Approximate value of magnitude and error of approximations definition. Exact and approximate values ​​of quantities

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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 number is an approximate value of the quantity Δ = , then the difference a - a" called*.

* Δ approximation error ε - Greek letter; read: delta. Next comes another Greek letter

(read: epsilon). Δ 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 - 3,8 = -0,044.

= 3.756 - 3.7 = 0.056. If we take 3.8 as an approximate value, then the error will be equal to: Δ In practice, the approximation error is most often used Δ , and the absolute value of this error | |. In what follows we will simply call this absolute value of error absolute error
|Δ . 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 | = |0,056| = 0,056.

| = | - 0.044| =0.044, and for the second | If is an approximate value of the number Numberε up toε :

|, then the difference | < ε .

, if the absolute error of this approximation is less than< 0,1.

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

a ≈ a" If < is an approximate value of the number Similarly, - 3/2 can be considered as an approximation of the number - 8/5 to within 1/5, since If , That is an approximate value of the number called the approximate value of the number.

with a disadvantage If > is an approximate value of the number Similarly, - 3/2 can be considered as an approximation of the number - 8/5 to within 1/5, since If , That is an approximate value of the number If

in abundance.< 3,671, а - 3 / 2 есть приближенное значение числа - 8 / 5 c избытком, так как - 3 / 2 > - 8 / 5 .

For example, 3.6 is an approximate value of the number 3.671 with a disadvantage, since 3.6 is an approximate value of the number If instead of numbers we And b If If instead of numbers we add up their approximate values b" , then the result a" + b" will be an approximate value of the sum . 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:

|will be an approximate value of the sum | < |a | + |And |.

End of work -

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Methodological manual for performing practical work in the discipline of mathematics, part 1

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Explanatory note
The methodological manual is compiled in accordance with work program in the discipline "Mathematics", developed on the basis of the Federal State educational standard third generation n

Proportions. Interest.
Lesson objectives: 1) Summarize theoretical knowledge on the topic “Percentages and Proportions.”

2) Consider the types and algorithms for solving problems involving percentages, drawing up proportions and solving them
Proportion.

Proportion (from Latin proportio - ratio, proportionality), 1) in mathematics - equality between two ratios of four quantities a, b, c,
PRACTICAL WORK No. 2

“Equations and Inequalities” Lesson objectives: 1) Summarize theoretical knowledge on the topic: “Equations and Inequalities.”
2) Consider algorithms for solving tasks on the topic “Ur” Equations containing a variable under the modulus sign. The modulus of a number is determined as follows: Example: Solve the equation.

SOLUTION: If, then and
given equation

will take the form. You can write it like this:
Equations with a variable in the denominator. Let's consider equations of the form. (1) The solution to an equation of type (1) is based on the following statement: a fraction is equal to 0 if and only if its numerator is equal to 0 and its denominator is non-zero. Rational equations.

The equation f(x) = g(x) is called rational if f(x) and g(x)
-rational expressions

. Moreover, if f(x) and g(x) are integer expressions, then the equation is called an integer;
An equation is called irrational in which the variable is contained under the sign of the root or under the sign of raising to a fractional power. One of the methods for solving such equations is the vozm method.

Interval method
Example: Solve an inequality.

Solution. ODZ: where we have x [-1; 5) (5; +) Solve the equation The numerator of the fraction is equal to 0 at x = -1, this is the root of the equation.
Exercises for independent work.

3x + (20 – x) = 35.2, (x – 3) - x = 7 – 5x.
(x + 2) - 11(x + 2) = 12. x = x, 3y = 96, x + x + x + 1 = 0, – 5.5n(n – 1)(n + 2.5)( n-

PRACTICAL WORK No. 4
“Functions, their properties and graphs” Lesson objectives: 1) Summarize theoretical knowledge on the topic: “Functions, properties and graphs”.

2) Consider the algorithm
It would be a grave mistake if, when drawing up a drawing, you carelessly allow the graph to intersect with an asymptote.

Example 3 Construct the right branch of a hyperbola We use the pointwise construction method, in which case it is advantageous to select the values ​​so that they are divisible by an integer:
Graphs of inverse trigonometric functions

Let's build a graph of the arcsine Let's build a graph of the arccosine Let's build a graph of the arctangent Just an inverted branch of the tangent. Let's list the main

Mathematical portraits of proverbs

Modern mathematics knows many functions, and each has its own unique appearance, just as the unique appearance of each of the billions of people living on Earth is unique. However, despite all the dissimilarity of one person
Construct graphs of functions a)y=x2,y=x2+1,y=(x-2)2 b)y=1/x, y=1/(x-2),y=1/x -2 on one coordinate plane.

