The rule for rounding numbers to a given digit. Rules for rounding natural numbers

The Microsoft Excel program also works with numerical data. When performing division or working with fractional numbers, the program performs rounding. This is primarily due to the fact that absolutely exact fractional numbers are rarely needed, but it is not very convenient to operate with a cumbersome expression with several decimal places. In addition, there are numbers that, in principle, do not exactly round off. But, at the same time, insufficiently accurate rounding can lead to gross errors in situations where precision is required. Fortunately, in Microsoft Excel, it is possible for users to set how numbers will be rounded.

All numbers that Microsoft Excel works with are divided into exact and approximate. Numbers up to 15 digits are stored in memory, and are displayed up to the digit that the user himself indicates. But, at the same time, all calculations are performed according to the data stored in memory, and not displayed on the monitor.

With the rounding operation, Microsoft Excel discards a number of decimal places. Excel uses the conventional rounding method where a number less than 5 is rounded down, and a number greater than or equal to 5 is rounded up.

Rounding with Ribbon Buttons

The easiest way to change the rounding of a number is to select a cell or a group of cells, and being in the "Home" tab, click on the button "Increase bit depth" or "Decrease bit depth" on the ribbon. Both buttons are located in the "Number" toolbox. In this case, only the displayed number will be rounded, but for calculations, if necessary, up to 15 digits of numbers will be involved.

When you click on the "Increase bit depth" button, the number of entered decimal places is increased by one.

When you click on the "Decrease bit depth" button, the number of digits after the decimal point is reduced by one.

Rounding Through Cell Format

You can also set rounding using the cell format settings. To do this, you need to select a range of cells on the sheet, right-click, and select "Format Cells" from the menu that appears.

In the cell format settings window that opens, go to the "Number" tab. If the data format specified is not numeric, then you need to select the numeric format, otherwise you will not be able to adjust the rounding. In the central part of the window near the inscription "Number of decimal places" simply indicate the number of characters that we want to see when rounding. After that, click on the "OK" button.

Set calculation accuracy

If in previous cases, the set parameters only affected the external display of data, and more accurate indicators (up to 15 digits) were used in the calculations, now we will tell you how to change the very accuracy of the calculations.

The Excel Options window opens. In this window, go to the "Advanced" subsection. We are looking for a block of settings called "When recalculating this book." The settings in this section apply not to a single sheet, but to the entire book as a whole, that is, to the entire file. Put a check next to the "Set accuracy as on screen" option. Click on the "OK" button located in the lower left corner of the window.

Now, when calculating the data, the displayed value of the number on the screen will be taken into account, and not the one that is stored in Excel's memory. Setting the displayed number can be done in any of the two ways that we talked about above.

Application of functions

If you want to change the rounding value when calculating relative to one or more cells, but do not want to reduce the accuracy of calculations for the document as a whole, then in this case, it is best to use the opportunities provided by the ROUND function and its various variations, as well as some other features.

Among the main functions that regulate rounding, the following should be highlighted:

ROUND - rounds to the specified number of decimal places, according to generally accepted rounding rules;
ROUNDUP - rounds up to the nearest number up by the modulo;
ROUNDDOWN - rounds down to the nearest number in modulo;
ROUND - rounds a number with a given precision;
ROUNDUP - rounds a number with a given precision up in modulus;
ROUNDDOWN - rounds the number down modulo with the specified precision;
OTBR - rounds the data to an integer;
EVEN - rounds data to the nearest even number;
ODD - rounds the data to the nearest odd number.

For the ROUND, ROUNDUP, and ROUNDDOWN functions, the input format is: “Function name (number;number_digits). That is, if you, for example, want to round the number 2.56896 to three digits, then use the ROUND(2.56896; 3) function. The output is 2.569.

For the ROUND, ROUNDUP, and ROUNDUP functions, the following rounding formula is used: "Function name (number, precision)". For example, to round the number 11 to the nearest multiple of 2, enter the function ROUND(11;2). The output is 12.

The FIND, EVEN, and ODD functions use the following format: "Function name (number)". In order to round the number 17 to the nearest even number, use the EVEN(17) function. We get the number 18.

