Mathematical abbreviations of numbers in which direction. Some important rules for rounding numbers

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

What is a round number anyway? It's the one that ends in 0 (for the most part). In everyday life, the ability to round numbers greatly facilitates shopping trips. Standing at the checkout, you can roughly estimate the total cost of purchases, compare how much a kilogram of the same product costs in packages of different weights. With numbers reduced to a convenient form, it is easier to make mental calculations without resorting to the help of a calculator.

Why are numbers rounded up?

A person tends to round 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 a southern fruit has, he may be considered not a very interesting interlocutor. Phrases like "So I bought a three-kilogram melon" sound much more concise without delving into 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 infinite fractions, which have the form 3.33333333 ... 3, then this becomes impossible. Therefore, the most logical option would be to simply round them. As a rule, the result after that is distorted slightly. So how do you round numbers?

Some important rules for rounding numbers

So, if you want 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 perform this action, you need to know a few important rules:

If the number of the required digit is in the range of 5-9, rounding up is carried out.
If the number of the desired digit is between 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. That's the answer to the question of how to round numbers. Now let's consider special cases. Actually, we figured out how to round a number to tens using this example. Now it remains only to put this knowledge into practice.

How to round a number to integers

It often happens that there is a need to round, 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 place, and we get 6.0. And since zeros in decimals are usually 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 rounding up to 6 becomes legal. 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 approximately the same way. If the digit that is in the second position after the decimal point is within 5-9, then it is generally removed, 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" goes away, and one is added to the five. But when rounding 4.41, the unit is omitted, and the four remains unchanged.

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

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

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

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

Understand the meaning of numbers in decimals. In any number, different digits represent different digits. For example, in the number 1872, one represents thousands, eight represents hundreds, seven represents tens, and two represents ones. If there is a decimal point in the number, then the numbers to the right of it reflect fractions of a whole number.

Determine the decimal place to which you want to round it. The first step in rounding decimals is determining the place to which you want to round a number. If you do homework, then this is usually determined by the task condition. Often, the condition may indicate the need to round the answer to tenths, hundredths, or thousandths of a decimal point.

For example, if the task is to round the number 12.9889 to thousandths, you should start by identifying the location of these thousandths. Count the decimal places as tenths, hundredths, thousandths, followed by ten thousandths. The second eight will be just what you need (12.98 8 9).
Sometimes a condition may specify where to round (for example, "round to three decimal places" means the same as "round to thousandths").

Look at the number to the right of where you want to round off. Now you should find out the number that is to the right of the place to which you are rounding. Depending on this figure, you will round up or down (up or down).

In the example of the number (12.9889) taken earlier, it is necessary to round to thousandths (12.98 8 9), so now you should look at the number to the right of the thousandth, namely the last nine (12.988 9 ).

If this figure is greater than or equal to five, then rounding up is performed. For greater clarity, if the number 5, 6, 7, 8 or 9 is to the right of the rounding point, then rounding up is performed. In other words, it is necessary to increase the digit at the rounded place by one, and discard the remaining digits to the right of it.

In the example taken (12.9889), the last nine is greater than five, so we will round the thousandths to the big side. The rounded number will appear as 12,989 . Note that after the rounding point, the figures are discarded.

If this figure is less than five, then rounding down is performed. That is, if the number 4, 3, 2, 1 or 0 is to the right of the rounding point, then rounding down is performed. Which means the need to leave the figure in place of the rounding in the form in which it is, and discard the numbers to the right of it.

You cannot round down 12.9889 because the last nine is not a four or less. However, if the number in question were 12.988 4 , then it could be rounded up to 12,988 .
Does the procedure sound familiar? This is due to the fact that integers are rounded in the same way, and the presence of a comma does not change anything.

Use the same method to round decimals to whole numbers. Often the task establishes the need to round the answer to integers. In this case, you must use the above method.

In other words, find the location of the integer units of the number, look at the number on the right. If it is greater than or equal to five, then round the whole number up. If it is less than or equal to four, then round the whole number down. The presence of a comma between the integer part of the number and its decimal fraction does not change anything.
For example, if you want to round the above number (12.9889) to integers, you would start by locating the integer units of the number: 1 2 .9889. Since the nine to the right of this place is greater than five, we round up to 13 whole. Since the answer is represented by an integer, there is no need to write a comma anymore.

