In a wide variety of theoretical and applied research, mathematical modeling methods are widely used, which reduce the solution of problems in a given area of research to the solution of adequate (or approximately adequate) problems. mathematical problems. It is necessary to bring the solution of these problems to obtain a numerical result (calculation of various types of quantities, solution of various types of equations, etc.). The goal of computational mathematics is to develop algorithms for the numerical solution of a wide range of mathematical problems. Methods must be designed so that they can be effectively implemented using modern computing technology. As a rule, the problems under consideration do not allow an exact solution, so we are talking about developing algorithms that give an approximate solution. To be able to replace an unknown exact solution to a problem with an approximate one, it is necessary that the latter be sufficiently close to the exact one. In this regard, there is a need to assess the proximity of the approximate solution to the exact one and to develop approximate methods for constructing approximate solutions that are as close to the exact ones as desired.
Schematically, the computational process is as follows: for a given value x(numeric, vector, etc.) calculate the value of some function A(x). The difference between the exact and approximate values of a quantity is called error. Accurate value calculation A(x) usually impossible, and forces you to replace the function (operation) A her approximate representation à , which can be calculated: calculating the quantity A(x), is replaced by the calculation- Ã(x) A(x) - Ã(x) called method error. A method for estimating this error must be developed together with the development of a method for calculating the value Ã(x). From possible methods When constructing an approximation, you should use the one that, given the available means and capabilities, gives the smallest error.
Value value x, that is, the initial data, in real problems is obtained either directly from measurements, or as a result of the previous stage of calculations. In these cases, only an approximate value is determined x o quantities x. Therefore, instead of the value Ã(x) only an approximate value can be calculated Ã(x o). The resulting error A(x) - Ã(x o) called irreparable. As a result of roundings inevitable during calculations, instead of the value Ã(x o) its “rounded” value is calculated, which leads to the appearance rounding errors Ã(x o)- . The total calculation error turns out to be equal to A(x) - .
Let us represent the total error in the form
A(x) - = [A(x) - ] + [ - Ã(x o)] +
+ [Ã(x o) - ] (1)
The last equality shows that the total calculation error is equal to the sum of the method error, the fatal error and the rounding error. The first two components of the error can be estimated before starting the calculations. The rounding error is assessed only during calculations.
Let's consider the following tasks:
a) characteristic of the accuracy of approximate numbers
b) assessment of the accuracy of the result given the known accuracy of the initial data (estimate of the fatal error)
c) determining the required accuracy of the initial data to ensure the specified accuracy of the result
d) matching the accuracy of source data and calculations with the capabilities of available computing tools.
4 Measurement errors
4.1 True and actual values of physical quantities. Measurement error. Causes of measurement errors
When analyzing measurements, two concepts should be clearly distinguished: the true values of physical quantities and their empirical manifestations - the results of measurements.
True values of physical quantities - these are the values, in an ideal way reflecting the properties of a given object both quantitatively and qualitatively. They do not depend on the means of measurement and are the absolute truth to which they strive when making measurements.
On the contrary, the results of measurements are products of cognition. Representing approximate estimates of the values of quantities found as a result of measurements, they depend on the measurement method, measuring instruments and other factors.
Measurement error the difference between the measurement result x and the true value Q of the measured quantity is called:
Δ= x – Q (4.1)
But since true meaning Q of the measured quantity is unknown, then to determine the measurement error, the so-called real value is substituted in formula (4.1) instead of the true value.
Under actual value of the measured quantity its meaning is understood to be one found experimentally and so close to the true value that for a given purpose it can be used instead.
The causes of errors are: imperfection of measurement methods, measuring instruments and the observer’s senses. Reasons related to the influence of measurement conditions should be combined into a separate group. The latter manifest themselves in two ways. On the one hand, all physical quantities that play any role in measurements depend on each other to one degree or another. Therefore, with changes in external conditions, the true values of the measured quantities change. On the other hand, the measurement conditions influence both the characteristics of the measuring instruments and the physiological properties of the observer’s sense organs and, through them, become a source of measurement errors.
4.2 Classification of measurement errors depending on the nature of their change
The described causes of errors are a combination large number factors under the influence of which the total measurement error is formed. They can be combined into two main groups.
The first group includes factors that appear irregularly and suddenly disappear or appear with an intensity that is difficult to predict. These include, for example, small fluctuations of influencing quantities (temperature, pressure environment and so on.). The share, or component, of the total measurement error arising under the influence of factors of this group determines the random measurement error.