A function can be entered both in a cell and in a line of functions, having previously selected the cell in which it will be located. Each function must be preceded by an "=" sign.

There is a slightly different way to introduce rounding functions. It is especially useful when you have a table with values that need to be converted to rounded numbers in a separate column.

To do this, go to the Formulas tab. Click on the "Math" button. Next, in the list that opens, select the desired function, for example, ROUND.

After that, the function arguments window opens. In the "Number" field, you can enter a number manually, but if we want to automatically round the data of the entire table, then click on the button to the right of the data entry window.

The function arguments window is minimized. Now we need to click on the topmost cell of the column whose data we are going to round. After the value is entered in the window, click on the button to the right of this value.

The function arguments window opens again. In the field "Number of digits" we write the bit depth to which we need to reduce fractions. After that, click on the “OK” button.

As you can see, the number has been rounded. In order to round all other data of the desired column in the same way, hover over the lower right corner of the cell with the rounded value, click on the left mouse button, and drag it down to the end of the table.

After that, all values in the desired column will be rounded.

As you can see, there are two main ways to round the visible display of a number: using the button on the ribbon, and by changing the cell format options. In addition, you can change the rounding of actually calculated data. This can also be done in two ways: by changing the settings of the book as a whole, or by using special functions. The choice of a particular method depends on whether you are going to apply this kind of rounding to all data in the file, or only to a certain range of cells.

In approximate calculations, it is often necessary to round some numbers, both approximate and exact, that is, to remove one or more final digits. In order to ensure that a single rounded number is as close as possible to the number being rounded, certain rules must be followed.

If the first of the separated digits is greater than the number 5, then the last of the remaining digits is strengthened, in other words, it increases by one. Gain is also assumed when the first of the removed digits is 5 , followed by one or more significant digits.

The number 25.863 is rounded off as - 25.9. In this case, the digit 8 will be strengthened to 9 , since the first cut off digit 6 is greater than 5 .

The number 45.254 is rounded off as - 45.3. Here, the digit 2 will be boosted to 3 because the first digit to cut off is 5 , followed by the significant digit 1 .

If the first of the cut off digits is less than 5 , then no amplification is performed.

The number 46.48 is rounded off as - 46. The number 46 is closest to the rounded number than 47 .

If the digit 5 is cut off, and there are no significant digits behind it, then rounding is performed to the nearest even number, in other words, the last remaining digit remains unchanged if it is even, and amplifies if it is odd.

The number 0.0465 is rounded off as - 0.046. In this case, no amplification is done, since the last remaining digit 6 is even.

The number 0.935 is rounded off as - 0.94. The last digit left, 3, is reinforced because it is odd.

Rounding numbers

Numbers are rounded when full precision is not needed or possible.

Round number to a certain digit (sign), it means to replace it with a number close in value with zeros at the end.

Natural numbers are rounded up to tens, hundreds, thousands, etc. The names of the digits in the digits of a natural number can be recalled in the topic of natural numbers.

Depending on the digit to which the number should be rounded, we replace the digit with zeros in the digits of units, tens, etc.

If the number is rounded to tens, then zeros replace the digit in the unit digit.

If a number is rounded to the nearest hundred, then zero must be in both the units and tens places.

The number obtained by rounding is called the approximate value of this number.

Record the rounding result after the special sign "≈". This sign is read as "approximately equal".

When rounding a natural number to some digit, you must use rounding rules.

Underline the digit to which you want to round the number.
Separate all digits to the right of this digit with a vertical bar.
If the number 0, 1, 2, 3 or 4 is to the right of the underlined digit, then all digits that are separated to the right are replaced by zeros. The digit of the category to which rounding is left unchanged.
If the number 5, 6, 7, 8 or 9 is to the right of the underlined digit, then all the digits that are separated to the right are replaced by zeros, and 1 is added to the digit of the digit to which they were rounded.

Let's explain with an example. Let's round 57,861 to the nearest thousand. Let's follow the first two points from the rounding rules.

After the underlined digit is the number 8, so we add 1 to the thousands digit (we have it 7), and replace all the digits separated by a vertical bar with zeros.

