Many people are interested in how to round numbers. This necessity often occurs in people who associate their lives with accounting or other activities where calculations are required. Rounding can be made to whole, tenths and so on. And you need to know how to do it correctly so that the calculations are less accurate.

What is a round number? This is the one that ends with 0 (for the most part). In everyday life, the ability to round numbers greatly facilitates shopping hikes. Standing at the box office, you can approximately estimate the total cost of purchases, compare how much a kilogram of the same name in various packets by weight. With numbers shown in a convenient form, it is easier to produce oral calculations without resorting to the calculator.

Why are the numbers round?

Any figures of the person are inclined to round in cases where need more simplified operations. For example, melon weighs 3,150 kilograms. When a person tells his acquaintances about how many grams have southern fruit, he can not have a very interesting interlocutor. Significantly concisely sound phrases like "I bought a three-kilogram melon" without understanding to all sorts of 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 endless fractions, which have the form of 3,33333333 ... 3, it becomes impossible. Therefore, the most logical option will be the usual rounding of them. As a rule, the result after that is distorted slightly. So how to round numbers?

Several important rules when rounding numbers

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

1. If the number of the desired discharge is within 5-9, the rounding is carried out in a majority.
2. If the number of the desired discharge is in the range of 1-4, the rounding is performed in a smaller side.

For example, we have a number 59. We need to round it. To do this, you need to take the number 9 and add a unit to it to work out 60. That's the answer to the question, how to round numbers. And now consider special cases. Actually, we figured out how to round the number up to dozen with this example. Now it remains only to use this knowledge in practice.

How to round the number to the whole

It often happens so that there is a need to round, for example, the number 5.9. This procedure It does not make much difficulty. It is necessary to omit the comma to begin with, and the number 60 is already familiar to our eyes when rounding the number 60. And now we put a comma in place, and we get 6.0. And since zeros in decimal fractions, as a rule, fall, then we finish the figure 6.

Similar operation can be made with more complex numbers. For example, how to round the numbers of type 5.49 to the whole? It all depends on what goals you put in front of you. In general, according to the rules of mathematics, 5.49 is still not 5.5. Therefore, it is impossible to round it in a major side. But it is possible to round it up to 5.5, after which the rounding is already the rounding up to 6. But such a trick does not always work, so you need to be extremely careful.

In principle, the example of the correct rounding number to the tenths was already considered above, so it is now important to display only the main subnie. In fact, everything is happening in approximately the same way. If the digit that is in the second point after the comma is within 5-9, then it is generally removed, and the number that standing in front of it increases by one. If less than 5, this figure is removed, and the previous one remains in its place.

For example, at 4.59 to 4.6 digit "9" goes, and the unit is added to the top five. But when rounding 4.41, the unit is lowered, and the Four remains in an uneplement form.

How do marketers use the inability of the mass consumer rounding the numbers?

It turns out that most people in the world do not have the habit of evaluating the real value of the product, which is actively exploiting marketers. Everyone knows the slogans of shares of the type "Buy in just 9.99". Yes, we deliberately understand that this is essentially ten dollars. Nevertheless, our brain is designed so that perceives only the first digit. So the non-sutomy operation of bringing the number in comfortable view Must get into the habit.

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

You can bring great amount Examples when the ability to round is incredibly useful. It is important to show ingenuity and whenever possible on loading unnecessary information. Then success will be immediate.

§ 4. Rounding results

The processing of measurement results in laboratories are carried out on calculators and PCs, and it is simply amazing how magically acts on many students of long row of numbers after the comma. "So more precisely," they say. However, it is easy to see, for example, that the record A \u003d 2.8674523 ± 0.076 is meaningless. In case of error 0.076, the last five digits of the number does not mean exactly nothing.

If we admit a mistake in hundredths, then the thousandths, even more than ten-thousand owners there are no faith. A competent result of the result would be 2.87 ± 0.08. It is always necessary to produce the necessary roundings so that there is no false impression of more than it is in fact, the accuracy of the results.

Rules of rounding

1. The measurement error is round to the first meaning digit, always increasing it by one.
   Examples:

   8.27 ≈ 9	0.237 ≈ 0.3
   0.0862 ≈ 0.09	0.00035 ≈ 0.0004
   857.3 ≈ 900	43.5 ≈ 50

2. The measurement results are rounded with an accuracy "before the error", i.e. The last meaning figure as a result should be in the same discharge as in the error.
   Examples:

   243.871 ± 0.026 ≈ 243.87 ± 0.03;
   243.871 ± 2.6 ≈ 244 ± 3;
   1053 ± 47 ≈ 1050 ± 50.

