The order in which numbers are rounded after the decimal point. How to round numbers up and down using Excel functions

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

Examples.

Round to whole numbers:

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

Solution. We underline the number in the units (integer) place and look at the number behind it. If this is the number 0, 1, 2, 3 or 4, then we leave the underlined number unchanged, and discard all the numbers after it. If the underlined number is followed by the number 5 or 6 or 7 or 8 or 9, then we will increase the underlined number by one.

1) 12 ,5≈13;

2) 28 ,49≈28;

3) 0 ,672≈1;

4) 547 ,96≈548;

5) 3 ,71≈4.

Round to the nearest tenth:

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

Solution. We underline the number in the tenths place, and then proceed according to the rule: we discard everything after the underlined number. If the underlined number was followed by the number 0 or 1 or 2 or 3 or 4, then we do not change the underlined number. 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. Behind nine there is a six, therefore, we increase nine by 1. (9+1=10) we write zero, 1 goes to the next digit and it will be 19. We just can’t write 19 in the answer, since it should be clear that we rounded to tenths - the number must be in the tenths place. Therefore, the answer is: 19.0.

Round to the nearest hundredth:

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

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

11) 2, 04 5≈2,05;

12) 32,09 3≈32,09;

13) 0, 76 89≈0,77;

14) 543, 00 8≈543,01;

15) 67, 38 2≈67,38.

Important: the last answer should contain a number in the digit to which you rounded.

Mathematics. 6 Class. Test 5 . Option 1 .

1. Infinite decimal non-periodic fractions are called... numbers.

A) positive; IN) irrational; WITH) even; D) odd; E) rational.

2 . When rounding a number to any digit, all digits following this digit are replaced with zeros, and if they are after the decimal point, they are discarded. If the first digit replaced by a zero or discarded is 0, 1, 2, 3 or 4, then the digit preceding it is not changed. If the first digit replaced by a zero or discarded is 5, 6, 7, 8 or 9, then the digit preceding it is increased by one. Round number to tenths 9,974.

A) 10,0;B) 9,9; C) 9,0; D) 10; E) 9,97.

3. Round number to tens 264,85 .

A) 270; B) 260;C) 260,85; D) 300; E) 264,9.

4 . Round to whole numbers 52,71.

A) 52; B) 52,7; C) 53,7; D) 53; E) 50.

5. Round to the nearest thousand 3, 2573 .

A) 3,257; B) 3,258; C) 3,28; D) 3,3; E) 3.

6. Round number to hundreds 49,583 .

A) 50;B) 0; C) 100; D) 49,58;E) 49.

7. An infinite periodic decimal fraction is equal to an ordinary fraction whose numerator is the difference between the entire number after the decimal point and the number after the decimal point before the period; and the denominator consists of nines and zeros, and there are as many nines as there are digits in the period, and as many zeros as there are digits after the decimal point before the period. 0,58 (3) to ordinary.

8. Convert an infinite periodic decimal fraction 0,3 (12) to ordinary.

9. Convert an infinite periodic decimal fraction 1,5 (3) into a mixed number.

10. Convert an infinite periodic decimal fraction 5,2 (144) into a mixed number.

11. Any rational number can be written Write down the number 3 as an infinite periodic decimal fraction.

A) 3,0 (0);IN) 3,(0); WITH) 3;D) 2,(9); E) 2,9 (0).

12 . Write a common fraction ½ as an infinite periodic decimal fraction.

A) 0,5; B) 0,4 (9); C) 0,5 (0); D) 0,5 (00); E) 0,(5).

You will find answers to the tests on the “Answers” page.

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Methods

Different areas may use different rounding methods. In all these methods, “extra” signs are reset (discarded), and the sign preceding them is adjusted according to some rule.