Graph functions c
Integers

Properties of addition and multiplication of natural numbers
a + b = b + a - commutative property of addition (a + b) + c = a + (b +c) - associative property of addition ab = ba

Signs of divisibility of natural numbers
If each term is divisible by a number, then the sum is divisible by that number. If in a product at least one of the factors is divisible by a certain number, then the product is also divisible.

Scales and coordinates
The lengths of the segments are measured with a ruler. There are strokes on the ruler (Fig. 19). They break the ruler into equal parts. These parts are called divisions. In Figure 19 the length ka is another form of writing a fraction with a denominator. For example, . If the factorization of the denominator of a fraction into prime factors contains only 2 and 5, then this fraction can be written as dec

Root of 2
Let us assume the opposite: it is rational, that is, it is represented in the form of an irreducible fraction, where is an integer, and - natural number. Let's square the supposed equality: .

From here
The absolute value of the sum of any two numbers does not exceed the sum of their absolute values.< a, то величина a называется

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 basic level of

identity transformations
expressions. The first type: the transformation that needs to be performed is explicitly specified.

identity transformations
For example. 1

identity transformations
Problems to solve independently

Mark the number of the correct answer: The result of simplifying the expression is 1. ; 4. ; 2. ; 5. . 3. ;
The value of the expression is 1) 4; 2) ; 3)

Find the value of expression 1. .2. . 2. . 3. . 4. . 5. .7. . 6.. at. 7.. at. 8.. at. 9. at. 1

Question 1. Find the logarithm of 25 to base 5. Question 2. Find the logarithm to base 5. Question 3.

PRACTICAL WORK No. 17

“Axioms of stereometry and consequences from them” Purpose of the lesson: 1) Summarize theoretical knowledge

Page 2

Mathematical operations on approximate values ​​of quantities are called approximate calculations. To date, a whole science of approximate calculations has been created, a number of provisions of which we will become familiar with later.  

The measurement result always gives an approximate value of the quantity. This is due to the inaccuracy of the measurements themselves and the imperfect accuracy of the measuring instruments.   What is called the relative error of the approximate value of a quantity.   In table Figure 25 shows the approximate values ​​of /Ci/ - d at various amplitudes Um0 for a 6X6 diode loaded with a resistance R0 5 mg. This table was compiled by Prof.  

If in a given exact or approximate value the number of digits is greater than is necessary for practical reasons, then this number is rounded. The operation of rounding numbers consists of discarding several low-order digits and replacing them with zeros; in this case, the last retained digit is left unchanged if the first discarded digit is less than 5; if it is equal to or greater than 5, then the digit of the last held digit is increased by one.  

Let us agree to assume that in the approximate value of a quantity all figures are correct if its absolute error does not exceed half a unit of the last digit.  

With this rounding, the number characterizing the approximate value of the quantity consists of correct digits, and the lowest digit of this number (the last one in the record) has an accuracy of 1 of the same digit. For example, the entry t3 68 kg means t3 68 0 01 kg, and the entry t3 680 kg means t3 680 0 001 kg.  

From the equation it is clear that the sum of the approximate values ​​of the quantities A and the sum of their errors are the approximate value of the sums of the quantities X and their absolute error.  

N) in (1) denotes the approximate value of the quantity y (xi, x0, g / o) obtained by the method under consideration.  

Calculations, as a rule, are made with approximate values ​​of quantities - approximate numbers. A reasonable estimate of the error in calculations allows you to indicate the optimal number of digits that should be retained during calculations, as well as in the final result.  

As a result of the calculation, you can obtain either an exact or approximate value of a quantity. In this case, a sufficient sign that the counting result is close is the presence of different answers during repeated calculations.  

In fact, the arithmetic mean X will give him only an approximate value of the value a xf, and if the very scheme of his experiment was unsatisfactory or the instruments were poorly tested (for example, a measuring ruler instead of 1 m is equal to 0 999 mm), then no matter how accurately our observer finds value a, he has no reason to believe that X or a correspond to the true value of the speed of sound, which can be observed in a wide variety of other experiments. The main assumption that should justify the use of the arithmetic mean method is physical measurements of this kind, consists in the assumption that the unknown quantity a xf or, in other words, that the measurement (or calculation) is carried out without systematic error.  