3. Rounding the measurement result is achieved by a simple discard of numbers if the first of the discarded numbers is less than 5.
   Examples:

   8.337 (rounded to tenths) ≈ 8.3;
   833.438 (round up to the integer) ≈ 833;
   0.27375 (rounded to hundredths) ≈ 0.27.

4. If the first of the discarded numbers is greater or equal to 5, (and after it, one or more digits are different from zero), then the latter of the remaining numbers increases by one.
   Examples:

   8.3351 (rounded for hundredths) ≈ 8.34;
   0.2510 (round up to tenths) ≈ 0.3;
   271.515 (round up to the integer) ≈ 272.

5. If the discarded digit is equal to 5, and there are no significant digits behind it (or there are alone zeros), then the last left digit increases per unit when it is odd, and leave unchanged when it is even.
   Examples:

   0.875 (rounded to hundredths) ≈ 0.88;
   0.5450 (rounded to hundredths) ≈ 0.54;
   275.500 (rounded to whole) ≈ 276;
   276.500 (round up to the integer) ≈ 276.

Note.

1. The most significant numbers call the numbers, except for zeros standing ahead of the number. For example, 0.00807 - there are three meaningful numbers: 8, zero between 8 and 7 and 7; The first three zero is insignificant.
   8.12 · 10 3 - In this number 3 significant figures.
2. Records 15.2 and 15,200 are different. Recording 15,200 means that the hundredths and thousands of stories are true. In the record of 15.2 - the entire and tenths are true.
3. The results of physical experiments are recorded only by meaningful numbers. The comma is put immediately after different numbers from zero, and the number is multiplied by ten to the appropriate degree. Zeros, standing at the beginning or end of the number, as a rule, do not write. For example, the numbers 0.00435 and 234000 are written as follows: 4.35 ° and 2,34 · 10 5. Such a record simplifies calculations, especially in the case of formulas that are convenient for logarithming.

Task number 1. Rounding measurement results

When performing measurements, it is important to comply with certain rules rounding and record their results in technical documentationSince, with non-compliance with these rules, significant errors are possible in the interpretation of measurement results.

Rules for recording numbers

1. The meaningful numbers of this number are all numbers from the first left, not equal to zero, to the last right. At the same time, zeros, the following from the multiplier 10, do not take into account.

Examples.

a) Number12,0 It has three meaning digits.

b) Number30 It has two meaning digits.

c) number12010 8 It has three meaning digits.

d)0,51410 -3 It has three meaning digits.

e)0,0056 It has two meaning digits.

2. If it is necessary to specify that the number is accurate, after the number indicate the word "accurate" or the last meaning digit is printed in bold. For example: 1 kW / h \u003d 3600 J (exactly) or 1 kW / h \u003d 360 0 J. .

3. There are records of approximate numbers by the number of significant digits. For example, the numbers of 2.4 and 2.40 are distinguished. Recording 2.4 means that only the whole and tenth shares are correct, the true value of the number may be, for example, 2.43 and 2.38. Recording 2.40 means that the hundredths are true: the true value of the number may be 2.403 and 2.398, but not 2.41 and not 2.382. Recording 382 means that all the numbers are correct: if it is impossible to vouch for the last digit, then the number must be recorded 3,8210 2. If there are only two first digits among the 4720, it must be recorded in the form: 4710 2 or 4,7 10 3.

4. The number for which the permissible deviation indicates should have the latest meaningful digit of the same discharge as the last digit of the deviation.

Examples.

a) right:17,0 + 0,2. Wrong:17 + 0,2 or17,00 + 0,2.

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

c) right:46,40+ 0,15. Wrong:46,4+ 0,15 or46,402+ 0,15.

5. The numeric values \u200b\u200bof the magnitude and its error (deviations) it is advisable to record with the indication of the same unit of magnitude. For example: (80,555 + 0.002) kg.

6. The intervals between numerical values \u200b\u200bof magnitudes sometimes it is advisable to record in text form, then the pretext "from" means "", the pretext "to" - "", the preposition "over" - "\u003e", the pretext "less" - "<":

"d.takes values \u200b\u200bfrom 60 to 100 "means" 60 d.100",

"d.takes values \u200b\u200bover 120 less than 150 "means" 120<d.< 150",

"d.takes values \u200b\u200bover 30 to 50 "means" 30<d.50".