Round to the nearest integer(English) rounding) - the most commonly used rounding, in which a number is rounded 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 to the Nth decimal place, the rule can be formulated as follows:
- If N+1 sign< 5 , then the Nth sign is retained, and N+1 and all subsequent ones are reset to zero;
- If N+1 character ≥ 5, then the Nth sign is increased by one, and N+1 and all subsequent ones are reset to zero;
For example: 11.9 → 12; −0.9 → −1; −1,1 → −1; 2.5 → 3.
Rounding down modulo(round to zero, integer English) fix, truncate, integer) is the “simplest” rounding, since after zeroing out the “extra” signs, the previous sign is retained. For example, 11.9 → 11; −0.9 → 0; −1,1 → −1).
Round up(round to +∞, round up, eng. ceiling) - if the zeroing signs are not equal to zero, the previous sign is increased by one if the number is positive, or retained if the number is negative. In economic jargon - rounding in favor of the seller, creditor(person receiving money). In particular, 2.6 → 3, −2.6 → −2.
Round down(round to −∞, round down, English. floor) - if the zeroing signs are not equal to zero, the previous sign is retained if the number is positive, or increased by one if the number is negative. In economic jargon - rounding in favor of the buyer, debtor(the person giving the money). Here 2.6 → 2, −2.6 → −3.
Rounding up modulo(round toward infinity, round away from zero) is a relatively rarely used form of rounding. If the zeroing signs are not equal to zero, the preceding sign is increased by one.

Options for rounding 0.5 to the nearest integer

Rounding rules require a separate description 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 is introduced of exactly 1/2 of the least significant digit . There are the following options for the rounding to the nearest integer rule for this case:

Mathematical rounding- rounding is always upward (the previous digit is always increased by one).
Bank rounding(English) banker's rounding) - rounding for this case occurs to the nearest even number, that is, 2.5 → 2, 3.5 → 4.
Random rounding- rounding occurs up or down in a random order, but with equal probability (can be used in statistics).
Alternate rounding- rounding occurs downward or upward alternately.

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

Mathematical rounding simply formally follows the general rounding rule (see above). Its disadvantage is that when rounding a large number of values, accumulation may occur. rounding errors. A typical example: rounding monetary amounts to whole rubles. So, if in a register of 10,000 lines there are 100 lines with amounts containing the value of 50 in kopecks (and this is a very realistic estimate), then when all such lines are rounded “up”, the “total” amount for the rounded register will be 50 rubles more than the exact one .

The other three options were invented precisely in order to reduce the overall error of the sum when rounding a large number of values. Rounding “to the nearest even” is based on the assumption that if there are a large number of rounded values that have a 0.5 remainder, on average half will end up to the left and half to the right of the nearest even number, thus canceling out rounding errors. 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, account amounts, 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 better.

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 represent real values measured in one way or another), the actually achievable accuracy of the calculations, or the desired accuracy of the result. In the past, rounding intermediate values and results was of practical importance (since 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 rounding, may be required to protect against computational errors associated with the finite 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 measurement methods and is estimated by the maximum relative or absolute deviation of the unknown real value from the measured one, which in the decimal representation of the value corresponds to either a certain number of significant digits or a certain position in the recording of a number, all the numbers after (to the right) of which are insignificant (are 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 accuracy 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 some of these digits are significant. The remaining numbers, while present in the values, do not actually reflect any physical reality and only take up 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 obtained values. In practice, it is usually recommended to store one more digit in intermediate values for long "chain" manual calculations. When using a computer, intermediate rounding in scientific and technical applications most often loses its meaning, and only the result is rounded.

So, for example, if a force of 5815 gf is given, accurate to the nearest gram of force, and the arm length is 1.4 m accurate to the 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, we find that the maximum 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 figure is completely reliable, even the second is already doubtful! It would be correct to round the calculation result to the first dubious digit, that is, to tenths: 8.1 kgf m, or, if it is necessary to more accurately indicate the scope of the error, present it in the form rounded to one or two decimal places indicating the error: 8.14 ± 0.06 kgf m.

Rules of thumb for 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 calculation using the formula, you can use a set of simple rules for rounded calculations:

All original values are rounded to actual measurement accuracy and written with the appropriate number of significant digits, so that in decimal notation all digits are reliable (the last digit is allowed to be doubtful). If necessary, values are written with significant right-hand zeros so that the record indicates the actual number of reliable characters (for example, if a length of 1 m is actually measured to the nearest centimeter, write “1.00 m” to show 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 to them).
Intermediate values are rounded with one “spare” digit.
When adding and subtracting, the result is rounded to the last decimal place of the least accurate parameter (for example, when calculating the value 1.00 m + 1.5 m + 0.075 m, the result is rounded to the nearest tenth of a meter, that is, to 2.6 m). In this case, 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 increasing order of their modules.
When multiplying and dividing, the result is rounded to the smallest number of significant figures that the parameters have (for example, when calculating the speed of uniform motion of a body at a distance of 2.5 10 2 m, in 600 s the result should be rounded to 4.2 m/s, since it is distance has two digits, and time has three, assuming that all digits in the entry are significant).
When calculating the function value f(x) it is required to estimate the modulus of the derivative of this function in the vicinity of the calculation point. If (|f"(x)| ≤ 1), then the function result is accurate to the same decimal place as the argument. Otherwise, the result contains fewer exact decimal places by the amount log 10 (|f"(x)|), rounded up to the nearest whole number.

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

Errors

Abuse of non-round numbers is quite common. For example:

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

Notes

Literature

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

Having learned to multiply multi-digit numbers “in a column”, we became convinced that this is a very dreary task. Fortunately, we won't be doing this for long. Soon we will do all any complex calculations using a calculator. Now we practice counting solely for educational purposes, in order to better understand and feel the “behavior” of numbers. However, understanding and instinct can be honed with no less success on approximate calculations, which are much simpler. We will now proceed to them.

Let's say we want to buy five chocolates for 19 rubles. We look at our wallet and want to quickly figure out whether we have enough money for this. We reason like this: 19 is approximately 20, and 20 multiplied by 5 is 100. Here we have just over a hundred rubles in our wallet. So there is enough money. A mathematician would say that we rounded nineteen to twenty and did some approximation. But let's start from the beginning.

First of all, let's make a reservation that at first we will be rounding only positive numbers. This can be done in different ways. For example, like this:

The “≈” symbol is read as “approximately equal.” Here, as they say, we rounded the numbers down and, accordingly, received a lower estimate. This is done very simply: we leave the first digit of the number as it is, and replace all subsequent ones with zeros. It is clear that the result of such rounding is always less than or equal to the original number.

On the other hand, numbers can also be rounded up, thus obtaining an upper estimate:

With this rounding, all digits, starting from the second, turn to zero, and the first digit increases by one. A special case arises when the first digit is equal to nine, which is replaced by two digits at once, 1 and 0:

The result of rounding up is always greater than or equal to the original number.

Thus, we have a choice in which direction to round: up or down. Usually they round in the direction that is closest. Obviously, in most cases it is better to round 11 to 10, and 19 to 20. The formal rules are as follows: if the second digit of our number is in the range from zero to 4, then we round down. If this figure is in the range from 5 to 9, then up. Thus:

98 765 ≈ 100 000.

Separately, we should note the situation when the second digit of a number is five, and all subsequent digits are equal to zero, for example 1500. This number is at the same distance from both 2000 and 1000:

2000 − 1500 = 500,

1500 − 1000 = 500.

Therefore, it would seem that it doesn’t matter which way to round it. However, it is customary to round it not anywhere, but only up - so that the rounding rules can be formulated as simply as possible. If we see a five in second place, then this is already enough to make a decision about where to round: we don’t have to be at all interested in subsequent numbers.

Using the rounding of numbers, we can now quickly, albeit approximately, solve multiplication examples of any complexity. Suppose we need to calculate:

We round both factors and in a couple of seconds we get:

6879 ∙ 267 ≈ 7000 ∙ 300 = 2,100,000 ≈ 2,000,000 = 2 million.

For comparison, I will give the exact answer that we calculated when we learned to multiply by column:

6879 ∙ 267 = 1 836 693.

What needs to be done now to understand whether the approximate answer is close or far from the exact one? - Of course, round off the exact answer:

6879 ∙ 267 = 1,836,693 ≈ 2,000,000 = 2 million.

It turned out that after rounding, the exact answer became equal to the approximate one. So our approximate answer is not so bad. However, it should be noted that such accuracy is not always achieved. Let's say we need to calculate 1497∙143. Approximate calculations look like this:

1497 ∙ 143 ≈ 1000 ∙ 100 = 100,000 = 100 thousand.

And here is the exact answer (with subsequent rounding):

1497 ∙ 143 = 214,071 ≈ 200,000 = 200 thousand.

Thus, the exact answer after rounding turned out to be 2 times larger than the approximate one. This, of course, is not very good. But I admit honestly: I deliberately took one of the worst cases. Usually the accuracy of approximate calculations is still better.

However, we have so far rounded numbers and made approximate calculations only in the most, so to speak, rough form. Of all the digits of the number, we left only one unzeroed - the most significant one. They say that we rounded numbers to one significant figure. However, we can round more accurately, for example, to two significant figures:

The rule here is almost the same as before. All digits except the two most senior ones are zeroed. If the first of the zeroed digits contained a number ranging from zero to 4, then we do nothing more. If this figure was in the range from 5 to 9, then add one to the last of the non-zero digits. Note that if there is a nine in the digit to which a unit is added, then this digit is overflowed and reset to zero, and the higher digit “inherits” the one. That is, this is what happens:

195 ≈ 190 + 10 = 200,

or even:

995 ≈ 990 + 10 = 1000.

Rounding to three significant figures, and so on, is defined in the same way.

Let's return to our example. Let's see what happens if we round numbers not to one, but to two significant figures:

1497 ∙ 143 ≈ 1500 ∙ 140 = 210,000 = 210 thousand.

And let’s compare it again with the exact answer:

1497 ∙ 143 = 214,071 ≈ 210,000 ≈ 210 thousand.

Isn't it true that our approximate calculation has become noticeably more accurate?

And here is another familiar example, for which we will write two versions of approximate answers and compare them with the exact answer:

6879 ∙ 267 ≈ 7 000 ∙ 3 00 = 2 100 000 ≈ 2 000 000,

6879 ∙ 267 ≈ 69 00 ∙ 27 0 = 1 863 000 ≈ 1 9 00 000,

6879 ∙ 267 = 1836693 ≈ 1 8 00 000 ≈ 2 000 000.

This is the time to mention this rule: If the factors are rounded to one significant figure, then the approximate answer should be immediately rounded to one significant figure. If the factors are rounded to two significant figures, then the answer must be rounded to two significant figures. In general, as many significant figures as the factors have, the same number of significant figures must remain in the product. Therefore, in the first line, having barely received 2,100,000, we immediately rounded this number to 2,000,000. Likewise in the second line: we did not stop at the intermediate result of 1,863,000, but immediately rounded it to 1,9,00,000 . Why is that? Because in the number 2,100,000, all digits except the very first are still calculated incorrectly. Likewise, in the number 1,863,000, all digits except the first two are incorrectly calculated. Let's take a look at the corresponding calculations done "in a column":

Here, the exact calculations are reproduced on the left, and the approximate calculations on the right, performed after rounding the factors to two significant figures. Instead of zeros, we wrote circles to emphasize that in fact behind these circles-zeros there are some other numbers that, after rounding, became unknown to us. Without knowing all the numbers in the first two lines, we also cannot calculate all the numbers in the subsequent lines - that’s why there are circles there too. Now let's take a closer look: in the two highest ranks we don't see any circles anywhere. This means that in the response line these bits are calculated more or less accurately. But already in the third highest rank there is one circle, which means a figure unknown to us. Therefore, we actually cannot calculate the third digit in the response line. This is especially true for the fourth and subsequent categories. It is these digits with unknown values that must be set to zero during subsequent rounding.

But what, I wonder, will happen if one of the factors is rounded to three significant figures, and the other - to only one? Let's see what the calculation will look like in this case:

We see that only the most significant digit is determined with any certainty, so the answer must be rounded to one significant figure:

6879 ∙ 267 ≈ 6880 ∙ 3 00 = 2 064 000 ≈ 2 000 000

We also see that the significant figure (in this case, 2) may differ from the true figure (in this case, 1), but, as a rule, by no more than one.

In general, we should focus on the factor with the smallest number of significant digits: we should round the answer to exactly the same number of significant digits.

So far we have only talked about approximate multiplication. What about addition? - Of course, addition can also be approximate. Just rounding the terms, preparing them for approximate addition, is not necessary in exactly the same way as we rounded the factors, preparing them for approximate multiplication. Let's look at an example:

61 238 + 349 = 61 587.

To begin with, let’s round each of the terms to one significant figure:

61 238 + 349 ≈ 60 000 + 300 = 60 300 ≈ 60 000.

Or, if you write it in a column:

61 238 + 349 ≈ 60 000 + 000 = 60 000.

Here we can write 0 instead of the second term, or, as they say, completely neglect it in comparison with the first term. Let's try to increase the accuracy of our calculations. Now round to two significant figures:

61 238 + 349 ≈ 61 000 + 350 = 61 350 ≈ 61 000.

Again, we could immediately neglect the second term and write:

61 238 + 349 ≈ 61 000 + 0 = 61 000.

Only when we increase the rounding precision to three significant figures does the second term begin to play some role:

61 238 + 349 ≈ 61 200 + 349 = 61 549 ≈ 61 500.

However, we again overdid it with the accuracy of the second term: for it, one significant figure would have been enough:

61 238 + 349 ≈ 61 200 + 300 = 61 500.

The following rule applies here: terms, unlike factors, should be rounded not to the same number of significant figures, but to the same digit. To round to the tens place means to round so that the last significant digit of the rounding result is in the tens place. When rounding to the hundreds place, the last significant digit is in the hundreds place, and so on. The approximate answer is immediately rounded to the required accuracy and does not require further rounding. Let's write out our example again, calculating it with varying accuracy:

61,238 + 349 = 61,587 (exact calculation),

61,238 + 349 ≈ 61,240 + 350 = 61,590 (rounded to the nearest ten),

61,238 + 349 ≈ 61,200 + 300 = 61,500 (up to hundreds),

61,238 + 349 ≈ 61,000 + 0 = 61,000 (up to thousands),

61,238 + 349 ≈ 60,000 + 0 = 60,000 (up to tens of thousands),

61,238 + 349 ≈ 100,000 + 0 = 100,000 (up to hundreds of thousands).

It should be noted that when rounding the second term (349) to thousands (and, especially, to higher digits), the result is zero. Here in the last line we also encounter another remarkable case:

61 238 ≈ 100 000,

when a number is rounded to a higher place than those contained in itself - and yet the result of such rounding turns out to be different from zero.

Let us now consider approximate subtraction. We know that subtraction can be thought of simply as a form of addition. Therefore, the rules for approximate subtraction generally coincide with the rules for approximate addition. However, a special situation is possible here, which arises when we calculate the difference between numbers that are close to each other. Let's say you want to roughly estimate what the value of the expression is:

After roughly rounding the difference terms we get:

Let's face it, it didn't turn out very well. The exact value, as can be easily calculated, is:

7654 − 7643 = 11.

Still, there is a considerable difference between zero and eleven! Therefore, even with the roughest estimates, it is customary to round off the difference terms to such a level that the result is still different from zero:

7654 − 7643 ≈ 7650 − 7640 = 10.

Here's another problem that can happen during approximate subtraction:

We got as much as a thousand in the answer, while the exact value of the difference is only one! Here we must look carefully and not allow what is called a formalist approach.

