Calculation decimal-based

The decimal-based monetary system in the United States yields calculations that are just like calculations with metric units, which are introduced in the next section. 

Much of modem life revolves around the use of the decimal system of numbers, although for many purposes it is far from ideal. This system (dec = ten) takes 10 as the basic unit of operation, and we say that we are working to base 10. However, the base 10 is not the only one possible the previous British and many other countries coinage was founded on a base of 12. The meaning of a base for purposes of calculation is illustrated in Figure 42.1. 

The system of units used in chemistry is the SI system (Systeme International), which is related to the metric system. There are base units for length, mass, etc. and decimal prefixes that modify the base unit. Since most of us do not tend to think in these units, it is important to be able to convert back and forth from the English system to the SI system. These three conversions are useful ones, although knowing the others might allow you to simplify your calculations ... 

Corporations pay an income tax based upon gross earnings, as shown in Table 9-19. Most major corporations pay the federal tax rate of 34 percent on their annual gross earnings. In addition, some states have a stepwise corporate income tax rate. State income tax is deductible as an expense item before the calculation of the federal tax. If 7) is the incremental tax rate and Tr is the incremental federal tax, both expressed as decimals, then the combined incremental rate T is... 

Although the value of e has converged to 18 decimal places, it is still not exact the addition of more terms causes the calculated value to change beyond the eighteenth decimal place. Likewise, attempts to calculate n are all based on the use of formulae with an infinite number of terms ... 

The procedure is very similar to the one set out in Toolbox 10.1. The only difference is that the acid or base is now an ion added as a salt. Although we shall calculate pH and pfC values to the number of significant figures appropriate to the data, the answers are often considerably less reliable than that. For instance, we might calculate the pH of a solution as 8.82, but in practice the answer is unlikely to be reliable to more than one decimal place (pH = 8.8). One reason for this poor reliability is that we are ignoring interactions between the ions in solution when we use concentrations in place of activities. 

The number of particles in one mole of a substance (the Avogadro constant) is often rounded down to 602 000 000 000 000 000 000 000. When performing calculations, however, this number is still too unwieldy. To simplify it, you can express it as a number between 1 and 10, multiplied by factors of 10. To do this, move the decimal point behind the left-most non-zero digit, counting the number of places that the decimal point moves. The number of places moved is the exponent for the base 10. 

Ultimately all arithmetic is based upon the natural numbers. If one multiplies 1.72 by. 047, for example, the mnltiplication is done with the natural numbers 172 and 47 then the result is converted to a decimal fraction by inserting a decimal point in the proper place-a process that is also done by counting. If one adds the fractions 1/3 and 2/7, the addition is not done directly, but only after converting the fractions to 7/21 and 6/21. Then the numerators are added, using natural-number arithmetic, and the denominators copied. Even computers and calculators reduce their complex and lightening-fast computations to simple steps involving only natural numbers. 

How interesting Now, we have the exact opposite of the problem that we encountered when we were calculating the volume of the wood block Now, we have measurements that are accurate to the thousandth place, but we have an answer that is based on those measurements that only claims accuracy to the tenth place. What do we do We follow the rule for addition and subtraction, and show an answer with as many decimal places as the measurement with the least number of decimal places. Each of the measurements show three digits past the decimal, so our answer becomes 10.100 g. Notice, we didn t change the value of the number, we simply reported the correct degree of certainty. 

Calculate the change in pH that occurs when 1.00 mmol of a strong base is added to 100 mL of the solutions listed in Problem 14-34. Calculate values to three decimal places. 

The following table gives conversion factors from varions nnits of measnre to SI nnits. It is reprodnced from NIST Special Pnblication 811, Guide for the Use of the International System of Units (SI). The table gives the factor by which a qnantity expressed in a non-SI nnit shonld be mnltiplied in order to calculate its valne in the SI. The SI valnes are expressed in terms of the base, snpplementary, and derived units of SI in order to provide a coherent presentation of the conversion factors and facilitate computations (see the table International System of Units in this section). If desired, powers of ten can be avoided by nsing SI prefixes and shifting the decimal point if necessary. 

Since three unit systems are in common use, it is often necessary to convert the magnitudes of quantities from one system to another. This is accomplished by using conversion factors. Only the defined conversion factors for the base units are required since conversion factors for all other units can be calculated from them. Interconversions between the SI and cgs systems are simple. Both use the same standards for time, temperature, and the mole, and only the decimal conversions defined by Eqs. (1.16) and (1.17) are needed. The SI and fps systems also use the second as the standard for time the three conversion factors defined for mass, length, and temperature by Eqs. (1.26), (1.27), and (1.28), respectively, are sufficient for all conversions of units between these two systems. 

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