Rounding down to 1000. Rounding a number to the nearest fractional value

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:

44 71≈4000 45 71≈5000

43 71≈4000 46 71≈5000

42 71≈4000 47 71≈5000

41 71≈4000 48 71≈5000

40 71≈4000 49 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.

For example:

1) Round to the tens place of 364.

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

36 4 ≈360

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:

47 8 1≈48 00

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:

215 9 36≈216 000

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:

130 2 894≈130 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) 4 573 426 ≈ 4 573 000 digit of thousands c) 16 7 841 ≈17 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) 5,999,994 ≈5,999,990 b) 5,999,99 4≈6,000,000 6,000,000.

Methods

Different fields may use different methods of 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 “up” or “down” - in both cases an error is introduced exactly in 1/2 of the least significant digit . There are the following variants of the rounding rule to the nearest integer for this case:

• Mathematical rounding- rounding is always up (the previous digit is always increased by one).
• Bank rounding(English) banker's rounding) - rounding for this case occurs to the nearest even number, i.e. 2.5 → 2, 3.5 → 4.
• Random rounding- rounding up or down 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 simply formally corresponds to the general rounding rule (see above). Its disadvantage is that when rounding a large number of values, accumulation can occur. rounding errors. 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 when rounding a large number of values. Rounding "to the nearest even" is based on the assumption that with a large number of 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, thus 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 about half of the special values ​​are rounded one way and half the other. But the implementation of such methods in practice requires additional efforts to organize the computational process.

Applications

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

Using rounding when working with numbers of limited precision

Real physical quantities are always measured with some finite accuracy, which depends on the instruments and methods of measurement and is estimated by the maximum relative or absolute deviation of the unknown actual value from the measured one, which in decimal representation of the value corresponds either to a certain number of significant digits, or to a certain position in the 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 rule of error of the multiplication operation (when multiplying approximate values, the relative errors add up) will be 7,3 10 −3 , which corresponds to the maximum absolute error of the result ±0.059 kgf m! That is, in reality, taking into account the error, the result can be from 8.082 to 8.200 kgf m, thus, in the calculated value of 8.141 kgf m, only the first digit is completely reliable, even the second is already doubtful! It will be correct to round the calculation result to the first doubtful digit, that is, to tenths: 8.1 kgf m, or, if necessary, a more accurate indication of the margin of error, present it in a form rounded to one or two decimal places with an indication of the error: 8.14 ± 0.06 kgf m.

Empirical rules of arithmetic with rounding

In cases where there is no need to accurately take into account computational errors, but only 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:

1. All raw values ​​are rounded up 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).
2. Intermediate values ​​are rounded off with one spare digit.
3. When adding and subtracting, the result is rounded to the last decimal place of the least accurate of the parameters (for example, when calculating a value of 1.00 m + 1.5 m + 0.075 m, the result is rounded to tenths of a meter, that is, to 2.6 m). At the same time, it is recommended to perform calculations in such an order as to avoid subtracting numbers that are close in magnitude and to perform operations on numbers, if possible, in ascending order of their modules.
4. When multiplying and dividing, the result is rounded 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).
5. When calculating the function value f(x) it is required to estimate the value of the modulus of the derivative of this function in the vicinity of the calculation point. If a (|f"(x)| ≤ 1), then the result of the function is exact to the same decimal place as the argument. Otherwise, the result contains fewer exact decimal places by the amount log 10 (|f"(x)|), rounded to the nearest integer.

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

Mistakes

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

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

see also

• Observation Processing
• Rounding errors

Notes

Literature

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

Rounding numbers is the simplest mathematical operation. To be able to correctly round numbers, you need to know three rules.

Rule 1

When we round a number to a certain digit, we must get rid of all the digits to the right of that digit.

For example, we need to round the number 7531 to the nearest hundred. This number is five hundred. To the right of this category are the numbers 3 and 1. We turn them into zeros and get the number 7500. That is, rounding the number 7531 to hundreds, we got 7500.

When rounding fractional numbers, everything happens in the same way, only the extra digits can simply be discarded. Let's say we need to round the number 12.325 to tenths. To do this, after the decimal point, we must leave one digit - 3, and discard all the numbers to the right. The result of rounding the number 12.325 to tenths is 12.3.

Rule 2

If to the right of the remaining digit the discarded digit is 0, 1, 2, 3 or 4, then the digit we leave does not change.

This rule worked in the previous two examples.