Thus, random measurement error - component of the measurement error that changes randomly during repeated measurements of the same quantity.
When creating measuring instruments and organizing the measurement process as a whole, the intensity of the manifestation of the factors that determine the random measurement error can be reduced to a general level, so that they all influence more or less equally on the formation of the random error. However, some of them, for example, a sudden drop in voltage in the power supply network, may appear unexpectedly strong, as a result of which the error will take on dimensions that clearly go beyond the limits determined by the course of the measuring experiment. Such errors within the random error are called rude . Closely adjacent to them misses - errors that depend on the observer and are associated with improper handling of measuring instruments, incorrect readings, or errors in recording results.
The second group includes factors that are constant or change naturally during the measurement experiment, for example, smooth changes in influencing quantities. The component of the total measurement error arising under the influence of factors of this group determines the systematic measurement error.
Thus, systematic measurement error - a component of measurement error that remains constant or changes naturally with repeated measurements of the same quantity.
During the measurement process, the described error components appear simultaneously, and the total error can be represented as a sum
, (4.2)
Where 
- random, and Δ s - systematic errors.
To obtain results that differ minimally from the true values of quantities, multiple observations of the measured quantity are carried out, followed by processing of the experimental data. That's why great importance has the study of error as a function of observation number, i.e. time A(t). Then individual error values can be interpreted as a set of values of this function:
Δ 1 = Δ(t 1), Δ 2 = Δ(t 2),..., Δ n = Δ(t n).
In the general case, the error is a random function of time, which differs from the classical functions of mathematical analysis in that it cannot be said what value it will take at time t i. You can only indicate the probability of the occurrence of its values in a particular interval. In a series of experiments consisting of a number of repeated observations, we obtain one implementation of this function. When repeating the series with the same values of the quantities characterizing the factors of the second group, we inevitably obtain a new implementation that differs from the first. Realizations differ from each other due to the influence of factors of the first group, and factors of the second group, which are equally manifested when receiving each realization, give them some common features(Figure 4.1).
The measurement error corresponding to each time moment t i is called the cross section random functionΔ(t). In each section, you can find the average error value Δ s (t i), around which the errors in different implementations are grouped. If a smooth curve is drawn through the points Δ s (t i) obtained in this way, then it will characterize the general trend of changes in the error over time. It is easy to see that the average values Δ s (tj) are determined by the action of factors of the second group and represent a systematic measurement error at time t i, and deviations Δ j (t j) from the average value in the section t i, corresponding jth implementation, give the value of the random error. Thus, the equality holds
(4.3)
Figure 4.1
Let us assume that Δ s (t i) = 0, i.e. systematic errors are excluded in one way or another from the observation results, and we will consider only random errors, the average values of which are equal to zero in each section. Let us assume that random errors in different sections do not depend on each other, i.e. knowledge of the random error in one section does not give us any additional information about the value taken by this realization in any section, and that all the theoretical and probabilistic features of random errors, which are the values of one realization in all sections, coincide with each other. Then the random error can be considered as a random variable, and its values for each of the multiple observations of the same physical quantity can be considered as the results of independent observations of it.
Under such conditions, the random measurement error is defined as the difference between the corrected measurement result XI (a result that does not contain a systematic error) and the true value Q of the measured quantity:
Δ = X AND –Q 4.4)
Moreover, the corrected measurement result will be from which systematic errors will be excluded.
Such data is usually obtained when checking measuring instruments by measuring previously known quantities. When carrying out measurements, the goal is to estimate the true value of the measured quantity, which is unknown before the experiment. In addition to the true value, the measurement result also includes a random error, therefore, it is itself a random variable. Under these conditions, the actual value of the random error obtained during verification does not yet characterize the accuracy of the measurements, so it is unclear what value to take as the final measurement result and how to characterize its accuracy.
The answer to these questions can be obtained by using methods of mathematical statistics that deal specifically with random variables when processing observational results.
4.3 Classification of measurement errors depending on the reasons for their occurrence
Depending on the reasons for their occurrence, the following groups of errors are distinguished: methodological, instrumental, external and subjective.
In many measurement methods it is possible to detect methodological error , which is a consequence of certain assumptions and simplifications, the use of empirical formulas and functional dependencies. In some cases, the impact of such assumptions turns out to be insignificant, i.e. much less than the permissible measurement errors; in other cases it exceeds these errors.