Rules rounding numbers

1. The rounding of the number is the discarding of the meaning digits to the right to a certain discharge with a possible change in the number of this discharge.

2. In the event that the first of the discarded numbers (counting from left to right) is less than 5, then the last saved figure does not change.

Example: rounding number12,23 up to three meaning digits gives12,2.

3. In the event that the first of the discarded numbers (counting from left to right) is 5, then the last stored digit is increased by one.

Example: rounding number0,145 up to two digits gives0,15.

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

4. If the discarded digit is obtained as a result of rounding to a smaller side, the last remaining digit is increased by one (with the transition when necessary in the following discharges), otherwise - on the contrary. This also applies to fractional and integers.

Example: rounding number0,25 (resulting as a result of the previous rounding number0,252) gives0,3.

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

Example: rounding number0,156 up to two meaning digits gives0,16.

5. The rounding is performed immediately to the desired number of meaningful numbers, and not steps.

Example: rounding number565,46 up to three meaning digits gives565.

6. The integers are rounded in the same rules as fractional.

Example: rounding number23456 up to two meaning digits gives2310 3

The numeric value of the measurement result should be ended in the figure of the same discharge as the error value.

Example:Number235,732 + 0,15 must be rounded to235,73 + 0,15but not up235,7 + 0,15.

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

Example: 442,749+ 0,4 Rounded before442,7+ 0,4.

8. If the first of the discarded numbers is greater than or equal to five, then the last saved digit increases by one.

Example:37,268 + 0,5 Rounded before37,3 + 0,5; 37,253 + 0,5 must be rounded before37,3 + 0,5.

9. Rounding should be performed immediately until the desired number of significant digits, the phased rounding can lead to errors.

Example: phased rounding measurement result220,46+ 4 gives at the first stage220,5+ 4 And on the second221+ 4, while the correct rounding result220+ 4.

10. If the measurement error is indicated in total with one or two significant numbers, and the calculated value of the error is obtained with a large number of characters, in the final value of the calculated error, only the first one or two significant digits should be left accordingly. At the same time, if the resulting number begins with numbers 1 or 2, then discarding the second sign leads to a very large error (up to 3050%), which is unacceptable. If the obtained number begins with the figures 3 and more, for example, from the number 9, then the mainmark of the second sign, i.e. An indication of error, for example, 0.94 instead of 0.9, is disinformation, since the initial data does not provide such accuracy.

Based on this, this rule was established in practice: if the resulting number begins with a meaningful digit equal to or greater than 3, then it is stored only one; If it starts with significant digits smaller 3, i.e. With numbers 1 and 2, then it retains two meaning digits. In accordance with this rule, the normalized values \u200b\u200bof measurement errors are also established: two significant digits are indicated in numbers 1.5 and 2.5%, but in numbers 0.5; four; 6% indicate only one meaning digit.

Example:On the voltmeter class accuracy2,5 With the limit of measurements x TO = 300 In was obtained a countdown of the measured voltage x \u003d267,5 Q. In what form should the measurement result be recorded in the report?

The calculation of the error is more convenient to lead in the following order: first it is necessary to find an absolute error, and then relative. Absolute Error  h. =  0 h. TO / 100, for the above voltmeter error  0 \u003d 2.5% and measurement limits (measurement range) of the device h. TO \u003d 300 V:  h.\u003d 2.5300 / 100 \u003d 7.5 V ~ 8 V; Relative error  \u003d  h.100/h. = 7,5100/267,5 = 2,81 % ~ 2,8 % .

Since the first meaning number of the absolute error value (7.5 V) is more than three, this value must be rounded by the usual rounding rules to 8 B, but in the value of the relative error (2.81%) the first significant number of less than 3, so here Two decimal discharges should be saved in response and indicated  \u003d 2.8%. Received h.\u003d 267.5 V should be rounded to the same decimal discharge, which ends the rounded value of the absolute error, i.e. to whole units volts.

Thus, in the final response, it must be reported: "Measurement is made with a relative error  \u003d 2.8%. Measured voltage H.= (268+ 8) B.

At the same time, more clearly indicate the limits of the uncertainty interval of the measured value in the form H.\u003d (260276) in or 260 VX276 V.

In some cases, the exact number in dividing a certain amount to a specific number cannot be determined in principle. For example, when dividing 10 to 3, we turn out 3,3333333333 ... ..3, that is, this number cannot be used to count specific items and in other situations. Then this number should lead to a certain discharge, for example, to an integer or to a number with a decimal discharge. If we give 3.33333333333 ... ..3 to an integer, then we get 3, and leading 3,3333333333 ... ..3 to a number with a decimal discharge, we get 3.3.

