Physical quantities are characterized by the concept of “error accuracy”. There is a saying that by taking measurements you can come to knowledge. This way you can find out the height of the house or the length of the street, like many others.
Introduction
Let us understand the meaning of the concept of “measure a quantity”. The measurement process is to compare it with homogeneous quantities, which are taken as a unit.
Liters are used to determine volume, grams are used to calculate mass. To make calculations more convenient, the SI system of international classification of units was introduced.
For measuring the length of the stick in meters, mass - kilograms, volume - cubic liters, time - seconds, speed - meters per second.
When calculating physical quantities, it is not always necessary to use the traditional method; it is enough to use the calculation using a formula. For example, to calculate indicators such as average speed, you need to divide the distance traveled by the time spent on the road. This is how the average speed is calculated.
When using units of measurement that are ten, one hundred, thousand times higher than the accepted measurement units, they are called multiples.
The name of each prefix corresponds to its multiplier number:
- Deca.
- Hecto.
- Kilo.
- Mega.
- Giga.
- Tera.
In physical science, powers of 10 are used to write such factors. For example, a million is written as 10 6 .
In a simple ruler, length has a unit of measurement - centimeters. It is 100 times less than a meter. A 15 cm ruler is 0.15 m long.
A ruler is the simplest type of measuring instrument for measuring lengths. More complex devices are represented by a thermometer - to a hygrometer - to determine humidity, an ammeter - to measure the level of force with which electric current propagates.
How accurate will the measurements be?
Take a ruler and a simple pencil. Our task is to measure the length of this stationery.
First you need to determine what the division price indicated on the scale of the measuring device is. On the two divisions, which are the closest strokes of the scale, numbers are written, for example, “1” and “2”.
It is necessary to count how many divisions are between these numbers. If counted correctly it will be "10". Let us subtract from the number that is larger the number that will be smaller and divide by the number that is the division between the digits:
(2-1)/10 = 0.1 (cm)
So we determine that the price that determines the division of stationery is the number 0.1 cm or 1 mm. It is clearly shown how the price indicator for division is determined using any measuring device.
When measuring a pencil with a length that is slightly less than 10 cm, we will use the knowledge gained. If there were no fine divisions on the ruler, it would be concluded that the object has a length of 10 cm. This approximate value is called the measurement error. It indicates the level of inaccuracy that can be tolerated when making measurements.
By determining the parameters of the length of a pencil with a higher level of accuracy, with a larger division price, greater measurement accuracy is achieved, which ensures a smaller error.
In this case, absolutely accurate measurements cannot be taken. And the indicators should not exceed the size of the division price.
It has been established that the measurement error is ½ of the price, which is indicated on the graduations of the device used to determine the dimensions.
After taking measurements of a pencil of 9.7 cm, we will determine its error indicators. This is the interval 9.65 - 9.85 cm.
The formula that measures this error is the calculation:
A = a ± D (a)
A - in the form of a quantity for measuring processes;
a is the value of the measurement result;
D - designation of absolute error.
When subtracting or adding values with an error, the result will be equal to the sum of the error indicators, which is each individual value.
Introduction to the concept
If we consider depending on the method of its expression, we can distinguish the following varieties:
- Absolute.
- Relative.
- Given.
The absolute measurement error is indicated by the letter “Delta” in capital. This concept is defined as the difference between the measured and actual values of the physical quantity that is being measured.
The expression of absolute measurement error is the units of the quantity that needs to be measured.
When measuring mass, it will be expressed, for example, in kilograms. This is not a measurement accuracy standard.
How to calculate the error of direct measurements?
There are ways to depict measurement errors and calculate them. To do this, it is important to be able to determine a physical quantity with the required accuracy, to know what the absolute measurement error is, that no one will ever be able to find it. Only its boundary value can be calculated.
Even if this term is used conventionally, it indicates precisely the boundary data. Absolute and relative measurement errors are indicated by the same letters, the difference is in their spelling.
When measuring length, the absolute error will be measured in the units in which the length is calculated. And the relative error is calculated without dimensions, since it is the ratio of the absolute error to the measurement result. This value is often expressed as a percentage or fraction.