So, when rounding the number 7531 to hundreds, the closest figure to the discarded figure was a three. Therefore, the number we left - 5 - has not changed. The rounding result is 7500.

Similarly, when 12.325 was rounded to tenths, the digit we dropped after the three was a two. Therefore, the rightmost of the remaining digits (three) did not change during rounding. It turned out 12.3.

Rule 3

If the leftmost of the discarded digits is 5, 6, 7, 8, or 9, then the digit to which we round is increased by one.

For example, you need to round the number 156 to tens. There are 5 tens in this number. In the digit of units, which we are going to get rid of, is the number 6. So, we should increase the tens digit by one. Therefore, when rounding the number 156 to tens, we get 160.

Consider an example with a fractional number. For example, we are going to round 0.238 to the nearest hundredth. By rule 1, we must discard the eight, which is to the right of the hundredth place. And according to rule 3, we have to increase the three in the hundredth place by one. As a result, rounding the number 0.238 to hundredths, we get 0.24.

Let's look at examples of how to round up to tenths of a number using the rounding rules.

Rule for rounding numbers to tenths.

To round a decimal to tenths, you must leave only one digit after the decimal point, and discard all other digits following it.

If the first of the discarded digits is 0, 1, 2, 3, or 4, then the previous digit is not changed.

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

Examples.

Round to tenths:

To round a number to tenths, leave the first digit after the decimal point, and discard the rest. Since the first discarded digit is 5, we increase the previous digit by one. They read: "Twenty-three point seventy-five hundredths is approximately equal to twenty-three point eight."

To round this number to tenths, leave only the first digit after the decimal point, discard the rest. The first discarded digit is 1, so the previous digit is not changed. They read: "Three hundred and forty-eight point thirty-one hundredth is approximately equal to three hundred and forty-one point three."

Rounding to tenths, we leave one digit after the decimal point, and discard the rest. The first of the discarded digits is 6, which means that we increase the previous one by one. They read: "Forty-nine point, nine hundred and sixty-two thousandths is approximately equal to fifty point, zero tenths."

We round up to tenths, so after the comma we leave only the first of the digits, the rest are discarded. The first of the discarded digits is 4, which means we leave the previous digit unchanged. They read: "Seven point twenty-eight thousandths is approximately equal to seven point zero tenths."

To round to tenths, this number leaves one digit after the decimal point, and discard all following after it. Since the first discarded digit is 7, therefore, we add one to the previous one. They read: "Fifty-six point eight thousand seven hundred and six ten-thousandths is approximately equal to fifty-six point nine-tenths."

And a couple more examples for rounding to tenths:

Fractional numbers in Excel spreadsheets can be displayed to varying degrees. accuracy:

• most simple method - on the tab " home» press the buttons « Increase bit depth" or " Decrease bit depth»;
• click right click by cell, in the drop-down menu, select " Cell Format...”, then the tab “ Number", select the format" Numerical”, determine how many decimal places there will be after the decimal point (2 decimal places are suggested by default);
• click the cell, on the tab " home» choose « Numerical", or go to " Other number formats...” and configure there.

Here's what the fraction 0.129 looks like if you change the number of decimal places in the cell format:

Please note that A1,A2,A3 have the same meaning, only the form of representation changes. In further calculations, not the value visible on the screen will be used, but original. For a novice spreadsheet user, this can be a bit confusing. To really change the value, you need to use special functions, there are several of them in Excel.

Rounding formula

One of the commonly used rounding functions is ROUND. It works according to standard mathematical rules. Select a cell, click the " Insert function”, category “ Mathematical", we find ROUND

We define the arguments, there are two of them - herself fraction and amount discharges. We click " OK' and see what happens.

For example, the expression =ROUND(0.129,1) will give a result of 0.1. The zero number of digits allows you to get rid of the fractional part. Choosing a negative number of digits allows you to round the integer part to tens, hundreds, and so on. For example, the expression =ROUND(5,129,-1) will give 10.

Round up or down

Excel provides other tools that allow you to work with decimals. One of them - ROUNDUP, gives the closest number, more modulo. For example, the expression =ROUNDUP(-10,2,0) will give -11. The number of digits here is 0, which means we get an integer value. nearest integer, greater in modulus, - just -11. Usage example:

ROUNDDOWN similar to the previous function, but returns the closest value that is smaller in absolute value. The difference in the work of the above means can be seen from examples: