One of the most significant and memorable events in life caring owner cats is considered the appearance of offspring in an animal. Especially in a situation if everything is happening for the first time. At the same time, when a pet gives birth, she will definitely need the help, love and care of her owners.

If an animal should bring kittens from day to day, then each caring owner should know the signs of impending birth in a cat. However, most of the animals able to give birth without any intervention, therefore, the role of a person in this matter mainly consists of observing from the side and helping if it is really necessary. But you need to prepare for possible surprises, and you need to know how to induce labor in a cat in advance.

Since pregnancy in cats is quite fleeting, it is important to know the signs of labor so as not to miss the time of birth of a new life. The first symptoms begin and fade a couple of weeks before the crucial moment arrives. The offspring in the womb begins vigorous activity, and the cat itself scours the apartment in search of a secluded place. In turn, a few hours before the kittens are born the animal's habits change dramatically:

• the cat becomes worried and begins to rush around the apartment, licking its tummy and genitals;
• at the moment when contractions appear, the pet begins to breathe quickly and heavily;
• indicators rectal temperature fall below 37 ºC, which is normal;
• there is swelling of the genital organs, accompanied by discharge;
• Colostrum is released through the swollen mammary glands.

Almost all cats, before giving birth, meow loudly and hide in a secluded place where no one will disturb them.

A healthy animal does not require human intervention during pregnancy and childbirth. However, some small breeds cats complications may occur. In European countries, obstetrics for animals has been developing for many years. But in our case, all responsibility is always placed on the shoulders of the pet owner and veterinarians general profile. Normal pregnancy in an animal it lasts for 8–9 weeks. At the same time, veterinarians there are 3 main stages.

1. During the first 3 weeks, mating occurs, immediately after which the animal may experience swelling of the mucous membrane of the genital organs, although this is not a 100% indicator that the cat is pregnant. You can clearly determine the piquant position of your pet after 3 weeks. The animal constantly becomes sleepy, appetite and activity decrease. The cat requires more attention and affection. Vomiting and enlargement of the uterus may appear, which can only be determined by a veterinarian during palpation.
2. Occurs from 4 to 6 weeks accelerated growth kittens as a result of which the cat’s stomach becomes noticeably rounder. By the end of 6 weeks, movement of the cubs may be observed. Kittens can be felt by probing the cat with your fingers. The animal eats and sleeps a lot.
3. At 7–9 weeks, the kittens’ movement becomes very energetic, which can be clearly observed even with the naked eye, especially during the animal’s sleep. The cat becomes anxious and is constantly looking for a den.

Finding a secluded place may not occur in every pet. Some are especially social active cats They will hope that their owner will take care of the secluded refuge, who will help in the process when the pet gives birth. Pets that are too attached to their owners must be monitored with increased care. Hormonal imbalance and the playful nature of the animal can cause rash actions in the cat, as a result of which labor failure or premature contractions may occur.

The first birth in cats is considered the most difficult. In this case, it will be difficult not only for the pet, but also for its owner. In the event of any deviations during childbirth, the life of the offspring, as well as the mother in labor, will be in the hands of the animal’s owners. If it is not possible to call an experienced veterinarian, then you will have to do everything yourself. But a few weeks before the cat gives birth, it is better stock up on everything you need:

Usage medical supplies advisable only in emergency situations. Oxytocin is used to further stimulate contractions. They give an injection of the drug into the muscle, 0.2 ml, but only after the second kitten has come out. Interval between injections should be from 40 minutes to an hour. Complications during childbirth in cats are rare, but you need to prepare for them in advance.

1. The baby's head has appeared, but its full birth does not take place for a long time. In such a situation, the animal’s genitals and the kitten’s head are lubricated with Vaseline. This will normalize the process.
2. After the baby appears in the bubble, the cat does not try to free it from there. You cannot wait more than 1 minute; it is important to cut the bubble and remove the fruit in time. In order for a newborn creature to begin to breathe, it must be bent so that the hind legs touch the chest. If necessary, this procedure must be done repeatedly.
3. After the birth of the cub, the placenta did not come out. In this case, it is removed manually.
4. In case of mechanical blockade, when the fetus is placed incorrectly or is too large, you must call a veterinarian. All independent attempts to help to a pet may lead to irreversible consequences.
5. If uterine atony is observed, when the contraction force is not enough for the fetus to come out on its own, the process is stimulated by an injection of oxytocin.

If during childbirth the cat develops intense discharge rich blood color, the animal loses orientation, and sometimes consciousness, the temperature rises sharply, and unpleasant odors You should immediately call a veterinarian. If this does not work for some reason, you need to ask for help by phone. A specialist will always tell you how to help an animal in a given situation.

The first harbinger imminent birth the cat has sudden change in behavior animal a day before the birth of offspring. The main changes begin in the animal's uterus and may be accompanied by painful sensations. At the same time, the same symptoms accompany premature birth, and even a miscarriage. The only difference between pathology during childbirth is the rapid occurrence of all processes.

Very often, before immediate birth, a mucous clot may come out of the cervix. At the same time, it is difficult to notice the rejection of the cork. Often it comes out at the moment of urination, and given that the cat is constantly licking itself, the owners may simply not pay attention to this. Signs that a cat is about to give birth include strong contractions that are very noticeable, even visually, as the cat begins to settle down and arch her back. In this case, the first cub should appear within 5–60 minutes.

When all the signs that the animal will give birth are present, need to prepare the place where a cat and kittens will live. This secluded nest must be protected from drafts and prying eyes. A regular box or a special cat house is suitable for this. It is also worth worrying about the soft bedding on the bottom of the house. But you shouldn’t use expensive fabric, as it will have to be thrown away after the birth.

Different animals during fights may behave differently. For example, breed Scottish cats before giving birth he begins to thrash about. Your beloved pet may climb onto a closet, hide under the bed, or hide in the nightstand. In such a situation, the owners should calm and caress the animal, moving it to a cozy house in which it should get used to it. If the animal was nevertheless born outside cat house, then you need to take it and the offspring that appear and move it to its place.

There is a lot of talk now about childbirth that takes place in full accordance with the laws of nature. What is it - natural childbirth? Why did this concept arise when it would seem that it is already clear that childbirth is a natural process? What stages of childbirth are usually distinguished in this process? What does it take to successfully give birth yourself? These many other questions are answered by a body therapist, medical psychologist and mother of seven children Yulia Gusakovskaya-Starovoitova.

40 weeks: should we give birth “on schedule”?

In our tradition, the doctor determines the due date by counting 40 weeks from the day the last menstruation began. As soon as this date has arrived, he is obliged to send the woman to the maternity hospital, since his observation period has ended. But from a physiological point of view, the date of birth does not depend on the doctor’s calculations, but on a number of factors.

The first one is woman's cycle, the duration of which affects the development of the child. If it is classic - 28 days, then pregnancy really often lasts about 40 weeks. But if the cycle is shorter or longer, then there is a tendency for the duration of pregnancy to change accordingly. In this case 37-38 weeks will be normal- with a cycle of 25-27 days and 41-43 weeks- with a cycle of 29-32 days. In general, it is considered physiological if birth occurs between 37 and 43 weeks.

Harbingers of labor: how to determine the approaching moment

Precursors are a number of symptoms that indicate that the body has begun preparing for birth process. They appear in about 2-3 weeks before the onset of labor. What applies to them?

The first harbinger: With it becomes easier to breathe

The diaphragm puts less pressure on the lungs and stomach, and heartburn disappears. This happens because the body begins to more actively produce special hormones prostaglandins and under their influence the muscles of the lower abdomen begin to relax. Due to this, the child drops lower and his head enters the pelvis.

As a result, other difficulties arise - it becomes harder to walk, the feeling of pressure on the pelvis and organs lying under the uterus increases, and urination becomes more frequent. There is a feeling of fullness, and since the child is also putting pressure on the intestines, diarrhea may occur out of the blue.

Harbinger two: release of mucus plug

This is a clot of fairly dense mucus, about the size of a tablespoon. The traffic jam is always there, but it can come out in different ways. Sometimes it comes out gradually, and then the woman doesn’t notice it at all. If the plug comes off a few days before giving birth, it is most often colored brownish, and if this happens immediately in front of them, the mucus has a bloody tint.

Pay attention! Sometimes, after a doctor’s examination on the eve of childbirth, a woman experiences symptoms within 24 hours. spotting. This is not a sign of impending labor, but only a reaction to examination, since the tissues of the cervix become very loose and easily vulnerable.

The third harbinger: nesting instinct intensifies

The woman begins to work more actively around the house, the list of things to do before giving birth triples or quadruples, You may feel a panicky feeling that you are not getting anything done. At the same time, the desire to give birth as quickly as possible and to free oneself from one’s burden intensifies.

Harbinger Four: the baby begins to move less

His head is in the pelvis, he seems to calm down, only from time to time making local movements with his arms or legs.

The fifth harbinger: false contractions

They differ from genuine ones by the lack of dynamics. woman can seize every 3-7 minutes, but no enhancement is noted, i.e. the force of muscle compression is constant. The period of false contractions, as a rule, starts at the same time: either they begin in the afternoon and subside in the evening, or in the morning, disappearing by noon. False contractions may continue for two to three weeks.

Yulia Gusakovskaya - Starovoitova,

Dear readers! In your opinion, is natural childbirth right? How was your birth? We are waiting for your answers in the comments.

Ovulation is called the most important stage female menstrual cycle when the follicle ruptures and from the ovary to fallopian tube an egg arrives, ready for fertilization. Only the release of an egg - healthy and mature - makes pregnancy possible, and therefore the ovulation period is so important for planning conception.

When does ovulation occur?

The release of the egg is a kind of peak of the menstrual cycle. Ovulation occurs approximately in the middle of the cycle. If it is standard and fluctuates somewhere between 27-29 days, then the egg is released 12-16 days before the start of the next menstruation. As a rule, with a clear 28-day cycle, which is observed in most women, this is the 14th day.

Since everyone’s indicators are individual, and cycle failure due to many external factors happens quite often, it is possible to understand that ovulation has occurred only by minor changes in well-being.

Conception is possible not only directly on the day of ovulation. The lifespan of an egg is about 24 hours, and the lifespan of a young sperm is healthy man- from 72 hours or more. It doesn’t matter at all when their fusion occurs, and therefore any of the sexual acts 3-4 days before ovulation and a day after it can result in conception.

By the way, it is precisely because of this instability and relativity of timing that you should not use ovulation calculation as a method of contraception - the probability of error here is very high. In addition, there are many exceptions: for some women, ovulation occurs twice during a cycle, often the cycle shifts, and quite unexpectedly, by several days, and it is quite difficult to find out about such failures. Why then do you need to know the time when ovulation occurs?

Why determine ovulation?

• The release of the egg marks the beginning of the period when you can conceive a child. For families who carefully plan their future, as well as for those who have problems conceiving, knowing this period is very important.
• The timely onset of ovulation, as well as other stages of the cycle, is an indicator of a woman’s health. If for several months a couple makes attempts to conceive a child, including having sex during the period of ovulation, but pregnancy does not occur, then there is some kind of failure. It will be possible to determine what exactly was the root of the problem after undergoing an examination, and both partners must undergo it.
• It also happens that a woman’s body functions normally and there are no abnormalities or diseases, but her husband’s spermogram is not normal. Spermatozoa are immobile, not tenacious, and are produced in small quantities. In this case, knowing the date of ovulation is especially important: only sexual intercourse 1-2 days before it or on the day the egg is released can lead to conception. In this case, it is advisable to refrain from making love 3-4 days before ovulation: then sperm accumulate, and thanks to them high concentration the chances of a successful conception become much greater.

There is also an opinion that by calculating the day of ovulation you can plan the gender of the child. Sperm with a “male” Y chromosome are usually more mobile, but last relatively short time, while “female” X chromosomes live longer. In this case, theoretically, if sexual intercourse takes place on the day of ovulation, the “male” sperm will be the first to enter the egg, and if 2-3 days before it, only the “female” part will survive. Doctors do not confirm this theory, and statistics speak in their favor. This method, like many others, has only a 50% effect, which is similar to the “randomness” that nature provides.

How to determine ovulation

Symptoms of ovulation are always quite subjective; some women experience them all at the same time, while others do not recognize any of them. However, in most cases, if the cycle is normal and there are no diseases, you can determine the approaching release of the egg by the following symptoms.

1. Change in the nature of discharge

Even between menstruation, a small amount of cervical mucus is secreted from the uterus - a characteristic odorless, transparent or white. If ovulation does not occur, the mucus changes its character: it becomes thick white, dense, thick. In case of failure, discharge may be completely absent - this is the same problem as its atypical appearance or smell. If everything is normal, the released liquid has a neutral odor and is transparent. Heavy discharge Signs of this nature may indicate the approach of ovulation or its beginning.

This method of determination can be used as an auxiliary one; it is not very reliable, since a change in the nature of the discharge is also observed when

• Vaginal infections;
• Venereal diseases;
• Strong sexual arousal;
• Regular use of a lubricant for successful sexual intercourse.

The method has its pros and cons.

Advantages:

• Does not require financial costs.
• Allows you to determine the release of the egg quickly and in a timely manner.
• Quite accurate, since discharge is always present during ovulation.

Flaws:

• Heavy discharge, which can be mistakenly taken as a signal of the onset of ovulation, is accompanied by a host of other processes.
• Discharge decreases when treated with Clomid or antihistamines, and ovulation occurs as usual. In this case, cervical fluid cannot serve as an indicator.

2. Noticeable increase in sexual desire

Oddly enough, nature took care of procreation on its own. During periods of increased fertility (that is, when successful conception can occur), a woman begins to experience especially strong sexual desire. Medical research also confirm this fact. Approaching ovulation suggests a change hormonal levels. Hormones are the factor that makes a woman want to make love more often.

Advantages of the method:

• No special devices, analyzes or tests are required - just listen to your feelings.
• The general emotional uplift has a good effect on the body.

Cons:

• Inaccuracy of the method: sexual desire in many women, especially young women, is associated not only with physiological, but also with emotional factors, and therefore can be experienced at other times.
• Stress, depression, and tension can interfere with determining the deadline. Sexual desire is a priori reduced if a diagnosis of “infertility” has already been made; it may also be absent due to more trivial reasons: a cold, runny nose, overwork.

3. Changes in the basal temperature chart

This method is one of the most reliable, as it takes into account the unconditional factor of ovulation, which is always present, namely an increase basal temperature. It should be measured in within three complete cycles, after which analysis of the information will show the normal temperature and how it changes during ovulation. The measurements themselves are not difficult to carry out, but creating suitable conditions for them is much more difficult.

The basal temperature indicator will be correct only if it is measured after 6 hours of sleep (more is possible, less is not possible) while still in bed. Measurements must be taken every day of the cycle; 1-2 missed days can significantly affect the schedule.

When ovulation occurs, the temperature rises by a few tenths of a degree, and as long as it remains there is a chance to make a successful attempt to conceive a child. Usually, throughout the cycle it stays within 36.7 - 36.9 degrees, and during ovulation it rises to 37.2 - 37.3. Basal temperature provides clearer information than monitoring discharge or emotional state.

Advantages of the method:

• High information content.
• A clear definition of the time of the egg release process.

The only expense that needs to be made to create a schedule is the purchase of a special electronic thermometer - it is inexpensive and will last a long time. And some doctors believe that electronic thermometers often lie due to a hypersensitive reaction from contact with the walls of the anus - after all, it is quite difficult to always maintain the same angle, and failure occurs. Therefore, they recommend using a regular mercury thermometer for measurements - however, it is more difficult to handle in bed, you need to be careful not to crush it.
Disadvantage of the method can be called unexpected results of temperature changes due to the following factors:

• Alcohol intoxication.
• Sexual contact less than 3-4 hours before temperature measurement.
• Infectious diseases with general increase body temperature.
• Inflammatory process in the genitourinary system.
• Failure in the daily routine, due to which there is a decrease or a strong increase in sleep duration.
• Temperature changes in the room - both extreme cold and heat.
• Starting or stopping taking hormonal medications.

Upon learning of any of these factors, it should be noted dubious result in the graph (for example, in a different color) and do not take this data into account.

4. Changes in the position and density of the cervix

With increased fertility, the cervix rises, becomes softer, more open; Other changes also occur that are not noticeable without special inspection. At home, you can recognize the changes that have begun by the swollen and more elastic inner labia.

Dignity The method is that the cervix changes in any case, and therefore even the absence of cervical fluid cannot mislead you.

Flaw is that the position of the cervix can be influenced various diseases, besides, to recognize changes, you need some practice and the ability to feel your body.

5. Soreness and hardening of the mammary glands

Preparation for the release of an egg automatically means preparation for a potential pregnancy, and therefore the body produces the hormones necessary in this case, which will ensure the fusion of sperm and egg and the formation of a zygote. The release of hormones that occurs before ovulation also affects the condition of the breasts - touching them becomes painful, the nipples become a little rougher. If you know how to listen to your own body well, this condition will definitely not go unnoticed.

Advantage of the method- determined quickly, without additional time and money.

Flaw- inaccurate, since breast soreness and hardening of the nipples are also observed during menstruation, during cycle failure, and hormonal imbalances.

6. Ovulation test

Its principle is similar to a pregnancy test, only it determines not hCG, but a different set of hormones, showing exactly when ovulation occurs. Having calculated what week you are supposed to start ovulating, you will need to do a test once a day during this period - just drop urine on a strip.

Advantages method:

• The test shows the onset of ovulation, regardless of the presence or absence of cervical discharge, swelling of the labia, and so on.
• It does not require lengthy, inconvenient measurements and charting, as in the case of basal temperature - it takes at most 5 minutes a day, and then for a week, and not several cycles in a row.

Flaws method:

• The test kit is quite expensive; for one cycle you will need such test strips for 15-20 dollars, or even more. If you calculate the costs for the year, it will be a considerable amount.
• Irregular ovulation requires large quantities test kits.
• The test, despite its accuracy, can be deceived - there are a number of reasons why the level of the lutein hormone changes regardless of ovulation (for example, polycystic ovary syndrome). But in this case, the test will reveal the disease.

Sometimes ovulation is also determined by painful pulling sensations in the lower abdomen - they are similar to those that appear during menstruation, but appear weaker. Observation of own body and the use of at least one of the methods listed above will allow you to determine when ovulation begins and plan your pregnancy.

When solving problems involving calculations, numerical results are obtained, which are often not accurate, because Errors arise when setting the problem and during calculations.

Sources of error are:

1) errors in the source data;

2) rounding errors of intermediate and final results;

3) errors in the approximate method for solving the problem.

When performing operations on approximate numbers, you must:

1) knowing the accuracy of the source data, be able to assess the accuracy of the result;

2) take the source data with such accuracy as to ensure the specified accuracy of the result.

2.1 Errors in approximate numbers

Let the number x be an exact value, and the number a be an approximate value of some quantity.

Definition. The difference between the number x and its approximate value a is called the error of the approximate number a: Δ = |x-a |.

Let x=10.5, a=10, then Δ=10.5-10=0.5.

Let x=9.5, a=10, then Δ=9.5-10=-0.5.

