Inexact numbers

Two kinds of numbers are encountered in scientific work exact numbers (those whose values are known exactly) and inexact numbers (those whose values have some uncertainty). Most of the exact numbers we will encounter in this book have defined values. For example, there are exactly 12 e s in a dozen, exactly 1000 g in a kilogram, and exactly 2.54 cm in an inch. The number 1 in any conversion factor, such as 1 m = 100 cm or 1 kg = 2.2046 lb, is an exact number. Exact numbers can also result from counting objects. For example, we can count the exact number of marbles in a jar or the exact munber of people in a classroom. 

Measured numbers are inexact. Numbers obtained by counting or that are part of a definition are exact numbers. 

To summarize, the Carnot cycle or the Caratheodory principle leads to an integrating denominator that converts the inexact differential 8qrev into an exact differential. This integrating denominator can assume an infinite number of forms, one of which is the thermodynamic (Kelvin) temperature T that is equal to the ideal gas (absolute) temperature. The result is... 

Note Exact conversions 1 m = 100 cm, 1 km = 1000 m. Inexact conversion 1 mile = 1.609 km to 4 significant figures. The number of significant figures in the answer is set by the data (4 sig. figs.) but the answer has extra source of error since the conversion from kilometers to miles is only good to 4 sig. figs. 

For non-numeric, inexact and incomplete information I = 01 U II used for the reduction of the set W (m, n) of all possible vectors w to obtain the discrete components set W (I n, m) is defined a number of constrain reflecting non-numeric information about mutual relation among the criteria under consideration. 

Zakharov [126] criticized Desvergnes s view, which he considered to be inexact in a number of points. 

A number of inexact applications have been made, some of which have been discussed in detail. As an example, in some work (147) on diastereoisomeric salts of (+)bromocamphorsulphonicacid, (+)tartaricacid, and of other resolving agents, with complexes of the type [Co(en)2(a)]2+, comparisons were made where a is glycinate, L-alaninate, L-leucinate, and L-phenylalaninate, which may not be justified, as owing to the differences in steric requirements of the amino acid, there is no reason to believe that these less soluble diastereoisomers will be isomorphous. 

In the derivation of Eq. (152), we neglected the quantity of the species produced or consumed at the electrode. In practice this is true for microscale electrochemical experiments but is necessarily inexact for macroscale or exhaustive electrolysis. To derive the general equation, let us consider the total equation for the bulk solution. The chemical contribution is given in Eq. (152). On the other hand, a flux of the species at the convection layer limit (5conv must be considered. This flux corresponds to an algebraic number dNgiec of moles of the species produced, given in Eq. (154),... 

For centuries, units of measurement were fairly inexact. A person might mark off the boundaries of a property by walking and counting the number of steps. The passage of time could be estimated with a sundial or an hourglass filled with sand. Such estimates worked for ordinary tasks. Scientists, however, need to report data that can be reproduced by other scientists. They need standard units of measurement. In 1795, French scientists adopted a system of standard units called the metric system. In 1960, an international committee of scientists met to update the metric system. The revised system is called the Systeme Internationale d Unites, which is abbreviated SI. 

The most commonly used definition of efficiency is the overall efficiency, defined as the number of theoretical trays required for a separation divided by the number of actual trays. A theoretical tray is defined as a contacting stage for which the vapor and the liquid streams leaving would be in thermodynamic equilibrium. The overall efficiency is inexact because separation... 

These numbers should not be taken as being precise, given the somewhat arbitrary and inexact nature of the terms used to calculate them. However, it is probably sensible to round up rather than down. 

This has been attempted by a fairly large number of workers, with only partial success, however To get an idea of this method of treatment the reader should consult the paper of O Stem (Zeitsch physik Chem, 81, 441, 1912)1 Apart from the fact that the van der Waals equation is not a very accurate one for compressed, t e concentrated, gases, and therefore may be expected to be similarly inexact for solutions, it will be at once evident that the constants a and b m the case of solutions are much more complex quantities than in the case of a single gas,... 

The average human body contains 5.2 L of blood. What is this volume in quarts The unit analysis setup for this conversion is below. Identify whether each value in the setup is exact or not. Determine the number of significant figures in each inexact value, calculate the answer, and report it to the correct number of significant figures. 

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