Now let's round 756,485 to the nearest hundred.

Let's round 364 to tens.

3 6 |4 ≈ 360 - there is 4 in the units place, so we leave 6 in the tens place unchanged.

On the numerical axis, the number 364 is enclosed between two "round" numbers 360 and 370. These two numbers are called approximate values of the number 364 with an accuracy of tens.

The number 360 is approximate deficient value, and the number 370 is approximate excess value.

In our case, rounding 364 to tens, we got 360 - an approximate value with a disadvantage.

Rounded results are often written without zeros, adding the abbreviations "thousands." (thousand), "million" (million) and "billion." (billion).

8,659,000 = 8,659 thousand
3,000,000 = 3 million

Rounding is also used to roughly check the answer in calculations.

Before an exact calculation, we will estimate the answer by rounding the factors to the highest digit.

794 52 ≈ 800 50 ≈ 40,000

We conclude that the answer will be close to 40,000 .

794 52 = 41 228

Similarly, you can perform an estimate by rounding and when dividing numbers.

In some cases, the exact number when dividing a certain amount by a specific number cannot be determined in principle. For example, when dividing 10 by 3, we get 3.3333333333…..3, that is, this number cannot be used to count specific items in other situations. Then the given number should be reduced to a certain digit, for example, to an integer or to a number with a decimal place. If we convert 3.3333333333…..3 to an integer, we get 3, and if we convert 3.3333333333…..3 to a number with a decimal place, we get 3.3.

Rounding rules

What is rounding? This is the discarding of several digits that are the last in a series of exact numbers. So, following our example, we discarded all the last digits to get an integer (3) and discarded the digits, leaving only the tens digits (3,3). The number can be rounded to hundredths and thousandths, ten thousandths and other numbers. It all depends on how accurate the number needs to be. 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 of a gram can be fatal. If it is necessary to calculate the performance of students at school, then most often a number with a decimal or hundredth place is used.

Let's look at another example that uses rounding rules. For example, there is a number 3.583333, which must be rounded to thousandths - after rounding, we should have three digits behind the comma, that is, the result will be the number 3.583. If this number is rounded to tenths, then we get not 3.5, but 3.6, since after “5” there is the number “8”, which is already equal to “10” during rounding. Thus, following the rules for rounding numbers, you need to know that if the digits are greater than "5", then the last digit to be stored will be increased by 1. If there is a digit less than "5", the last stored digit remains unchanged. Such rules for rounding numbers apply regardless of whether they are up to an integer or up to tens, hundredths, etc. you need to round the number.

In most cases, if it is necessary to round a number in which the last digit is "5", this process is not performed correctly. But there is also a rounding rule that applies to just such cases. Let's look at an example. You need to round the number 3.25 to tenths. Applying the rules for rounding numbers, we get the result 3.2. That is, if there is no digit after “five” or there is zero, then the last digit remains unchanged, but only on condition that it is even - in our case, “2” is an even digit. If we were to round 3.35, the result would be 3.4. Since, in accordance with the rounding rules, if there is an odd digit before the “5” that needs to be removed, the odd digit is increased by 1. But only on the condition that there are no significant digits after the “5”. In many cases, simplified rules can be applied, according to which, if there are digits from 0 to 4 after the last stored digit, the stored digit does not change. If there are other digits, the last digit is incremented by 1.

5.5.7. Rounding numbers

To round a number to a certain digit, we underline the digit of this digit, and then we replace all the digits behind the underlined one with zeros, and if they are after the decimal point, we discard. If the first zero-replaced or discarded digit is 0, 1, 2, 3 or 4, then the underlined number leave unchanged. If the first zero-replaced or discarded digit is 5, 6, 7, 8 or 9, then the underlined number increase by 1.

Examples.

Round to whole:

1) 12,5; 2) 28,49; 3) 0,672; 4) 547,96; 5) 3,71.

Decision. We underline the number in the units (integer) category and look at the number behind it. If this is the number 0, 1, 2, 3 or 4, then the underlined number is left unchanged, and all the numbers after it are discarded. If the underlined number is followed by the number 5 or 6 or 7 or 8 or 9, then the underlined number will be increased by one.