Pay attention to rounding instructions. The above rounding instructions are generally accepted. However, there are situations where special rounding requirements are given, be sure to read them before resorting to the generally accepted rounding rules right away.

For example, if the requirements say to round down to tenths, then in the number 4.59 you will leave a five, despite the fact that a nine to the right of it should usually result in rounding up. This will give you the result 4,5 .
Similarly, if you are told to round the number 180.1 to whole to the big side, then you will succeed 181 .

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."

Lesson content

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 special mark. 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, the following 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 "rounding" numbers when dividing large 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 in the tens place instead of a 5.

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 finding the nearest number, you can apply ready-made rules that make rounding numbers much easier.

First rounding rule

From the previous examples, it became clear that when rounding a number to a certain digit, the 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 retained 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 retained 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.

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 approximate number 700.

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 stored 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, 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 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 the decimal fraction 123.456 - 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 deal with the rounding of 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 stored number 5 will increase by one, and the rest will be replaced by zeros

123,456 ≈ 123,460

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

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

How to round a number by cell format

Let's enter the value 76.575 in cell A1. By right-clicking, we call the "Format Cells" menu. You can do the same through the "Number" tool on the main page of the Book. Or press the hot key combination CTRL+1.

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

Rounding result:

You can assign the number of decimal places in the "monetary" format, "financial", "percentage".

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

The peculiarity of this option: the more digits after the decimal point we leave, the more accurate the result will be.

How to round a number correctly in Excel

Using the ROUND() function (rounds to the number of decimal places required by the user). To call the "Function Wizard" use the fx button. The desired function is in the "Math" category.

Arguments:

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

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

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

How to round a number in Excel to thousands?

An example of rounding a number to thousands:

Formula: =ROUND(A3,-3).

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

Suppose there is data on the price and quantity of goods. It is necessary to find the cost to the nearest ruble (round to the nearest whole number).

The first argument of the function is a numeric expression for finding the cost.

How to round up and down in Excel

To round up, use the ROUNDUP function.

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

The second argument: "0" - rounds the decimal fraction to the integer part, "1" - the function rounds, leaving one decimal place, etc.

Formula: =ROUNDUP(A1,0).

Result:

To round down in Excel, use the ROUNDDOWN function.

Formula example: =ROUNDDOWN(A1,1).

Result:

The ROUNDUP and ROUNDDOWN formulas are used to round expression values (products, sums, differences, etc.).

How to round to whole number in Excel?

To round up to a whole number, use the ROUNDUP function. To round down to a whole number, use the ROUNDDOWN function. The "ROUND" function and the cell format also allow rounding to an integer by setting the number of digits to "0" (see above).

Excel also uses the "SELECT" function to round to a whole number. It simply discards the decimal places. Basically, there is no rounding. The formula cuts off the numbers to the designated digit.

Compare:

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

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

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

You can round up to a whole number in Excel using the ROUNDUP and ROUNDDOWN functions. Rounding occurs up or down to the nearest whole number.

An example of using functions:

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

Rounding to the nearest even integer is performed by the "EVEN" function, to the nearest odd - "ODD".

An example of their use:

Why does Excel round large numbers?

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

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 character< 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 particular case is characterized by the fact that for a single rounding it is formally indifferent whether it is 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 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 randomly, 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 just formally corresponds to general rule rounding (see above). Its disadvantage is that when rounding a large number of values, accumulation can occur. rounding errors. A typical example: rounding up 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 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 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, 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 a certain 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 notation of a number, all the numbers after (to the right) of which are insignificant (they 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 precision 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 result of calculations to the first doubtful figure, 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 need to approximately estimate the number of exact numbers as a result of the calculation by the formula, you can use a set of simple rules for rounded calculations:

All raw values are rounded to the actual measurement accuracy and recorded with the 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, 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).
Intermediate values are rounded off with one "spare" digit.
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.
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 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 distance has two digits and time has three, assuming all digits in the entry are significant).
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 (|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".