Definition. The absolute value of the difference between the number x and its approximate value a is called the absolute error of the approximate number a: Δa = |x-a|

Let x=10.5, a=10, then Δa =|10.5-10|=0.5.

Let x=9.5, a=10, then Δa=|9.5-10|=0.5.

Often the exact number x is unknown. Then it is impossible to find Δa = |x-a|, so they use an estimate of the absolute error - the maximum absolute error Δa ≥ Δa =x-a|. In this case, the number x is contained within the boundaries:

a - Δ a  x  a + Δ a or briefly: x = a ± Δ a.

Read: x is equal to a to within Δ a.

In order to determine the quality of the calculations performed, it is necessary to determine what proportion the absolute error of the measured value is. For this purpose, relative error is used.

Definition. The relative error δa of the approximate number a is the ratio of the absolute error Δa to the absolute value of the number x:

or
.

The assessment of the relative error ba is the maximum relative error:

Example. The number x=0.4287 and its approximate value a=0.4264 are given. Find the absolute and relative errors of the number a.

Solution. Let's calculate the absolute error of the number a:

Δa=|0.4287-0.4264| = 0.0023.

Let's calculate the relative error of the number a:

or 5.4%.

Notes. 1. When recording errors, it is customary to leave 1-2 significant figures s. Errors are always rounded upward. In this case, the boundaries of the exact number x expand.

2. If the number x is unknown, then the number a is used to find the relative error.

3. Relative error is often expressed as a percentage by multiplying it by 100%.

2.2. Significant and true digits of an approximate number

To assess the accuracy of an approximate number a, it is customary to write it as a decimal fraction. The accuracy of a calculation is determined not by the number of decimal places (digits after the decimal point), but by the number of correct significant digits of the result.

Definition. The significant digits of a number are all its digits, except for the zeros written before the first digit other than zero, and the zeros at the end of the recording if they serve to preserve the digit or precision of the number.

Example. Determine the significant digits of a.

a = 0.02701 => significant figures: 2.7,0.1.

a = 0.0270 => significant figures: 2.7.0.

a = 2700 => significant figures: 2,7,0,0.

Definition. The digit α i of an approximate number a is called a true significant digit in the broad sense (in the strict sense) if the maximum absolute error of the number a does not exceed one (half a unit) of the digit in which the digit α i is written: Δ а 10 i (Δ а  0.5∙10 i).

Example. Determine the correct numbers of the approximate number a = 0.7264, if the absolute error is Δ a = 0.0023.

Solution. Absolute error Δ a = 0.0023  0.005 = 0.5∙10 -2. Consequently, numbers 7 and 2 are correct in the strict sense, numbers 6 and 4 are incorrect (doubtful). Since Δ a  = 0.0023< 0,01 = 10 -2 , то цифры 7 и 2 являются верными в широком смысле.

Notes. 1. In mathematical tables, all significant figures are true in the strict sense.

2. It is customary to leave only correct numbers in the final result. Then the maximum absolute error of the number a is determined by the unit of the least significant digit. For example, let a = 127.38, then Δ a = 0.01 if all numbers are correct in the strict sense, and Δ a = 0.5∙0.01 = 0.005 if all numbers are correct in the broad sense.

Example. Determine which equality is more accurate: 13/19 = 0.684 or
=7,21?

Solution. Let's denote a =0.684, b =7.21. Let's find the absolute errors of these numbers. To do this, take 13/19 and
With a large number decimal places: 13/39=0.68421...,
=7,2111...

Then Δ a =|0.68421...-0.684|< 0,00022, Δ в = |7,2111...-7,21| < 0,0012.

Let's find the relative errors:

or 0.033%.

or 0.017%.

The second equality is more accurate, since
.

2.3. Rounding numbers

In approximate calculations, it is often necessary to round numbers, both approximate and exact, that is, discard one or more last digits. When we round a number, we replace it with an approximate number with fewer significant digits, resulting in a rounding error. To keep this error to a minimum, you need to adhere to some rounding rules.