1) 1 2 ,5≈13;

2) 2 8 ,49≈28;

3) 0 ,672≈1;

4) 54 7 ,96≈548;

5) 3 ,71≈4.

Round to tenths:

6) 0, 246; 7) 41,253; 8) 3,81; 9) 123,4567; 10) 18,962.

Decision. We underline the number that is in the category of tenths, and then we act according to the rule: we discard all those after the underlined number. If the underlined digit was followed by the number 0 or 1 or 2 or 3 or 4, then the underlined digit is not changed. If the underlined number was followed by the number 5 or 6 or 7 or 8 or 9, then the underlined number will be increased by 1.

6) 0, 2 46≈0,2;

7) 41, 2 53≈41,3;

8) 3, 8 1≈3,8;

9) 123, 4 567≈123,5;

10) 18, 9 62≈19.0. There is a six behind the nine, therefore, we increase the nine by 1. (9 + 1 \u003d 10) we write zero, 1 goes to the next digit and it will be 19. We just cannot write 19 in the answer, since it should be clear that we rounded up to tenths - the figure in the category of tenths should be. Therefore, the answer is: 19.0.

Round to hundredths:

11) 2, 045; 12) 32,093; 13) 0, 7689; 14) 543, 008; 15) 67, 382.

Decision. We underline the number in the hundredth place and, depending on which digit is after the underlined one, leave the underlined number unchanged (if it is followed by 0, 1, 2, 3 or 4) or increase the underlined number by 1 (if it is followed by 5, 6, 7, 8 or 9).

11) 2, 0 4 5≈2,05;

12) 32,0 9 3≈32,09;

13) 0, 7 6 89≈0,77;

14) 543, 0 0 8≈543,01;

15) 67, 3 8 2≈67,38.

Important: the last digit in the answer should be the digit in the digit to which you rounded.

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How to round a number to an integer

Applying the rounding rule for numbers, let's look at specific examples of how to round a number to an integer.

Rule for rounding a number to an integer

To round a number to an integer (or round a number to units), you must discard the comma and all numbers after the decimal point.

If the first of the discarded digits is 0, 1, 2, 3, or 4, then the number will not change.

If the first of the discarded digits is 5, 6, 7, 8, or 9, the previous digit must be increased by one.

Round a number to an integer:

To round a number to an integer, we discard the comma and all the numbers after it. Since the first discarded digit is 2, the previous digit is not changed. They read: "eighty-six point twenty-four hundredths is approximately equal to eighty-six whole."

Rounding the number to an integer, we discard the comma and all the numbers following it. Since the first of the discarded digits is 8, the previous one is increased by one. They read: "Two hundred and seventy-four point eight hundred and thirty-nine thousandths is approximately equal to two hundred and seventy-five whole."

When rounding a number to an integer, we discard the comma and all the numbers behind it. Since the first of the discarded digits is 5, we increase the previous one by one. They read: "Zero point fifty-two hundredths is approximately equal to one whole."

We discard the comma and all the numbers after it. The first of the discarded digits is 3, so we do not change the previous digit. They read: "Zero point three hundred ninety-seven thousandths is approximately equal to zero point."

The first of the discarded digits is 7, which means that we increase the digit in front of it by one. They read: "Thirty-nine point seven hundred and four thousandths is approximately equal to forty point." And a couple more examples for rounding a number to integers:

27 Comments

Incorrect theory about if the number 46.5 is not 47 but 46 it is also called banking rounding to the nearest even rounded if after the decimal point 5 and there is no number after it

Dear ShS! Perhaps (?), In banks, rounding occurs according to other rules. I don't know, I don't work in a bank. This site is about the rules that apply in mathematics.

how to round the number 6.9?

To round a number to an integer, you must discard all numbers after the decimal point. We discard 9, so the previous number should be increased by one. So 6.9 is approximately equal to seven integers.

In fact, the figure really does not increase if after the decimal point 5 in any financial institution

Um. In this case, financial institutions in matters of rounding are guided not by the laws of mathematics, but by their own considerations.