An example of methodological errors are the errors in the method of measuring electrical resistance using an ammeter and voltmeter (Figure 4.2). If the resistance R x is determined by the formula of Ohm's law R x =U v /I a, where U v is the voltage drop measured by a voltmeter V; I a is the current strength measured by ammeter A, then in both cases methodological measurement errors will be allowed.
In Figure 4.2a, the current strength I a, measured by an ammeter, will be greater than the current strength in resistance R x by the value of the current strength I v in a voltmeter connected in parallel with the resistance. Resistance R x calculated using the above formula will be less than the actual one. In Figure 4.2.6, the voltage measured by the voltmeter V will be greater than the voltage drop U r in the resistance R x by the value U a (voltage drop across the resistance of the ammeter A). The resistance calculated using the formula of Ohm's law will be greater than the resistance R x by the value R a (the resistance of the ammeter). Corrections in both cases can be easily calculated if you know the resistance of the voltmeter and ammeter. Corrections need not be made if they are significantly less than the permissible error in measuring resistance R x, for example, if in the first case the resistance of the voltmeter is significantly b
Larger than R x, and in the second case, R a is significantly less than R x.
Figure 4.2
Another example of the occurrence of a methodological error is the measurement of the volume of bodies, the shape of which is assumed to be geometrically correct, by measuring the dimensions in one or in an insufficient number of places, for example, measuring the volume of a room by measuring the length, width and height in only three directions. To accurately determine the volume, it would be necessary to determine the length and width of the room along each wall, at the top and bottom, measure the height at the corners and in the middle, and, finally, the corners between the walls. This example illustrates the possibility of a significant methodological error occurring when the method is unjustifiably simplified.
As a rule, methodological error is a systematic error.
Instrumental error - this is a component of error due to imperfection of measuring instruments. A classic example of such an error is the error of a measuring instrument caused by inaccurate calibration of its scale. It is very important to clearly distinguish between measurement errors and instrumental errors. The imperfection of measuring instruments is only one of the sources of measurement error and determines only one of its components - instrumental error. In turn, the instrumental error is total, the components of which - errors of functional units - can be both systematic and random.
External error - component of the measurement error caused by the deviation of one or more influencing quantities from normal values or their exit beyond the normal range (for example, the influence of temperature, external electric and magnetic fields, mechanical influences, etc.). As a rule, external errors are determined by additional errors of the measuring instruments used and are systematic. However, if the influencing quantities are unstable, they can become random.
Subjective (personal) error due to individual characteristics experimenter and can be either systematic or random. When using modern digital measuring instruments, subjective error can be neglected. However, when taking readings from pointer instruments, such errors can be significant due to incorrect reading of tenths of a scale division, asymmetry that occurs when setting a stroke in the middle between two marks, etc. For example, the errors that an experimenter makes when estimating tenths of a division of an instrument scale can reach 0.1 division. These errors are manifested in the fact that for different tenths of division, different experimenters are characterized by different frequencies of estimates, and each experimenter maintains his characteristic distribution for a long time. Thus, one experimenter more often than not refers the readings to the lines forming the edges of the division and to the value of 0.5 divisions. The other is to the values of 0.4 and 0.6 divisions. The third prefers values of 0.2 and 0.8 divisions, etc. In general, keeping in mind a random experimenter, the distribution of errors in counting tenths of a division can be considered uniform with boundaries of ±0.1 divisions.
4.4 Forms for representing measurement error. Accuracy of measurements
The measurement error can be represented in the form absolute error expressed in units of the measured value and determined by formula (4.1), or relative error, defined as the ratio of the absolute error to the true value of the measured value:
δ = Δ/Q. (4.5)
In the case of expressing the random error as a percentage, the ratio Δ/Q is multiplied by 100%. In addition, in formula (4.5) it is allowed to use the result of measuring x instead of the true value of Q.
The concept is also widely used accuracy of measurements − a characteristic that reflects the closeness of their results to the true value of the measured value. In other words, high accuracy corresponds to small measurement errors. Therefore, the measurement accuracy can be quantitatively assessed by the reciprocal of the modulus of the relative error
3.2. Rounding
One source for obtaining approximate numbers is O rounding. Both exact and approximate numbers are rounded.
Rounding given number up to a certain digit is called replacing it with a new number, which is obtained from the given one by discarding all his numbers written down to the right digits of this digit, or by replacing it with zeros. These zeros usually underline or write them smaller. To ensure the closest proximity of the rounded number to the rounded one, you should use the following rules:
To round a number to one of a certain digit, you need to discard all the digits after the digit of this digit, and replace them with zeros in the whole number. The following are taken into account:
1 ) if the first (left) of the discarded digits less than 5, then the last digit left is not changed (rounding with disadvantage);
2 ) if the first digit to be discarded greater than 5 or equal to 5, then the last digit left is increased by one (rounding with excess).*
For example:
Round:Answers:
A) to tenths 12.34; 12.34 ≈ 12.3;
b) to hundredths 3.2465; 1038.785; 3.2465 ≈ 3.25; 1038.785 ≈ 1038.79;
V) to thousandths 3.4335; 3.4335 ≈ 3.434;
G) up to thousands 12,375, 320,729. 12,375 ≈ 12 000 ; 320 729 ≈ 321 000.
(* Several years ago, in the case of discarding only one digit 5 enjoyed "even number rule": the last digit was left unchanged if it was even, and increased by one if it was odd. Now "even digit rules" Not adhere to: if one digit is discarded 5 , then one is added to the last digit left, regardless of whether it is even or odd).
3.3. Absolute and relative error of approximate values
Absolute value differences between the approximate and exact (true) value of a quantity is called absolute error approximate value. For example, if the exact number 1,214 round to the nearest tenth, we get an approximate number 1,2 . IN in this case the absolute error of the approximate number will be 1,214 – 1,2 = 0,014 .
But in most cases exact value the quantity under consideration is unknown, but only approximate. Then the absolute error is unknown. In these cases indicate border, which it does not exceed. This number is called limiting absolute error. They say that the exact value of a number is equal to its approximate value with an error less than the marginal error. For example, number 23,71 is an approximate value of the number 23,7125 up to 0,01 , since the absolute approximation error is equal to 0,0025 and less 0,01 . Here the limiting absolute error is equal to 0,01 .*
(* Absolute The error can be both positive and negative. For example,1,68 ≈ 1,7 . The absolute error is 1 ,68 – 1,7 ≈ - 0,02 .Boundary the error is always positive).
Boundary absolute error of the approximate number " A » is indicated by the symbol Δ A . Record
X ≈ a (Δa)
should be understood as follows: the exact value of the quantity X is between the numbers A +Δ A And A –Δ A, which are called accordingly bottom And upper limitX and denote N G X And IN G X .
For example, If X
≈ 2,3 (
0,1),
That 2,2 <
X
< 2,4
.
On the contrary, if 7,3 < X < 7,4 , That X ≈ 7,35 ( 0,05).
Absolute or marginal absolute error Not characterize the quality of the measurement performed. The same absolute error can be considered significant and insignificant depending on the number with which the measured value is expressed.
For example, if we measure the distance between two cities with an accuracy of one kilometer, then such accuracy is quite sufficient for this measurement, but at the same time, when measuring the distance between two houses on the same street, such accuracy will be unacceptable.
Consequently, the accuracy of the approximate value of a quantity depends not only on the magnitude of the absolute error, but also on the value of the measured quantity. That's why the measure of accuracy is the relative error.
Relative error is called the ratio of the absolute error to the value of the approximate number. The ratio of the limiting absolute error to the approximate number is called limit relative error; denote it like this: Δ a/a . Relative and marginal relative errors are usually expressed as in percentages.
For example, if measurements show that the distance between two points is greater 12.3 km, but less 12.7 km, then for approximate its meaning is accepted average these two numbers, i.e. their half the sum, Then boundary the absolute error is half-differences these numbers. In this case X ≈ 12,5 ( 0,2). Here is the boundary absolute the error is equal to 0.2 km, and the boundary relative:
Absolute and relative errors
Absolute measurement error is a quantity determined by the difference between the measurement result x and the true value of the measured quantity x 0:
Δ x = |x – x 0 |.
Value δ, equal to the ratio absolute measurement error to the measurement result is called relative error:
Example 2.1. The approximate value of π is 3.14. Then its error is 0.00159... . The absolute error can be considered equal to 0.0016, and the relative error equal to 0.0016 / 3.14 = 0.00051 = 0.051%.
Significant figures. If the absolute error of the value a does not exceed one place unit of the last digit of the number a, then the number is said to have all the correct signs. Approximate numbers should be written down, keeping only the correct signs. If, for example, the absolute error of the number 52,400 is 100, then this number should be written, for example, in the form 524 · 10 2 or 0.524 · 10 5. You can estimate the error of an approximate number by indicating how many correct significant digits it contains. When counting significant figures, the zeros on the left side of the number are not counted.
For example, the number 0.0283 has three valid significant figures, and 2.5400 has five valid significant figures.
Rules for rounding numbers. If the approximate number contains extra (or incorrect) digits, then it should be rounded. When rounding, an additional error occurs that does not exceed half a unit of the place of the last significant digit ( d) rounded number. When rounding, only the correct digits are retained; extra characters are discarded, and if the first discarded digit is greater than or equal to d/2, then the last digit stored is increased by one.
Extra digits in integers are replaced by zeros, and in decimals are discarded (as are extra zeros). For example, if the measurement error is 0.001 mm, then the result 1.07005 is rounded to 1.070. If the first of the digits modified by zeros and discarded is less than 5, the remaining digits are not modified. For example, the number 148,935 with a measurement precision of 50 has a rounding value of 148,900. If the first of the digits replaced by zeros or discarded is 5, and it is followed by no digits or zeros, then rounding is done to the nearest even number. For example, the number 123.50 is rounded to 124. If the first digit to be replaced by zeros or discarded is greater than or equal to 5 but followed by a significant digit, then the last remaining digit is increased by one. For example, the number 6783.6 is rounded to 6784.
Example 2.2. When rounding 1284 to 1300, the absolute error is 1300 – 1284 = 16, and when rounding to 1280, the absolute error is 1280 – 1284 = 4.
Example 2.3. When rounding the number 197 to 200, the absolute error is 200 – 197 = 3. The relative error is 3/197 ≈ 0.01523 or approximately 3/200 ≈ 1.5%.
Example 2.4. A seller weighs a watermelon on a scale. The smallest weight in the set is 50 g. Weighing gave 3600 g. This number is approximate. Exact weight watermelon unknown. But the absolute error does not exceed 50 g. The relative error does not exceed 50/3600 = 1.4%.
Errors in solving the problem on PC
Three types of errors are usually considered as the main sources of error. These are called truncation errors, rounding errors, and propagation errors. For example, when using iterative methods for searching for the roots of nonlinear equations, the results are approximate, in contrast to direct methods that provide an exact solution.
Truncation errors
This type of error is associated with the error inherent in the task itself. It may be due to inaccuracy in determining the source data. For example, if any dimensions are specified in the problem statement, then in practice for real objects these dimensions are always known with some accuracy. The same goes for any other physical parameters. This also includes the inaccuracy of calculation formulas and the numerical coefficients included in them.
Propagation errors
This type of error is associated with the use of one or another method of solving a problem. During calculations, error accumulation or, in other words, propagation inevitably occurs. In addition to the fact that the original data themselves are not accurate, a new error arises when they are multiplied, added, etc. The accumulation of errors depends on the nature and number of arithmetic operations used in the calculation.
Rounding errors
This type of error occurs because the true value of a number is not always accurately stored by the computer. When a real number is stored in computer memory, it is written as a mantissa and exponent in much the same way as a number is displayed on a calculator.
Now that a person owns a powerful arsenal of computer equipment (various calculators, computers, etc.), compliance with the rules of approximate calculations is especially important so as not to distort the reliability of the result.
When performing any calculations, you should remember the accuracy of the result that can or should (if established) be obtained. Thus, it is unacceptable to perform calculations with greater accuracy than is specified by the data of the physical problem or required by the experimental conditions1. For example, when performing mathematical operations with numerical values of physical quantities that have two reliable (significant) digits, you cannot write down the result of calculations with an accuracy that goes beyond the limits of two reliable digits, even if in the end we have more of them.
The value of physical quantities must be written down, noting only the signs of a reliable result. For example, if the numerical value of 39,600 has three reliable digits (the absolute error of the result is 100), then the result should be written as 3.96 104 or 0.396 105. When calculating reliable digits, the zeros to the left of the number are not taken into account.
In order for the calculation result to be correct, it must be rounded, leaving only the true value of the quantity. If the numeric value of a quantity contains extra (unreliable) digits that exceed the specified precision, then the last digit stored is increased by 1 provided that the excess (extra digits) is equal to or greater than half the value of the next digit of the number.