Rule I. If the first from the left of the discarded digits is greater than 5, then the last of the retained digits is strengthened, i.e. increases by one. Strengthening is also done when the first digit to the left of the discarded digits is 5, followed by non-zero digits.

Example. Rounding the number 73.473 to the nearest tenth, we get 73.5. The last of the remaining digits is strengthened, since 7 > 5.

Rule II. If the first of the discarded digits is less than 5, then the last of the remaining digits is not amplified, i.e. remains unchanged.

Example. Rounding the number 73.473 to the nearest hundredth, we get 73.47.

RuleIII. If the first left digit discarded is 5 and is not followed by non-zero digits, then the last remaining digit is strengthened if it is odd and left unchanged if it is even (even digit rule).

Example. Rounding the number 5.785 to hundredths, we get 5.78. We do not make any gains, since the last digit saved, 8, is even. Rounding the number 5.775 to the second decimal place, we have 5.78. The last digit stored, 7, is increased by one because it is odd.

When Rule III is applied to rounding a single number, we do not actually increase the accuracy of the calculation, but with multiple roundings, overnumbers are about as common as undernumbers. Mutual compensation of errors occurs, the result is more accurate.

Thus, when applying the rounding rules discussed above, the absolute rounding error does not exceed half a unit of digit determined by the last significant digit left.

If the exact number x is rounded to n significant figures, then the resulting approximate number a has an absolute error equal to the rounding error. In this case, the approximate number a has n valid significant digits in the narrow sense.

Example. Rounding the number x = 26.837 to three significant figures, we get a = 26.8, whence Δ a = |x-a | = | 26.837-26.8 |=0.037< 0,05, т. е. число а имеет три верные значащие цифры в узком смысле.

When rounding the approximate number a, we obtain a new approximate number a 1.

Definition. The number Δ a1 = Δ a + Δ env is called the rounding error.

The absolute error of the number a 1 consists of the absolute error of the original number Δ a and the rounding error Δ env, i.e.

Δ a1 = Δ a + Δ env.

Example. Round off the doubtful digits of the number x=34.124 ± 0.021. Determine the absolute error of the result.

Solution. The approximate number a=34.124 has three correct digits in the narrow sense: 3, 4, 1, since Δ a =0.021< 0,05. Применяя правила округления, найдем приближенное значение а 1 , сохранив десятые доли: а 1 = 34,1. Погрешность округления Δ окр =|34,124-34,1|=0,024. Тогда абсолютная погрешность числа а 1 равна Δ а1 =Δ а +Δ окр =0,021+0,024 = 0,045 < 0,05.

Thus, all significant digits of a 2 are correct (in the narrow sense).

So, x=34.1 ±0.045.

However, when rounding an approximate number a that has n correct significant digits (in the narrow sense) to n significant digits, it may turn out that the rounded number a 1 will have n correct significant digits in the broad sense.

Example. The approximate number a = 15.3654 (± 0.0018) has four correct significant figures in the narrow sense (1, 5, 3, 6), since Δ a = 0.0018< 0,005. При округлении до четырех значащих цифр получим а 1 = 15,37 и Δ а1 =Δ а +Δ окр =0,0018+|15,3654-15,37|=0,0064.

Obviously 0.005< 0,0064 < 0,01. Следовательно, число 15,37 (± 0.0064) has four correct digits in the broad sense.

So, x=15.37 ±0.0064.

Example. Round off the doubtful digits of the number a = 26.7245 (± 0.0026), leaving sure signs in the narrow sense. Determine the absolute error of the result.

Solution. According to the condition Δ a = 0.0026< 0,005, следовательно, в числе 26,7245 верными в узком смысле являются цифры 2, 6, 7, 2. Используя правила округления, найдем приближенное значение а 1 , сохранив сотые доли:

The resulting error is greater than 0.005 (0.005< 0,0071), поэтому уменьшим чис­ло цифр в приближенном числе до трех; а 2 = 26.7. We find Δ a2 = =Δ a +Δ env =0.0026+|26.7245-26.7|=0.0271< 0,05. Следовательно, оставшиеся три цифры верны в узком смысле.