Please tell me how to round 46.466667. confused

If you want to round a number to an integer, then you must discard all the digits after the decimal point. The first of the discarded digits is 4, so we do not change the previous digit:

Dear Svetlana Ivanovna, You are not familiar with the rules of mathematics.

Rule. If the digit 5 is discarded, and there are no significant figures behind it, then rounding is performed to the nearest even number, i.e. the last digit stored is left unchanged if it is even, and amplifies if it is odd.

And Accordingly: Rounding the number 0.0465 to the third decimal place, we write 0.046. We do not make amplifications, since the last saved digit 6 is even. The number 0.046 is as close to the given value as 0.047.

Dear guest! Let it be known to you, in mathematics there are various rounding methods for rounding a number. At school, they study one of them, which consists in discarding the lower digits of the number. I am glad for you that you know another way, but it would be nice not to forget school knowledge.

Thank you very much! It was necessary to round 349.92. It turns out 350. Thanks for the rule?

how to round 5499.8 correctly?

If we are talking about rounding to an integer, then discard all the numbers after the decimal point. The discarded figure is 8, therefore, we increase the previous one by one. So 5499.8 is approximately equal to 5500 integers.

Good day!
But this question arose seyas:
There are three numbers: 60.56% 11.73% and 27.71% How to round up to whole numbers? That in the sum that 100 remained. If you just round up, then 61+12+28=101 There is a problem. (If, as you wrote, according to the “banking” method, in this case it will work, but in the case, for example, 60.5% and 39.5%, something will fall again - we will lose 1%). How to be?

O! the method from "guest 02.07.2015 12:11" helped
Thanks to"

I don't know, they taught me this in school:
1.5 => 1
1.6 => 2
1.51 => 2
1.51 => 1.6

Maybe that's how you were taught.

0, 855 to hundredths please help

0, 855≈0.86 (discarded 5, increase the previous figure by 1).

Round 2.465 to whole number

2.465≈2 (the first discarded digit is 4. Therefore, we leave the previous one unchanged).

How to round 2.4456 to an integer?

2.4456 ≈ 2 (since the first discarded digit is 4, we leave the previous digit unchanged).

Based on the rounding rules: 1.45=1.5=2, therefore 1.45=2. 1,(4)5 = 2. Is it true?

No. If you want to round 1.45 to an integer, discard the first digit after the decimal point. Since it's 4, we don't change the previous digit. Thus, 1.45≈1.

Round numbers in Excel in several ways. Using cell format and using functions. These two methods should be distinguished as follows: the first is only for displaying values or printing, and the second 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 the 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" - rounding 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 you to round to an integer by setting the number of digits - "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 display of large numbers, you need to change the format of the cell with this large number to "Numeric" (the fastest way is to 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".

When rounding, only the correct characters are left, 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 digit 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:

No. p / p	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 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 error of the caliper equal to 0.1 mm (h is measured with a caliper), the boundaries of the confidence interval should be determined by the formula:

Hence:

Let us calculate the average value of the density ρ:

Let's find an expression for 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

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

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The rounding of a natural number is understood as replacing it with such a number closest in value, in which one or several last digits in its record are replaced by zeros.

Rounding rule:

To round a natural number, you need to select the digit in the number entry to which rounding is performed.

The number written in the selected digit:

does not change if the digit following it on the right is 0, 1, 2, 3 or 4;

All digits to the right of this bit are replaced by zeros.

Example: 14 3 ≈ 140 (rounded to the nearest tens);
56 71 ≈ 5700 (rounded to the nearest hundred).

If the digit to which rounding is performed contains the number 9 and it is necessary to increase it by one, then the digit 0 is written in this digit, and the digit in the adjacent high-order digit (on the left) is increased by 1.

Example: 79 6 ≈ 800 (rounded to tens);
9 70 ≈ 1000 (rounded to the nearest hundred).

Rounding decimals

To round a decimal fraction, you need to select the digit in the number entry to which rounding is performed. The number written in this category:

increments by one if the next digit to the right is 5,6,7,8, or 9.