In different numerical values, zero can be either a reliable or unreliable number. So, in example b) it is an unreliable figure, and in d) it is reliable and significant. In physics, if they want to emphasize the reliability of the digit of a numerical value of a physical quantity, they indicate “0” in its standard expression. For example, recording a mass value of 2.10 10-3 kg indicates three reliable digits of the result and the corresponding measurement accuracy, and a value of 2.1 10-3 kg only two reliable digits.
It should be remembered that the result of actions with numerical values of physical quantities is an approximate result that takes into account the calculation accuracy or measurement error. Therefore, when making approximate calculations, you should be guided by the following rules for calculating reliable numbers:
1. When performing arithmetic operations with numerical values of physical quantities, their result should be taken as many reliable signs as there are numerical values with the least number of reliable signs.
2. In all intermediate calculations, one more digit should be kept than the numerical value with the least number of reliable digits. Ultimately this "extra" figure is discarded by rounding.
3. If some data has more reliable signs than others, their values should first be rounded (you can save one “excess” digit) and then perform actions.
In most cases, the numerical data in problems are approximate. In task conditions, exact values may also appear, for example, the results of counting a small number of objects, some constants, etc.
To indicate the approximate value of a number, use the approximate equality sign; read like this: “approximately equal” (should not read: “approximately equal”).
Finding out the nature of numerical data is an important preparatory stage when solving any problem.
The following guidelines can help you recognize exact and approximate numbers:
| Exact values | Approximate values |
| 1. The values of a number of conversion factors for the transition from one unit of measurement to another (1m = 1000 mm; 1h = 3600 s) Many conversion factors have been measured and calculated with such high (metrological) accuracy that they are now practically considered accurate. | 1. Most of the values of mathematical quantities given in tables (roots, logarithms, values trigonometric functions, as well as the meaning of number and base used in practice natural logarithms(number e)) |
| 2. Scale factors. If, for example, it is known that the scale is 1:10000, then the numbers 1 and 10000 are considered accurate. If it is indicated that 1 cm is 4 m, then 1 and 4 are the exact length values | 2. Measurement results. (Some basic constants: the speed of light in a vacuum, the gravitational constant, the charge and mass of an electron, etc.) Tabulated values of physical quantities (density of matter, melting and boiling points, etc.) |
| 3. Tariffs and prices. (cost of 1 kWh of electricity – exact price) | 3. Design data are also approximate, because they are specified with some deviations, which are standardized by GOSTs. (For example, according to the standard, the dimensions of a brick are: length 250 6 mm, width 120 4 mm, thickness 65 3 mm) The same group of approximate numbers includes dimensions taken from the drawing |
| 4. Conditional values of quantities (Examples: absolute zero temperature -273.15 C, normal atmospheric pressure 101325 Pa) | |
| 5. Coefficients and exponents found in physical and mathematical formulas ( ; %; etc.). | |
| 6. Item counting results (number of batteries in the battery; number of milk cartons produced by the plant and counted by the photoelectric meter) | |
| 7. Setpoints quantities (For example, in the problem “Find the periods of oscillation of pendulums 1 and 4 m long,” numbers 1 and 4 can be considered the exact values of the length of the pendulum) |
Execute the following tasks, format your answer in table form:
1. Indicate which of the given values are exact and which are approximate:
1) Density of water (4 C)………..………………………..…………………1000kg/m 3
2) Speed of sound (0 C)………………………………………….332 m/s
3) Specific heat capacity of air….……………………………1.0 kJ/(kg∙K)
4) Boiling point of water…………….…………………………….100 C
5) Avogadro’s constant….…………………………………..…..6.02∙10 23 mol -1
6) Relative atomic mass of oxygen…………………………………..16
2. Find exact and approximate values in the following problems:
1) In a steam engine, a bronze spool, the length and width of which are 200 and 120 mm, respectively, experiences a pressure of 12 MPa. Find the force required to move the spool along the cast iron surface of the cylinder. The friction coefficient is 0.10.
2) Determine the resistance of the filament of an electric lamp using the following markings: “220V, 60 W.”
3. What answers – exact or approximate – will we get when solving the following problems?
1) What is the speed of a freely falling body at the end of the 15th second, assuming the time interval is specified exactly?
2) What is the speed of the pulley if its diameter is 300 mm and the rotation speed is 10 rps? Consider the data to be accurate.
3) Determine the modulus of force. Scale 1 cm – 50N.