Absolute and relative measurement errors have several different methods of calculation, depending on what physical quantity.
Concept of direct measurement
The absolute and relative errors of direct measurements depend on the accuracy class of the device and the ability to determine the weighing error.
Before we talk about how the error is calculated, it is necessary to clarify the definitions. Direct measurement is a measurement in which the result is directly read from the instrument scale.
When we use a thermometer, ruler, voltmeter or ammeter, we always carry out direct measurements, since we directly use a device with a scale.
There are two factors that influence the effectiveness of the readings:
- Instrument error.
- The error of the reference system.
The absolute error limit for direct measurements will be equal to the sum of the error that the device shows and the error that occurs during the counting process.
D = D (flat) + D (zero)
Example with a medical thermometer
The error indicators are indicated on the device itself. A medical thermometer has an error of 0.1 degrees Celsius. The counting error is half the division value.
D ots. = C/2
If the division value is 0.1 degrees, then for a medical thermometer you can make the following calculations:
D = 0.1 o C + 0.1 o C / 2 = 0.15 o C
On the back of the scale of another thermometer there is a specification and it is indicated that for correct measurements it is necessary to immerse the entire back of the thermometer. not specified. All that remains is the counting error.
If the scale division value of this thermometer is 2 o C, then it is possible to measure temperature with an accuracy of 1 o C. These are the limits of the permissible absolute measurement error and the calculation of the absolute measurement error.
A special system for calculating accuracy is used in electrical measuring instruments.
Accuracy of electrical measuring instruments
To specify the accuracy of such devices, a value called accuracy class is used. The letter “Gamma” is used to designate it. To accurately determine the absolute and relative measurement error, you need to know the accuracy class of the device, which is indicated on the scale.
Let's take an ammeter for example. Its scale indicates the accuracy class, which shows the number 0.5. It is suitable for measurements on direct and alternating current and belongs to electromagnetic system devices.
This is a fairly accurate device. If you compare it with a school voltmeter, you can see that it has an accuracy class of 4. You must know this value for further calculations.
Application of knowledge
Thus, D c = c (max) X γ /100
We will use this formula for specific examples. Let's use a voltmeter and find the error in measuring the voltage provided by the battery.
Let's connect the battery directly to the voltmeter, first checking whether the needle is at zero. When connecting the device, the needle deviated by 4.2 divisions. This state can be characterized as follows:
- It can be seen that the maximum U value for this item is 6.
- Accuracy class -(γ) = 4.
- U(o) = 4.2 V.
- C=0.2 V
Using these formula data, the absolute and relative measurement error is calculated as follows:
D U = DU (ex.) + C/2
D U (ex.) = U (max) X γ /100
D U (ex.) = 6 V X 4/100 = 0.24 V
This is the error of the device.
The calculation of the absolute measurement error in this case will be performed as follows:
D U = 0.24 V + 0.1 V = 0.34 V
Using the formula discussed above, you can easily find out how to calculate the absolute measurement error.
There is a rule for rounding errors. It allows you to find the average between the absolute and relative error limits.
Learning to determine weighing error
This is one example of direct measurements. Weighing has a special place. After all, lever scales do not have a scale. Let's learn how to determine the error of such a process. The accuracy of mass measurement is influenced by the accuracy of the weights and the perfection of the scales themselves.
We use lever scales with a set of weights that must be placed on the right pan of the scale. To weigh, take a ruler.
Before starting the experiment, you need to balance the scales. Place the ruler on the left bowl.
The mass will be equal to the sum of the installed weights. Let us determine the error in measuring this quantity.
D m = D m (scales) + D m (weights)
The error in mass measurement consists of two terms associated with scales and weights. To find out each of these values, factories producing scales and weights provide products with special documents that allow the accuracy to be calculated.
Using tables
Let's use a standard table. The error of the scale depends on what mass is put on the scale. The larger it is, the correspondingly larger the error.
Even if you put a very light body, there will be an error. This is due to the friction process occurring in the axes.