Rules of rounding

What is rounding? This discarding several digits that are the last in a number of exact numbers. So, following our example, we dropped all the last figures to get an integer (3) and dropped the numbers, leaving only tens (3.3) discharges. The number can be rounded to hundredths and thousandth, ten-thousand and other numbers. It all depends on how exactly the exact number must be obtained. For example, in the manufacture of medicines, the amount of each of the ingredients of the drug is taken with the greatest accuracy, since even a thousandth gram can lead to a fatal outcome. If you need to calculate, what the performance of students in school, then the number with a decimal or a hundredth discharge is most often used.

Consider a different example in which rounding rules apply. For example, there is a number of 3,583333, which must be rounded up to thousands - after rounding, for the comma, we should remain three digits, that is, the result will be the number 3,583. If this number is rounded to the tenths, then we will succeed in 3,5, and 3.6, since after "5" there is a number "8", which is already equivalent to "10" during rounding. Thus, following the rules of rounding numbers, it is necessary to know if the numbers are greater than "5", then the last figure that needs to be kept will be increased by 1. If there is a figure, less than "5", the last saved digit remains unchanged. Such rules rounding numbers are applied regardless of whether to an integer or up to dozen hundredths, etc. It is necessary to round the number.

In most cases, if necessary, rounding the number in which the last digit "5" is incorrectly performed. But there is also such a rounding rule, which concerns exactly such cases. Consider on the example. It is necessary to round the number 3.25 to the tenths. Applying the rules of rounding numbers, we obtain the result of 3.2. That is, if after "five" no digit or worth zero, then the last figure remains unchanged, but only if it is even - in our case, "2" is an even figure. If we needed rounding 3.35, then the result was the number 3.4. Since, in accordance with the rules of rounding, in the presence of odd figures before "5", which must be removed, an odd figure increases by 1. But only if there are no significant digits after "5". In many cases, simplified rules can be applied according to which, if there is a digit value from 0 to 4, the saved digit is not changed. If there are other numbers, the latter digit increases by 1.

The numbers with which we have to deal in real life are two types. Some exactly transmit the true size, others - only approximate. The first call accurate, second - approximated.

In real life, most often use approximate numbers instead of accurate, since the latter are usually not required. For example, approximated values \u200b\u200bare used when specifying such values \u200b\u200bas a length or weight. In many same cases, it is impossible to find the exact number.

Rules of rounding

To obtain an approximate value obtained as a result of any actions, the number needs to be rounded, that is, replace it with the nearest round number.

The numbers are always rounded to a certain discharge. Natural numbers are rounded up to dozen, hundreds, thousands, etc. When rounding the numbers to dozen, they are replaced by round numbers consisting only of whole dozens, in such numbers in the discharge of units are zeros. When rounding to hundreds, the numbers are replaced with more round, consisting of only hundreds, that is, zeros are already in the discharge of units, and in the discharge of dozens. Etc.

Decimal fractions can be rounded as well as natural numbers, that is, up to dozen, hundreds, etc. But they can also be rounded to tenths, hundredths, thousandth parts, etc. When rounding decimal places, the discharge is not filled with zeros, but Just discarded. In both cases, rounding is made according to a specific rule:

If the discarded number is greater than or equal to 5, then the previous one needs to be increased by one, and if less than 5, the previous digit does not change.

Consider several examples of rounding numbers:

• Round 43152 to thousands. Here it is necessary to discard 152 units, since the digit 1 is worth the right of thousands of thousands, then the previous number retains unchanged. The approximate value of Number 43152, rounded to thousands will be 43000.
• Round 43152 to hundreds. The first of the discarded numbers 5, so the previous figure is increasing by one: 43152 ≈ 43200.
• Round 43152 to tens: 43152 ≈ 43150.
• Round out 17.7438 to units: 17,7438 ≈ 18.
• Round out 17.7438 to the tenths: 17,7438 ≈ 17.7.
• Round out 17.7438 to hundredths: 17,7438 ≈ 17.74.
• Round out 17,7438 to thousandths: 17,7438 ≈ 17,744.

The sign is called the sign of approximate equality, it is read - "approximately equal."

If the number when rounding the number it turned out more initial value, then the value obtained is called approximate extension valueif less - approximate value with disadvantage:

7928 ≈ 8000, the number 8000 is an approximate extension value
5102 ≈ 5000, the number 5000 is an approximate value with a disadvantage