4) Determine the coefficient of static friction for a body located on an inclined plane if the body begins to slide uniformly along the slope at = 0.675, where is the angle of inclination of the plane.
If it is known that a< А, то а называют an approximate value of A with a disadvantage. If a > A, then a is called approximate value of A with excess.
The difference between the exact and approximate values of a quantity is called approximation error and is denoted by D, i.e.
D = A – a (1)
The approximation error D can be either a positive or negative number.
In order to characterize the difference between an approximate value of a quantity and an exact one, it is often sufficient to indicate the absolute value of the difference between the exact and approximate values.
The absolute value of the difference between the approximate A and accurate A the values of a number are called absolute error (error) of approximation and denoted by D A:
D A = ½ A – A½ (2)
Example 1. When measuring a segment l used a ruler, the scale division of which is 0.5 cm. We obtained an approximate value of the length of the segment A= 204 cm.
It is clear that during the measurement there could have been an error of no more than 0.5 cm, i.e. The absolute measurement error does not exceed 0.5 cm.
Usually the absolute error is unknown, since the exact value of the number A is unknown. Therefore, any assessment absolute error:
D A <= DA before. (3)
where D and before. – maximum error (number, more zero), given taking into account the reliability with which the number a is known.
The maximum absolute error is also called error margin. So, in the example given,
D and before. = 0.5 cm.
From (3) we get:
D A = ½ A – A½<= DA before. .
A– D A before. ≤ A≤ A+D A before. . (4)
a – D A before. will be an approximate value A with a disadvantage
a + D A before – approximate value A in abundance. The short notation is also used:
A= A± D A before (5)
From the definition of the maximum absolute error it follows that the numbers D A before, satisfying inequality (3), there will be an infinite set. In practice, they try to choose possibly less from numbers D and before, satisfying the inequality D A <= DA before.
Example 2. Let us determine the maximum absolute error of the number a=3.14, taken as an approximate value of the number π.
It is known that 3,14<π<3,15. It follows that
|A – π |< 0,01.
The maximum absolute error can be taken as the number D A = 0,01.
If we take into account that 3,14<π<3,142 , then we get a better rating: D A= 0.002, then π ≈3.14 ±0.002.
4. Relative error (error). Knowing only the absolute error is not enough to characterize the quality of the measurement.
Let, for example, when weighing two bodies the following results are obtained:
P 1 = 240.3 ±0.1 g.
P 2 = 3.8 ±0.1 g.
Although the absolute measurement errors of both results are the same, the measurement quality in the first case will be better than in the second. It is characterized by relative error.
Relative error (error) approaching number A called the absolute error ratio D a approaching the absolute value of the number A:
Since the exact value of a quantity is usually unknown, it is replaced by an approximate value and then:
(7)
Maximum relative error or boundary of the relative approximation error, is called the number d and before>0, such that:
d A<= d and before(8)
The maximum relative error can obviously be taken as the ratio of the maximum absolute error to the absolute value of the approximate value:
(9)
From (9) the following important relationship is easily obtained:
∆and before = |a| d and before(10)
The maximum relative error is usually expressed as a percentage:
Example. The base of natural logarithms for calculation is assumed to be equal to e=2.72. We took as the exact value e t = 2.7183. Find the absolute and relative errors of the approximate number.
D e = ½ e – e t ½=0.0017;
.
The magnitude of the relative error remains unchanged with a proportional change in the most approximate number and its absolute error. Thus, for the number 634.7, calculated with an absolute error of D = 1.3, and for the number 6347 with an error of D = 13, the relative errors are the same: d= 0,2.
The magnitude of the relative error can be approximately judged by the number true signifiers digits of numbers.
Sakhalin region
"Vocational School No. 13"
Guidelines To independent work students
Alexandrovsk-Sakhalinsky
Approximate values of quantities and approximation errors: Method indicated. / Comp.
GBOU NPO "Vocational School No. 13", - Aleksandrovsk-Sakhalinsky, 2012
Guidelines are intended for students of all professions studying mathematics courses
Chairman of the MK
Approximate value of magnitude and error of approximations.
In practice, we almost never know the exact values of quantities. No scale, no matter how accurate it may be, shows weight absolutely accurately; any thermometer shows the temperature with one error or another; no ammeter can give accurate readings of current, etc. In addition, our eye is not able to absolutely correctly read the readings of measuring instruments. Therefore, instead of dealing with the true values of quantities, we are forced to operate with their approximate values.