The second table is for a set of weights. It indicates that each of them has its own mass error. The 10 gram has an error of 1 mg, the same as the 20 gram. Let's calculate the sum of the errors of each of these weights taken from the table.
It is convenient to write the mass and mass error in two lines, which are located one below the other. The smaller the weights, the more accurate the measurement.
Results
In the course of the material reviewed, it was established that it is impossible to determine the absolute error. You can only set its boundary indicators. To do this, use the formulas described above in the calculations. This material is proposed for study at school for students in grades 8-9. Based on the knowledge gained, you can solve problems to determine the absolute and relative errors.
Absolute and relative error of numbers.
As characteristics of the accuracy of approximate quantities of any origin, the concepts of absolute and relative errors of these quantities are introduced.
Let us denote by a the approximation to the exact number A.
Define. The quantity is called the error of the approximate numbera.
Definition.
Absolute error 
approximate number a is called the quantity 
.
The practically exact number A is usually unknown, but we can always indicate the limits within which the absolute error varies.
Definition.
Maximum absolute error 
approximate number a is called the smallest of the upper bounds for the quantity 
, which can be found using this method of obtaining the numbera.
In practice, as 
choose one of the upper bounds for 
, quite close to the smallest.
Because the 
, That 
. Sometimes they write: 
.
Absolute error is the difference between the measurement result
and true (real) value measured quantity.
Absolute error and maximum absolute error are not sufficient to characterize the accuracy of measurement or calculation. Qualitatively, the magnitude of the relative error is more significant.
Definition.
Relative error 
We call the approximate number a the quantity:
Definition.
Maximum relative error 
approximate number a let's call the quantity
Because 
.
Thus, the relative error actually determines the magnitude of the absolute error per unit of measured or calculated approximate number a.
Example. Rounding the exact numbers A to three significant figures, determine
absolute D and relative δ errors of the obtained approximate
Given:
Find:
∆-absolute error
δ – relative error
Solution:
=|-13.327-(-13.3)|=0.027
,a 
0
*100%=0.203%
Answer:=0.027; δ=0.203%
2. Decimal notation of an approximate number. Significant figure. Correct digits of numbers (definition of correct and significant digits, examples; theory of the relationship between relative error and the number of correct digits).
Correct number signs.
Definition. The significant digit of an approximate number a is any digit other than zero, and zero if it is located between significant digits or is a representative of a stored decimal place.
For example, in the number 0.00507 = 
we have 3 significant figures, and in the number 0.005070= 
significant figures, i.e. the zero on the right, preserving the decimal place, is significant.
From now on, let us agree to write zeros on the right if only they are significant. Then, in other words,
All digits of a are significant, except for the zeros on the left.
In the decimal number system, any number a can be represented as a finite or infinite sum (decimal fraction):
Where 
,
- the first significant digit, m - an integer called the most significant decimal place of the number a.
For example, 518.3 =, m=2.
Using the notation, we introduce the concept of correct decimal places (in significant figures) approximately -
on the 1st day.
Definition.
It is said that in an approximate number a of the form n are the first significant digits 
,
where i= m, m-1,..., m-n+1 are correct if the absolute error of this number does not exceed half a unit of digit expressed by the nth significant digit:
Otherwise the last digit 
called doubtful.
When writing an approximate number without indicating its error, it is required that all written numbers
were faithful. This requirement is met in all mathematical tables.
The term “n correct digits” characterizes only the degree of accuracy of the approximate number and should not be understood to mean that the first n significant digits of the approximate number a coincide with the corresponding digits of the exact number A. For example, for the numbers A = 10, a = 9.997, all significant digits are different , but the number a has 3 valid significant digits. Indeed, here m=0 and n=3 (we find it by selection).
As mentioned earlier, when we compare the accuracy of a measurement of some approximate value, we use absolute error.
The concept of absolute error
The absolute error of the approximate value is the magnitude of the difference between the exact value and the approximate value.
Absolute error can be used to compare the accuracy of approximations of the same quantities, and if we are going to compare the accuracy of approximations of different quantities, then absolute error alone is not enough.
For example: The length of a sheet of A4 paper is (29.7 ± 0.1) cm. And the distance from St. Petersburg to Moscow is (650 ± 1) km. The absolute error in the first case does not exceed one millimeter, and in the second - one kilometer. The question is to compare the accuracy of these measurements.
If you think that the length of the sheet is measured more accurately because the absolute error does not exceed 1 mm. Then you are wrong. These values cannot be directly compared. Let's do some reasoning.
When measuring the length of a sheet, the absolute error does not exceed 0.1 cm per 29.7 cm, that is, in percentage terms it is 0.1/29.7 * 100% = 0.33% of the measured value.
When we measure the distance from St. Petersburg to Moscow, the absolute error does not exceed 1 km per 650 km, which as a percentage is 1/650 * 100% = 0.15% of the measured value. We see that the distance between cities is measured more accurately than the length of an A4 sheet.
The concept of relative error
Here, to assess the quality of the approximation, a new concept, relative error, is introduced. Relative error- this is the quotient of dividing the absolute error by the module of the approximate values of the measured value. Typically, the relative error is expressed as a percentage. In our example, we received two relative errors equal to 0.33% and 0.15%.
As you may have guessed, the relative error value is always positive. This follows from the fact that the absolute error is always a positive value, and we divide it by the module, and the module is also always positive.
In practice, usually the numbers on which calculations are performed are approximate values of certain quantities. For brevity, the approximate value of a quantity is called an approximate number. The true value of a quantity is called an exact number. An approximate number has practical value only when we can determine with what degree of accuracy it is given, i.e. estimate its error. Let us recall the basic concepts from the general mathematics course.
Let's denote: x- exact number (true value of the quantity), A- approximate number (approximate value of a quantity).
Definition 1. The error (or true error) of an approximate number is the difference between the number x and its approximate value A. Approximate number error A we will denote . So,
Exact number x most often it is unknown, so it is not possible to find the true and absolute error. On the other hand, it may be necessary to estimate the absolute error, i.e. indicate the number that the absolute error cannot exceed. For example, when measuring the length of an object with this tool, we must be sure that the error in the resulting numerical value will not exceed a certain number, for example 0.1 mm. In other words, we must know the absolute error limit. We will call this limit the maximum absolute error.
Definition 3. Maximum absolute error of the approximate number A is a positive number such that , i.e.
Means, X by deficiency, by excess. The following notation is also used:
| . | (2.5) |
It is clear that the maximum absolute error is determined ambiguously: if a certain number is the maximum absolute error, then any larger number is also the maximum absolute error. In practice, they try to choose the smallest and simplest number in writing (with 1-2 significant digits) that satisfies inequality (2.3).
Example.Determine the true, absolute and maximum absolute error of the number a = 0.17, taken as an approximate value of the number.
True error: 
Absolute error: 
The maximum absolute error can be taken as a number and any larger number. In decimal notation we will have: Replacing this number with a larger and possibly simpler notation, we accept:
Comment. If A is an approximate value of the number X, and the maximum absolute error is equal to h, then they say that A is an approximate value of the number X up to h.
Knowing the absolute error is not enough to characterize the quality of a measurement or calculation. Let, for example, such results be obtained when measuring length. Distance between two cities S 1=500 1 km and the distance between two buildings in the city S 2=10 1 km. Although the absolute errors of both results are the same, what is significant is that in the first case an absolute error of 1 km falls on 500 km, in the second - on 10 km. The measurement quality in the first case is better than in the second. The quality of a measurement or calculation result is characterized by relative error.
Definition 4. Relative error of the approximate value A numbers X is called the ratio of the absolute error of a number A to the absolute value of a number X:
Definition 5. Maximum relative error of the approximate number A is called a positive number such that .
Since , it follows from formula (2.7) that it can be calculated using the formula
| . | (2.8) |
For the sake of brevity, in cases where this does not cause misunderstandings, instead of “maximum relative error” we simply say “relative error”.
The maximum relative error is often expressed as a percentage.
Example 1. . Assuming , we can accept = . By dividing and rounding (necessarily upward), we get =0.0008=0.08%.
Example 2.When weighing the body, the result was obtained: p = 23.4 0.2 g. We have = 0.2. . By dividing and rounding, we get =0.9%.
Formula (2.8) determines the relationship between absolute and relative errors. From formula (2.8) it follows:
| . | (2.9) |
Using formulas (2.8) and (2.9), we can, if the number is known A, using a given absolute error, find the relative error and vice versa.
Note that formulas (2.8) and (2.9) often have to be applied even when we do not yet know the approximate number A with the required accuracy, but we know a rough approximate value A. For example, you need to measure the length of an object with a relative error of no more than 0.1%. The question is: is it possible to measure the length with the required accuracy using a caliper, which allows you to measure the length with an absolute error of up to 0.1 mm? We may not have measured an object with an exact instrument yet, but we know that a rough approximation of the length is about 12 cm. Using formula (1.9) we find the absolute error:
This shows that using a caliper it is possible to perform measurements with the required accuracy.
In the process of computational work, it is often necessary to switch from absolute to relative error, and vice versa, which is done using formulas (1.8) and (1.9).
No measurement is free from errors, or, more precisely, the probability of a measurement without errors approaches zero. The type and causes of errors are very diverse and are influenced by many factors (Fig. 1.2).
The general characteristics of the influencing factors can be systematized from various points of view, for example, according to the influence of the listed factors (Fig. 1.2).
Based on the measurement results, errors can be divided into three types: systematic, random and errors.
Systematic errors in turn, they are divided into groups due to their occurrence and the nature of their manifestation. They can be eliminated in various ways, for example, by introducing amendments.
rice. 1.2
Random errors are caused by a complex set of changing factors, usually unknown and difficult to analyze. Their influence on the measurement result can be reduced, for example, by repeated measurements with further statistical processing of the results obtained using the probability theory method.
TO misses These include gross errors that arise from sudden changes in experimental conditions. These errors are also random in nature and, once identified, must be eliminated.
The accuracy of measurements is assessed by measurement errors, which are divided according to the nature of their occurrence into instrumental and methodological and according to the calculation method into absolute, relative and reduced.
Instrumental The error is characterized by the accuracy class of the measuring device, which is given in its passport in the form of normalized main and additional errors.
Methodical the error is due to the imperfection of measurement methods and instruments.
Absolute the error is the difference between the measured G u and the true G values of a quantity, determined by the formula:
Δ=ΔG=G u -G
Note that the quantity has the dimension of the measured quantity.
Relative the error is found from the equality
δ=±ΔG/G u ·100%
Given the error is calculated using the formula (accuracy class of the measuring device)
δ=±ΔG/G norm ·100%
where G norms is the normalizing value of the measured quantity. It is taken equal to:
a) the final value of the instrument scale, if the zero mark is on the edge or outside the scale;
b) the sum of the final values of the scale without taking into account signs, if the zero mark is located inside the scale;
c) the length of the scale, if the scale is uneven.
The accuracy class of a device is established during its testing and is a standardized error calculated using the formulas
γ=±ΔG/G norms ·100%, ifΔG m =const
where ΔG m is the largest possible absolute error of the device;
G k – final value of the measuring limit of the device; c and d are coefficients that take into account the design parameters and properties of the measuring mechanism of the device.
For example, for a voltmeter with a constant relative error, the equality holds
δ m =±c
The relative and reduced errors are related by the following dependencies:
a) for any value of the reduced error
δ=±γ·G norms/G u
b) for the largest reduced error
δ=±γ m ·G norms/G u
From these relations it follows that when making measurements, for example with a voltmeter, in a circuit at the same voltage value, the lower the measured voltage, the greater the relative error. And if this voltmeter is chosen incorrectly, then the relative error can be commensurate with the value G n , which is unacceptable. Note that in accordance with the terminology of the problems being solved, for example, when measuring voltage G = U, when measuring current C = I, the letter designations in the formulas for calculating errors must be replaced with the corresponding symbols.