However, situations are possible when the difference value needs to be calculated with an accuracy to some predetermined digit, for example, to the thousand digit. In this case, it is quite acceptable to write exactly like this:

7654 − 7643 ≈ 8000 − 8000 = 0.

2500 − 2499 ≈ 3000 − 2000 = 1000.

Formally, we are absolutely right. We are mistaken in the thousands place by no more than one unit, and this is a completely common thing when we work with such precision that the last significant digit falls exactly in the thousands place. Likewise, to the nearest hundreds:

7654 − 7643 ≈ 7700 − 7600 = 100.

2500 − 2499 ≈ 2500 − 2500 = 0.

Although approximate calculations are a fairly simple thing, you cannot approach it completely thoughtlessly. Each time, the accuracy of the approximation must be chosen based on the task at hand and common sense.

We just have to consider approximate division. Looking ahead, I will say that division can be considered as a type of multiplication. Therefore, the rules for approximate division are the same as in the case of multiplication: the dividend and the divisor must be rounded to the same number of significant figures, and the same number of significant figures must remain in the answer.

But we still haven’t really gone through the division. We know how to divide by a whole and divide with a remainder, but we still cannot divide “in an adult way”, without a remainder, one arbitrary number by another. Therefore, for now we will develop, so to speak, temporary rules of approximate division that correspond to our current understanding of the subject. For now we will only divide roughly, with an accuracy of one significant figure.

Suppose we need to calculate approximately:

First of all, round the divisor (324) to one significant figure:

76 464 / 324 ≈ 76 464 / 300.

Now let's compare the only significant digit of the divisor (3) with the first digit of the dividend (7). Here, in principle, two cases are possible. The first case is when the first digit of the dividend is greater than or equal to the only significant digit of the divisor. We will now consider this case, because it is the one that is implemented in this example, since 7 ≥ 3. Now we zero all the digits of the dividend, except for the highest one, and round the value of the highest digit to the nearest number divisible by the significant digit of the divisor:

76 464 / 324 ≈ 76 464 / 300 ≈ 90 000 / 300.

Note that according to standard rounding rules, 76,464 ≈ 80,000, however, since 8 is not evenly divisible by 3, we “went even further up” so that we ended up with 76,464 ≈ 90,000. Next, the dividend and of the divisor, we simultaneously remove the same number of “extra zeros” from the tail:

76 464 / 324 ≈ 76 464 / 300 ≈ 90 000 / 300 = 900 / 3.

After this, division is not difficult:

76 464 / 324 ≈ 76 464 / 300 ≈ 90 000 / 300 = 900 / 3 = 300.

The approximate answer is ready. Let me give you the exact answer for comparison:

76 464 / 324 = 236 ≈ 200.

As you can see, the discrepancy in the only significant figure of the approximate answer is one unit, which is quite acceptable.

Let us now complete the following approximate calculations:

35 144 / 764 ≈ 35 144 / 800.

This is the second case we have mentioned where the first digit of the dividend is less than the only significant digit of the rounded divisor (3< 8). В этом случае мы зануляем все разряды делимого, кроме двух самых старших, а то число, которое образует эти два старших разряда, «подтягиваем» к ближайшему числу, которое можно поделить нацело на единственную значащую цифру делителя:

35 144 / 764 ≈ 35 144 / 800 ≈ 32 000 / 800.

(If you can “pull up” with equal success in both directions, then “pull up”, for definiteness, upwards.) Now we remove the “extra” zeros and perform division:

35 144 / 764 ≈ 35 144 / 800 ≈ 32 000 / 800 = 320 / 8 = 40.

The exact calculation is:

35 144 / 764 = 46 ≈ 50.

And again, the accuracy of the approximate result is quite acceptable.

It should be noted that even numbers that are not completely divisible by each other can be divided approximately. It is only important (for now) that the dividend be greater than or equal to the divisor.

