measurement error). The measured parameters themselves are recorded with such a number of digits that all digits are reliable, perhaps the last one is doubtful. The error in mathematical operations with numbers of limited precision is preserved and changed according to known mathematical laws, therefore, when intermediate values ​​and results with a large number of digits appear in further calculations, only part of these digits are significant. The rest of the numbers, being present in the values, actually do not 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 digits that reflects the real accuracy of the obtained values. In practice, it is usually recommended for long "chained" manual calculations to store one more digit in intermediate values. 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 specified 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 , the second - 1/140 ≈ 7,1 10 −3 , the relative error of the result according to the rule of error of the multiplication operation (when multiplying the approximate values, the relative errors are added) 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 figure is completely reliable, even the second is already doubtful! It will be correct to round the result of calculations to the first doubtful digit, that is, to tenths: 8.1 kgf m, or, if it is necessary to more accurately indicate 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.

Rules of thumb for rounding arithmetic

In cases where there is no need to accurately account for computational errors, but only need to roughly estimate the number of exact digits as a result of calculating by the formula, you can use a set of simple rules for rounded calculations:

1. All initial values ​​are rounded to the actual measurement accuracy and recorded with an appropriate number of significant digits, so that in decimal notation all digits are reliable (it is allowed that the last digit is doubtful). If necessary, the values ​​are written with significant right-hand zeros so that the record indicates the real number of reliable characters (for example, if the length of 1 m is actually measured with an accuracy of centimeters, write "1.00 m" so that it can be seen that two characters are reliable in the record after the decimal point), or the accuracy is clearly indicated (for example, 2500 ± 5 m - here only tens are reliable, and should be rounded 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 parameter (for example, when calculating the value 1.00 m + 1.5 m + 0.075 m, the result is rounded to tenths of a meter, that is, 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 actions on numbers, if possible, in ascending order of their modules.
4. When multiplying and dividing, the result is rounded to the smallest number of significant digits that the parameters have (for example, when calculating the speed of uniform body movement at a distance of 2.5 10 2 m, in 600 s the result should be rounded to 4.2 m / s, because two digits have a distance, and time has three, assuming that all the digits in the entry are significant).
5. When calculating the value of a function f (x) it is required to estimate the value of the modulus of the derivative of this function in the vicinity of the computation point. If (| 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 up to the nearest whole.

Despite the laxity, the above rules work quite well in practice, in particular, due to the rather high probability of mutual cancellation of errors, which is usually not taken into account when accurately accounting for errors.

Errors

Non-circular numbers are abused quite often. For example:

• Numbers that have low precision are recorded in an unrounded form. In statistics: if 4 people out of 17 answered “yes”, then they write “23.5%” (while “24%” is correct).
• Users of dial gauges sometimes think like this: "the arrow stopped between 5.5 and 6 closer to 6, let it be 5.8" - this is also prohibited (the calibration of the device usually corresponds to its real accuracy). In this case, you should say "5.5" or "6".

see also

• Processing observations
• Rounding errors

Notes (edit)

Literature

• Henry S. Warren, Jr. Chapter 3. Rounding to a power of 2// Algorithmic tricks for programmers = Hacker "s Delight. - M .:" Williams ", 2007. - P. 288. - ISBN 0-201-91465-4

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

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

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

1. RULES FOR RECORDING NUMBERS

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

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

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 (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 whole and tenth digits are correct; the true value of a number might be 2.43 and 2.38, for example. The record 2.40 means that the hundredths of the number are also correct; the true number can be 2.403 and 2.398, but not 2.421 or 2.382.

2. Record 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.

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 order as the last significant digit of the deviation.

Examples:

1.5. It is advisable to write the numerical values ​​of the 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 recorded:

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 should be indicated in the standards with the same number of digits, which is necessary to ensure the required performance and product quality. The recording of numerical values ​​of quantities up to the first, second, third, etc. decimal places for various standard sizes, types of brands of products 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 indicated with an accuracy of 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 same parameter, size, indicator or norm may have several steps (groups) and should be the same only within this step (group).

2. RULES OF ROUND

2.1. Rounding of a number is the casting of significant digits from 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 does not change.

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 digits gives 0.15.

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

1) if the discarded digit was the result of the previous rounding up, then the last stored digit is saved;

Example. Rounding to one significant digit of 0.15 (obtained after rounding 0.149) gives 0.1.

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

Example. Rounding 0.25 (resulting from the previous rounding of 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 done immediately to the desired number of significant figures, not step by step.

Example. Rounding 565.46 to three significant digits is done directly on 565. Rounding in steps would result in:

565.46 in stage I - to 565.5,

and in the second stage - 566 (wrongly).

2.6. Whole numbers are rounded off using the same rules as fractional numbers.

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

Topic 01.693.04-75.

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

4. Terms of the beginning of the application of the CMEA standard:

CMEA member countries	The term for the commencement of the application of the CMEA standard in contractual and legal relations on economic, scientific and technical cooperation	The beginning of the application of the CMEA standard in the national economy
NRB	December 1979	December 1979
Hungarian People's Republic	December 1978	December 1978
GDR	December 1978	December 1978
Republic of Cuba
Mongolia
Poland
CPP
the USSR	December 1979	December 1979
Czechoslovakia	December 1978	December 1978

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

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

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

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

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

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

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

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

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

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

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