The fact that A" is an approximate value of the number A , is written as follows:
a ≈ a" .
If A" is an approximate value of the quantity A , then the difference Δ = a - a" called approximation error*.
* Δ - Greek letter; read: delta. Next comes another Greek letter ε (read: epsilon).
For example, if the number 3.756 is replaced by an approximate value of 3.7, then the error will be equal to: Δ = 3.756 - 3.7 = 0.056. If we take 3.8 as an approximate value, then the error will be equal to: Δ = 3,756 - 3,8 = -0,044.
In practice, the approximation error is most often used Δ
, and the absolute value of this error | Δ
|. In what follows we will simply call this absolute value of error absolute error. One approximation is considered to be better than another if the absolute error of the first approximation is less than the absolute error of the second approximation. For example, the 3.8 approximation for the number 3.756 is better than the 3.7 approximation because for the first approximation
|Δ
| = | - 0.044| =0.044, and for the second | Δ
| = |0,056| = 0,056.
Number A" A up toε , if the absolute error of this approximation is less thanε :
|a - a" | < ε .
For example, 3.6 is an approximation of the number 3.671 with an accuracy of 0.1, since |3.671 - 3.6| = | 0.071| = 0.071< 0,1.
Similarly, - 3/2 can be considered as an approximation of the number - 8/5 to within 1/5, since
< A , That A" called the approximate value of the number A with a disadvantage.
If A" > A , That A" called the approximate value of the number A in abundance.
For example, 3.6 is an approximate value of the number 3.671 with a disadvantage, since 3.6< 3,671, а - 3/2 есть приближенное значение числа - 8/5 c избытком, так как - 3/2 > - 8/5 .
If instead of numbers we A And b add up their approximate values A" And b" , then the result a" + b" will be an approximate value of the sum a + b . The question arises: how to evaluate the accuracy of this result if the accuracy of the approximation of each term is known? The solution to this and similar problems is based on the following property of absolute value:
|a + b | < |a | + |b |.
The absolute value of the sum of any two numbers does not exceed the sum of their absolute values.
Errors
The difference between the exact number x and its approximate value a is called the error of this approximate number. If it is known that | x - a |< a, то величина a называется предельной абсолютной погрешностью приближенной величины a.
The ratio of the absolute error to the absolute value of the approximate value is called the relative error of the approximate value. The relative error is usually expressed as a percentage.
Example. | 1 - 20 | < | 1 | + | -20|.
Really,
|1 - 20| = |-19| = 19,
|1| + | - 20| = 1 + 20 = 21,
Exercises for independent work.
1. With what accuracy can lengths be measured using an ordinary ruler?
2. How accurate is the clock?
3. Do you know with what accuracy body weight can be measured on modern electric scales?
4. a) Within what limits is the number contained? A , if its approximate value with an accuracy of 0.01 is 0.99?
b) Within what limits is the number contained? A , if its approximate value with a disadvantage accurate to 0.01 is 0.99?
c) What are the limits of the number? A , if its approximate value with an excess of 0.01 is equal to 0.99?
5 . What is the approximation of the number π ≈ 3.1415 is better: 3.1 or 3.2?
6. Can an approximate value of a certain number with an accuracy of 0.01 be considered an approximate value of the same number with an accuracy of 0.1? What about the other way around?
7. On the number line, the position of the point corresponding to the number is specified A . Indicate on this line:
a) the position of all points that correspond to approximate values of the number A with a disadvantage with an accuracy of 0.1;
b) the position of all points that correspond to approximate values of the number A with excess with an accuracy of 0.1;
c) the position of all points that correspond to approximate values of the number A with an accuracy of 0.1.
8. In what case is the absolute value of the sum of two numbers:
a) less than the sum of the absolute values of these numbers;
b) equal to the sum of the absolute values of these numbers?
9. Prove inequalities:
a) | a-b | < |a| + |b |; b)* | a - b | > ||A | - | b ||.
When does the equal sign occur in these formulas?
Literature:
1. Bashmakov (basic level) 10-11 grades. – M., 2012
2. Bashmakov, 10th grade. Collection of problems. - M: Publishing center "Academy", 2008
3., Mordkovich: Reference materials: Book for students. - 2nd ed. - M.: Education, 1990
4. encyclopedic Dictionary young mathematician / Comp. .-M.: Pedagogy, 1989