At the end of this lesson, we just need to figure out how to round negative numbers and how to do approximate calculations with them. In fact, for any negative number we can always write something like this:

−3456 = −(+3456).

Here we have a positive number in brackets. We will round it according to the rules that we have developed for positive numbers. For example, if it needs to be rounded to two significant figures, then we get:

−3456 = −(+3456) ≈ −(+3500) = −3500.

It’s just as easy to replace all calculations with negative numbers with calculations involving only positive numbers. For example,

−234 − 567 = −(234 + 567) ≈ −(200 + 600) = −(800) = −800,

234 − 567 = −(567 − 234) ≈ −(600 − 200) = −(400) = −400,

234 ∙ (−567) = −(234 ∙ 567) ≈ −(200 ∙ 600) = −(120 000) = −120 000.

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

Rule 1

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

For example, we need to round the number 7531 to hundreds. This number includes five hundred. To the right of this digit 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 the same way, only the extra digits can simply be discarded. Let's say we need to round the number 12.325 to the nearest tenth. To do this, after the decimal point we must leave one digit - 3, and discard all the digits to the right. The result of rounding the number 12.325 to tenths is 12.3.

Rule 2

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

This rule worked in the two previous examples.

So, when rounding the number 7531 to hundreds, the closest digit to the one left was three. Therefore, the number we left - 5 - has not changed. The result of rounding was 7500.

Similarly, when rounding 12.325 to the nearest tenth, the digit we dropped after the three was the two. Therefore, the rightmost digit left (three) did not change during rounding. It turned out to be 12.3.

Rule 3

If the leftmost digit to be discarded 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 units place, which we are going to get rid of, there is a number 6. This means that we should increase the tens place by one. Therefore, when rounding the number 156 to tens, we get 160.

Let's look at an example with a fractional number. For example, we're going to round 0.238 to the nearest hundredth. According to Rule 1, we must discard the eight, which is to the right of the hundredths place. And according to rule 3, we will have to increase the three in the hundredths place by one. As a result, rounding the number 0.238 to hundredths, we get 0.24.

Many people are interested in how to round numbers. This need often arises among people who connect their lives with accounting or other activities that require calculations. Rounding can be done to whole numbers, 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? This is the one that ends in 0 (for the most part). In everyday life, the ability to round numbers makes shopping trips much easier. Standing at the checkout, you can roughly estimate the total cost of purchases and compare how much a kilogram of the same product costs in bags of different weights. With numbers reduced to a convenient form, it is easier to make mental calculations without resorting to a calculator.

Why are numbers rounded?

People tend to round any numbers in cases where it is necessary to perform more simplified operations. For example, a melon weighs 3,150 kilograms. When a person tells his friends about how many grams the southern fruit has, he may be considered a not very interesting interlocutor. Phrases like “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 possible. But 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 is then slightly distorted. So how do you round numbers?

Some important rules when rounding numbers

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

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

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

How to round a number to whole numbers

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 we round, the already familiar number 60 appears before our eyes. Now we put the comma in place, and we get 6.0. And since zeros in decimal fractions 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 it becomes legal to round up to 6. But this trick doesn’t always work, so you need to be extremely careful.

In principle, an example of correct rounding of a number to tenths has already been discussed above, so now it is important to display only the main principle. Essentially, everything happens in approximately the same way. If the digit that is in the second position after the decimal point is in the range of 5-9, then it is removed altogether, and the digit in front of it is increased by one. If it is 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” disappears, and one is added to the five. But when rounding 4.41, the unit is omitted, and the four remains unchanged.

How do marketers take advantage of the mass consumer's inability to round numbers?

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

Very often, rounding allows for a better assessment of intermediate successes expressed in numerical form. For example, a person began to earn $550 a month. An optimist will say that it is almost 600, a pessimist will say 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 a huge number of examples where the ability to round turns out to be incredibly useful. It is important to be creative and avoid loading yourself with unnecessary information whenever possible. Then success will be immediate.