In practice, when measuring areas, we most often use approximate values.  

Introduction

Absolute error- is an estimate of the absolute measurement error. Calculated different ways. The calculation method is determined by the distribution of the random variable. Accordingly, the magnitude of the absolute error depending on the distribution of the random variable may be different. If is the measured value and is the true value, then the inequality must be satisfied with a certain probability close to 1. If a random variable is distributed according to a normal law, then its standard deviation is usually taken as the absolute error. Absolute error is measured in the same units as the quantity itself.

There are several ways to write a quantity along with its absolute error.

· Usually the notation with the ± sign is used. For example, the 100 meter record, set in 1983, is 9.930±0.005 s.

· To record quantities measured with very high accuracy, another notation is used: the numbers corresponding to the error of the last digits of the mantissa are added in parentheses. For example, the measured value of Boltzmann's constant is 1.380 6488 (13)?10 ?23 J/K, which can also be written much longer as 1.380 6488?10 ?23 ±0.000 0013?10 ?23 J/K.

Relative error- measurement error, expressed as the ratio of the absolute measurement error to the actual or average value of the measured value (RMG 29-99):.

The relative error is a dimensionless quantity or measured as a percentage.

Approximation

With excess and insufficient? In the process of calculations, one often has to deal with approximate numbers. Let A- exact value of a certain value, hereinafter called exact number A. Under the approximate value A, or approximate numbers called number A, replacing the exact value of the quantity A. a ≈ a" is an approximate value of the number< A, That is an approximate value of the number called the approximate value of the number And for lack. a ≈ a" is an approximate value of the number> A,- That by excess. For example, 3.14 is an approximation of the number R by deficiency, and 3.15 by excess. To characterize the degree of accuracy of this approximation, the concept is used errors or errors.

Accuracy D is an approximate value of the number approximate number is an approximate value of the number called a difference of the form

D a = A-A,

Where A- the corresponding exact number.

From the figure it can be seen that the length of segment AB is between 6 cm and 7 cm.

This means that 6 is an approximate value of the length of segment AB (in centimeters) > with a deficiency, and 7 with an excess.

Denoting the length of the segment by the letter y, we get: 6< у < 1. Если a < х < b, то а называют приближенным значением числа х с недостатком, a b - приближенным значением х с избытком. Длина segment AB (see Fig. 149) is closer to 6 cm than to 7 cm. It is approximately equal to 6 cm. They say that the number 6 was obtained by rounding the length of the segment to whole numbers.

General information

Often, an exact number is represented by a limited number of digits, discarding “extra” digits, or rounding it to a certain digit. This number is called approximate.

The true error of the approximate number, i.e. the difference between exact and approximate numbers, when discarding digits, does not exceed one digit of the last stored digit, and when discarding with rounding, performed according to the rules established by the standard, half a unit of the digit of the stored digit.

An approximate number is characterized by the number of significant digits, which include all digits except the zeros on the left.

The numbers in the recording of an approximate number are called correct if the error does not exceed half a unit of the last digit.

Approximate numbers also include the results of measuring A, which evaluate the actual values ​​of A d of the measured value. Since the true error of the obtained result is unknown, it is replaced by the concept of the maximum absolute error Δ pr = | A - A d | or maximum relative error δ pr = Δ pr / A (more often indicated as a percentage δ pr = 100 Δ pr / A)

The maximum relative error of the approximate number can be estimated using the formula:

where δ is the number of correct significant figures;

n 1 – first from the left significant figure.

To determine the required number of correct signs providing a given limit relative error you should follow the rules:

if the first significant digit does not exceed three, then the number of correct digits must be one more than the modulus of the exponent |-q|

at 10 in a given relative error δ pr = 10 -q

if the first significant digit is 4 or more, then the modulus of the indicator q is equal to the number of correct digits.
)

(If δ pr = 10 - q, then S can be determined by the formula

Rules for calculations with approximate numbers

The result of summing (subtracting) approximate numbers will have as many correct signs as the summand with the least number of correct signs.

When multiplying (dividing), the resulting result will have as many significant correct digits as there are in the original number with the least number of correct digits.

When raising to a power (extracting the root) of any power, the result has as many correct signs as there are in the base.

The number and mantissa of its logarithm contain the same number of correct signs. In order to reduce rounding errors as much as possible, it is recommended that in those source data that allow this, as well as as a result, if it is involved in further calculations, one extra digit is retained in addition to what is determined by rules 1-4.