All digits to the right of this bit are replaced by zeros. If these zeros are in the fractional part of the number, then they are not written.

Example: 143,6 4 ≈ 143.6 (rounded to tenths);
5,68 7 ≈ 5.69 (rounded to hundredths);
27 .945 ≈ 28 (rounded to the nearest integer).

If the digit to which rounding is performed contains the number 9 and it is necessary to increase it by one, then the digit 0 is written in this digit, and the figure in the previous digit (on the left) is increased by 1.

Example: 8 9, 6 ≈ 90 (rounded to tens);
0,09 7 ≈ 0.10 (rounded to hundredths).

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Rounding numbers

1) Rules for rounding natural numbers. Natural numbers are rounded to units of a certain digit. To round a natural number to units of a certain digit means to establish how many units of this digit are contained in a given number. For example, we want to round the number 237456 to the nearest thousand. This means to find out how many thousands there are in this number. Obviously, it has 237 thousand. How did we know? To do this, we all the digits of a given number up to the thousands place, i.e. hundreds, tens and ones, replaced with zeros and got the number 237000, which can be written as follows: 237 thousand. But you can, knowing that 1000=10 3, write this rounded number like this: 237 * 10 3 .

So, 237456? 237 thousand or 237 456? 237*10 3 .

Please note that here we did not put the usual equal sign, but approximate equal sign (?).

Why such a sign? Yes, because the numbers 237,456 and 237 thousand are not equal, the second number is somewhat less than the first, namely, less than 456, therefore, replacing the number 237,456 with the number 237 thousand, we thereby make an error equal to 456, which means that the numbers 237,456 and 237,000 are only approximately equal. Therefore, the sign of approximate equality is put. Note that the error in rounding the number 237,456 to thousands was 456 units, which is less than half of one thousand. Therefore, if we need to round the number 237 873 to thousands, then it is more reasonable to take 237 thousand as the rounded value of the number 237 873, then let's make an error equal to 873, which is more than half a thousand, i.e. 500. If the rounded value is 238 thousand , then the error will be only 127, which is much less than half a thousand. From these examples, we can deduce the following the general rule for rounding natural numbers to units of a certain digit: replace all digits to the right of this digit with zeros. If the first digit on the left of those replaced by zeros is less than 5, then the rounding is completed and the resulting rounded number can be written in an abbreviated form. If it is equal to or greater than 5, then the digit of the digit to which rounding was performed is replaced by a larger one.

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Rounding natural numbers.

We often use rounding in everyday life. If the distance from home to school is 503 meters. We can say, by rounding up the value, that the distance from home to school is 500 meters. That is, we have brought the number 503 closer to the more easily perceived number 500. For example, a loaf of bread weighs 498 grams, then by rounding the result we can say that a loaf of bread weighs 500 grams.

rounding- this is the approximation of a number to a “lighter” number for human perception.

The result of rounding is approximate number. Rounding is indicated by the symbol ≈, such a symbol reads “approximately equal”.

You can write 503≈500 or 498≈500.

Such an entry is read as “five hundred three is approximately equal to five hundred” or “four hundred ninety-eight is approximately equal to five hundred”.

Let's take another example:

4 4 71≈4000 4 5 71≈5000

4 3 71≈4000 4 6 71≈5000

4 2 71≈4000 4 7 71≈5000

4 1 71≈4000 4 8 71≈5000

4 0 71≈4000 4 9 71≈5000

In this example, numbers have been rounded to the thousands place. If we look at the rounding pattern, we will see that in one case the numbers are rounded down, and in the other - up. After rounding, all other numbers after the thousands place were replaced by zeros.

Number rounding rules:

1) If the figure to be rounded is equal to 0, 1, 2, 3, 4, then the digit of the digit to which the rounding is going does not change, and the rest of the numbers are replaced by zeros.

2) If the figure to be rounded is equal to 5, 6, 7, 8, 9, then the digit of the digit up to which the rounding is going on becomes 1 more, and the remaining numbers are replaced by zeros.

1) Round to the tens place of 364.