Example 1.1. A voltmeter with values γ m = 1.0%, U n = G norms, G k = 450 V, measure the voltage U u equal to 10 V. Let us estimate the measurement errors.
Solution.
Answer. The measurement error is 45%. With such an error, the measured voltage cannot be considered reliable.
If the possibilities for choosing a device (voltmeter) are limited, the methodological error can be taken into account by an amendment calculated using the formula
Example 1.2. Calculate the absolute error of the V7-26 voltmeter when measuring voltage in a DC circuit. The accuracy class of the voltmeter is specified by the maximum reduced error γ m =±2.5%. The voltmeter scale limit used in the work is U norm = 30 V.
Solution. The absolute error is calculated using the known formulas:
(since the reduced error, by definition, is expressed by the formula 
, then from here you can find the absolute error: 
Answer.ΔU = ±0.75 V.
Important steps in the measurement process are processing of results and rounding rules. The theory of approximate calculations allows, knowing the degree of accuracy of the data, to evaluate the degree of accuracy of the results even before performing actions: to select data with the appropriate degree of accuracy, sufficient to ensure the required accuracy of the result, but not too great to save the calculator from useless calculations; rationalize the calculation process itself, freeing it from those calculations that will not affect the exact numbers and results.
When processing results, rounding rules are applied.
- Rule 1. If the first digit discarded is greater than five, then the last digit retained is increased by one.
- Rule 2. If the first of the discarded digits is less than five, then no increase is made.
- Rule 3. If the discarded digit is five and there are no significant digits behind it, then rounding is done to the nearest even number, i.e. the last digit stored remains the same if it is even and increases if it is not even.
If there are significant figures behind the number five, then rounding is done according to rule 2.
By applying Rule 3 to rounding a single number, we do not increase the precision of the rounding. But with numerous roundings, excess numbers will occur about as often as insufficient numbers. Mutual error compensation will ensure the greatest accuracy of the result.
A number that obviously exceeds the absolute error (or in the worst case is equal to it) is called maximum absolute error.
The magnitude of the maximum error is not entirely certain. For each approximate number, its maximum error (absolute or relative) must be known.
When it is not directly indicated, it is understood that the maximum absolute error is half a unit of the last digit written. So, if an approximate number of 4.78 is given without indicating the maximum error, then it is assumed that the maximum absolute error is 0.005. As a result of this agreement, you can always do without indicating the maximum error of a number rounded according to rules 1-3, i.e., if the approximate number is denoted by the letter α, then
Where Δn is the maximum absolute error; and δ n is the maximum relative error.
In addition, when processing the results, we use rules for finding an error sum, difference, product and quotient.
- Rule 1. The maximum absolute error of the sum is equal to the sum of the maximum absolute errors of the individual terms, but with a significant number of errors of the terms, mutual compensation of errors usually occurs, therefore the true error of the sum only in exceptional cases coincides with the maximum error or is close to it.
- Rule 2. The maximum absolute error of the difference is equal to the sum of the maximum absolute errors of the one being reduced or subtracted.
The maximum relative error can be easily found by calculating the maximum absolute error.
- Rule 3. The maximum relative error of the sum (but not the difference) lies between the smallest and largest of the relative errors of the terms.
If all terms have the same maximum relative error, then the sum has the same maximum relative error. In other words, in this case the accuracy of the sum (in percentage terms) is not inferior to the accuracy of the terms.
In contrast to the sum, the difference of the approximate numbers may be less precise than the minuend and subtrahend. The loss of precision is especially great when the minuend and subtrahend differ little from each other.
- Rule 4. The maximum relative error of the product is approximately equal to the sum of the maximum relative errors of the factors: δ=δ 1 +δ 2, or, more precisely, δ=δ 1 +δ 2 +δ 1 δ 2 where δ is the relative error of the product, δ 1 δ 2 - relative errors factors.
Notes:
1. If approximate numbers with the same number of significant digits are multiplied, then the same number of significant digits should be retained in the product. The last digit stored will not be completely reliable.