So, x=26.7 ±0.0271 => x=26.7 ±0.03, rounding the error to two digits.

Example. Round off the doubtful digits of the number a=22.7314, leaving the correct signs in the narrow sense. Determine the absolute error of the number if δ a = 0.2%.

Solution. Let us write δ a in the form of a decimal fraction: δа=0.002 and determine the absolute error. Since Δ a = =0.0455< 0,05, то верными в этом числе будут три цифры: 2, 2, 7. Округлим число 22,7314, сохранив в нем десятые доли: а 1 = 22.73. Then Δ a1 = =Δ a +Δ env =0.0455+|22.7314-22.73|=0.0769>0.05, so let’s reduce the number of digits in the approximate number to two: a 2 =23. We find Δ a2 = =Δ a +Δ env =0.0455+|22.7314-23|=0.3141< 0,05. Следовательно, оставшиеся две цифры верны в узком смысле.

So, x=23 ±0.3141 => x=23 ±0.32.

2.3. Rules for working with approximate numbers

Rule 1. The absolute error of an algebraic sum of several approximate numbers is equal to the sum of the absolute errors of these numbers:

Δ а±в =Δ а + Δ в

Rule 2. The relative error of the product of several approximate numbers is equal to the sum of the relative errors of these numbers:

δ aw = δ a + δ b.

Rule 3. The relative error of the partial approximate numbers is equal to the sum of the relative numbers: δ а/в = δ а +δ в.

Rule 4. The relative error of the degree of the approximate number a is equal to: δa n = nδ a.

Rule 5. The relative error of the root of the approximate number a is equal to:
.

Rule 6. When making calculations, if a strict calculation of errors is not carried out, it is recommended to use the rules for counting numbers. These rules indicate how results should be rounded to ensure the desired accuracy of the result without making calculations with extra digits.

The rules assume that the numbers being manipulated contain only correct digits and that the number of manipulations is small.

I. When adding and subtracting approximate numbers, the result should retain as many decimal places as there are in the number that has the fewest decimal places.

II. When multiplying and dividing, the result should retain as many significant digits as there are in the number with the fewest significant digits.

III. When raising an approximate number to a power, the result should retain as many significant digits as there are in the base of the power.

IV. When extracting a root from an approximate number, you should keep as many significant digits as there are in the radical number.

V. In intermediate results, you should save 1-2 digits more than recommended by rules I-IV. In the final result, the "spare digits" are discarded and the number is rounded.

VI. If some source data has more decimal places (for addition and subtraction) or more significant digits (for other operations) than others, then they should first be rounded, keeping only one "safe digit".

VII. To obtain a result with N correct digits, the source data should be taken with such a number of digits that, according to the previous rules, provide N+1 digits in the result.

Example. Let's find s=2.35+11.8 without taking into account errors. Applying rule I, we get s=14.15. We round the result to the number 11.8 with the least number of decimal places. We get: s =14.2.

Let's solve the problem taking into account errors. In the number s=14,15, only the correct numbers should be left. To do this, we will find the maximum absolute error of the sum s using rule 1. Considering that all the figures in the numbers 2.35 and 11.8 are correct, we obtain: Δ 14.15 = Δ 2.35 + Δ 11.8 = 0.01 +0.1=0.11< 0,5. Последняя верная цифра в числе 14,15 находится в разряде единиц. Поэтому число s=14,15 надо округлить: s=14 и найти абсолютную погрешность округленного числа. Погрешность округления равна: |14,15-14|=0,15. Тогда абсолютная погрешность округленного числа Δ 14 =0,11+0,15=0,26 <0,5. Окончательный результат примет вид: s=14 ± 0,26.

Problems are solved similarly when performing other operations on approximate numbers.