The digit of tens in this example is the number 6. After the six is the number 4. According to the rounding rule, the number 4 does not change the tens digit. We write zero instead of 4. We get:

2) Round to the hundreds place of 4781.

The hundreds digit in this example is the number 7. After the seven is the number 8, which affects whether the hundreds digit changes or not. According to the rounding rule, the number 8 increases the hundreds place by 1, and the rest of the numbers are replaced by zeros. We get:

3) Round to the thousands place of 215936.

The thousands place in this example is the number 5. After the five is the number 9, which affects whether the thousands place changes or not. According to the rounding rule, the number 9 increases the thousands place by 1, and the remaining numbers are replaced by zeros. We get:

21 5 9 36≈21 6 000

4) Round to the tens of thousands of 1,302,894.

The thousand digit in this example is the number 0. After zero, there is the number 2, which affects whether the tens of thousands digit changes or not. According to the rounding rule, the number 2 does not change the digit of tens of thousands, we replace this digit and all digits of the lower digits with zero. We get:

13 0 2 894≈13 0 0000

If the exact value of the number is not important, then the value of the number is rounded off and you can perform computational operations with approximate values. The result of the calculation is called estimation of the result of actions.

For example: 598⋅23≈600⋅20≈12000 is comparable to 598⋅23=13754

An estimate of the result of actions is used in order to quickly calculate the answer.

Examples for assignments on the topic rounding:

Example #1:
Determine to what digit rounding is done:
a) 3457987≈3500000 b) 4573426≈4573000 c) 16784≈17000
Let's remember what are the digits on the number 3457987.

7 - unit digit,

8 - tens place,

9 - hundreds place,

7 - thousands place,

5 - digit of tens of thousands,

4 - hundreds of thousands digit,
3 is the digit of millions.
Answer: a) 3 4 57 987≈3 5 00 000 digit of hundreds of thousands b) 4 57 3 426 ≈ 4 57 3 000 digit of thousands c) 1 6 7 841 ≈ 1 7 0 000 digit of tens of thousands.

Example #2:
Round the number to 5,999,994 places: a) tens b) hundreds c) millions.
Answer: a) 5 999 99 4 ≈5 999 990 b) 5 999 9 9 4 ≈6 000 000 994≈6,000,000.

Rules for rounding natural numbers

Rules for rounding natural numbers.
Rounding a number up to some digit.

From time to time, a population census is conducted in the country. Every day people are born, die, change their place of residence, so the number of inhabitants is constantly changing. Let's say that there are 34,489 inhabitants in one city. Accordingly, when people move in this number, the numbers of the digits of units, tens and even hundreds will change. Such numbers are replaced with zeros, and we get a simpler number. It can be said that he lives in the city approximately 34,000 inhabitants.

The number 34 489 was rounded up to 3 thousand 4 000.
If we want to round some number, then we apply the rule:
45|245 - the line shows to what digit we want to round.

If the first digit following the digit to which the number is rounded (to the right of the bar) is 5, 6, 7, 8, 9, then the last remaining digit is increased by 1, and the rest of the digits after the dash are replaced by zeros. In other cases, the last remaining digit is not changed.

The given number and the number obtained by rounding it approximately equal.This is written with the sign » » «.
45|245 » 45,000, since the digit following the thousands place is 2.
124 7 | 89 » 124 800, since the digit following the hundreds place is 8.

Round the numbers 12,344; 12,343; 12,342; 12 340; 12,341 to tens.
.

Rounding of natural numbers is used when calculating the price. Subtractions are made orally, an estimate of the result is made. For example:
358 56 = 20,048

For simplified multiplication, round each number:
358 » 400 and 56 » 60 400 x 60 = 24 000

It can be seen that this answer is approximately equal to the first answer.

1. Give examples where you can use rounding numbers..
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2. Explain to what digit the numbers are rounded. The first column has been rounded to the nearest tens. The second column has been rounded to the nearest thousand.

6789 » 6800 . 12 897 » 10 000 .
12 544 » 12 500 . 2 344 672 » 2 340 000 .
245 673 » 245 700 . 78 358 » 78 360 .
26 577 » 30 000 . 34 057 123 » 34 100 000 .