2. If some factors have more significant digits than others, then before multiplying, the first ones should be rounded, keeping in them as many digits as the least accurate factor or one more (as a spare), saving further digits is useless.
3. If it is required that the product of two numbers have a predetermined number that is completely reliable, then in each of the factors the number of exact digits (obtained by measurement or calculation) must be one more. If the number of factors is more than two and less than ten, then in each of the factors the number of exact digits for a complete guarantee must be two units more than the required number of exact digits. In practice, it is quite enough to take only one extra digit.
- Rule 5. The maximum relative error of the quotient is approximately equal to the sum of the maximum relative errors of the dividend and divisor. The exact value of the maximum relative error always exceeds the approximate one. The percentage of excess is approximately equal to the maximum relative error of the divider.
Example 1.3. Find the maximum absolute error of the quotient 2.81: 0.571.
Solution. The maximum relative error of the dividend is 0.005:2.81=0.2%; divisor – 0.005:0.571=0.1%; private – 0.2% + 0.1% = 0.3%. The maximum absolute error of the quotient will be approximately 2.81:0.571·0.0030=0.015
This means that in the quotient 2.81:0.571=4.92 the third significant figure is not reliable.
Answer. 0,015.
Example 1.4. Calculate the relative error of the readings of a voltmeter connected according to the circuit (Fig. 1.3), which is obtained if we assume that the voltmeter has an infinitely large resistance and does not introduce distortions into the measured circuit. Classify the measurement error for this problem.
rice. 1.3
Solution. Let us denote the readings of a real voltmeter by AND, and a voltmeter with infinitely high resistance by AND ∞. Required relative error 
notice, that
then we get
Since R AND >>R and R > r, the fraction in the denominator of the last equality is much less than one. Therefore, you can use the approximate formula 
, valid for λ≤1 for any α. Assuming that in this formula α = -1 and λ= rR (r+R) -1 R And -1, we obtain δ ≈ rR/(r+R) R And.
The greater the resistance of the voltmeter compared to the external resistance of the circuit, the smaller the error. But condition R< Answer. Systematic methodological error. Example 1.5.
The DC circuit (Fig. 1.4) includes the following devices: A – ammeter type M 330, accuracy class K A = 1.5 with a measurement limit I k = 20 A; A 1 - ammeter type M 366, accuracy class K A1 = 1.0 with a measurement limit I k1 = 7.5 A. Find the largest possible relative error in measuring current I 2 and the possible limits of its actual value, if the instruments showed that I = 8 ,0A. and I 1 = 6.0A. Classify the measurement. rice. 1.4 Solution. We determine the current I 2 from the readings of the device (without taking into account their errors): I 2 =I-I 1 =8.0-6.0=2.0 A. Let's find the absolute error modules of ammeters A and A 1 For A we have the equality Let's find the sum of absolute error modules: Consequently, the largest possible value of the same value, expressed in fractions of this value, is equal to 1. 10 3 – for one device; 2·10 3 – for another device. Which of these devices will be the most accurate? Solution. The accuracy of the device is characterized by the reciprocal of the error (the more accurate the device, the smaller the error), i.e. for the first device this will be 1/(1 . 10 3) = 1000, for the second – 1/(2 . 10 3) = 500. Note that 1000 > 500. Therefore, the first device is twice as accurate as the second one. A similar conclusion can be reached by checking the consistency of the errors: 2. 10 3 / 1. 10 3 = 2. Answer. The first device is twice as accurate as the second. Example 1.6.
Find the sum of the approximate measurements of the device. Find the number of correct characters: 0.0909 + 0.0833 + 0.0769 + 0.0714 + 0.0667 + 0.0625 + 0.0588+ 0.0556 + 0.0526. Solution. Adding up all the measurement results, we get 0.6187. The maximum maximum error of the sum is 0.00005·9=0.00045. This means that in the last fourth digit of the sum, an error of up to 5 units is possible. Therefore, we round the amount to the third digit, i.e. thousandths, we get 0.619 - a result in which all the signs are correct. Answer. 0.619. The number of correct digits is three decimal places.
for ammeter 