3. Accuracy class and its use for assessing the instrumental error of instruments

Accuracy class is a generalized characteristic used to assess the maximum values ​​of the main and additional errors.

The main error is the error of the device inherent in it under normal operating conditions.

Operating conditions are determined by the values ​​of quantities influencing the readings of devices that are not informative for a given device. Influencing quantities include the temperature of the environment in which measurements are performed, the position of the instrument scale, the frequency of the measured value (not for frequency meters), the strength of the external magnetic (or electric) field, the supply voltage of electronic and digital devices, etc.

The technical documentation of the device indicates the normal and operating ranges of the influencing quantities. Using the device with an influencing quantity outside the operating range is not permitted.

The accuracy class of the device is determined in the form:

absolute error limit Δ pr = ± a or Δ pr = ± (a + b A);

relative error limit δ pr = ± p or δ pr = ± ;

reduced error limit γ pr = ± k

Numbers a, b, p, c, d, k are chosen from row 1; 1.5; 2; 2.5; 4; 5; 6 10 n, where n = 1, 0, -1, -2, etc.

A – instrument readings;

And max is the upper limit of the used measurement range of the device.

Reduced error

,

where A n is the normalizing value conventionally accepted for a given device, depending on the shape of the scale.

The definitions of AN for the most frequently encountered scales are given below:

a) one-sided scale b) scale with zero inside

A n = A max A n = |A 1 | + A 2

c) scale without zero d) significantly uneven scale (for ohmmeters, phase meters)

A n = A 2 – A 1 A n = L

Rules and examples for designating accuracy classes are given in Table 3.1.

Table 3.1

Subject " ” is studied fluently in 9th grade. And students, as a rule, do not fully develop the skills to calculate it.

But with practical application relative error of the number , as well as with absolute error, we encounter at every step.

During the repair work, we measured (in centimeters) the thickness m carpeting and width n threshold. We got the following results:

m≈0.8 (with an accuracy of 0.1);

n≈100.0 (accurate to 0.1).

Note that the absolute error of each measurement data is no more than 0.1.

However, 0.1 is a significant part of the number 0.8. As fornumber 100 it represents insignificant his. This shows that the quality of the second dimension is much higher than the first.

To assess the quality of measurement it is used relative error of the approximate number.

Definition.

Relative error of the approximate number (values) is the ratio of the absolute error to the absolute value of the approximate value.

They agreed to express the relative error as a percentage.

Example 1.

Consider the fraction 14.7 and round it to whole numbers. We will also find relative error of the approximate number:

14,7≈15.

To calculate the relative error, in addition to the approximate value, as a rule, you also need to know the absolute error. The absolute error is not always known. Therefore calculate impossible. And in this case, it is enough to indicate an estimate of the relative error.

Let's remember the example that was given at the beginning of the article. The thickness measurements were indicated there. m carpet and width n threshold.

Based on the results of measurements m≈0.8 with an accuracy of 0.1. We can say that the absolute measurement error is no more than 0.1. This means that the result of dividing the absolute error by the approximate value (and this is the relative error) is less than or equal to 0.1/0.8 = 0.125 = 12.5%.

Thus, the relative approximation error is ≤ 12.5%.

In a similar way, we calculate the relative error in approximating the width of the sill; it is no more than 0.1/100 = 0.001 = 0.1%.

They say that in the first case the measurement was carried out with a relative accuracy of up to 12.5%, and in the second - with a relative accuracy of up to 0.1%.

Summarize.

Absolute error approximate number - this is the differencebetween the exact number x and its approximate value a.

If the difference modulus | x – a| less than some D a, then the value D a called |. In what follows we will simply call this absolute value of error approximate number a.

Relative error of the approximate number is the ratio of the absolute error D a to the modulus of a number a, that isD a / |a| = d a .

Example 2.

Let's consider the known approximate value of the number π≈3.14.

Considering its value with an accuracy of one hundred thousandths, you can indicate its error as 0.00159... (it will help to remember the digits of the number π )

The absolute error of the number π is equal to: | 3,14 – 3,14159 | = 0,00159 ≈0,0016.

The relative error of the number π is equal to: 0.0016/3.14 = 0.00051 = 0.051%.

Example 3.

Try to calculate it yourself relative error of the approximate number √2. There are several ways to remember the digits of a number “ Square root from 2″.