Calculating With Significant Digits

The confidence interval Cl(fi) serves the same purpose as Cl(Xmean) in Section 1.3.2 the quality of these average values is described in a manner that is graphic and allows meaningful comparisons to be made. An example from photometry, see Table 2.2, is used to illustrate the calculations (see also data file UV.dat) further calculations, comments, and interpretations are found in the appropriate Sections. Results in Table 2.3 are tabulated with more significant digits than is warranted, but this allows the reader to check... 

The programmable calculator or PC that is to be used must be able to work with the number of significant digits required by the algorithm rounding the coefficients can appreciably alter the results of an approximation. 

In practice, in numerical calculations with a computer, both rational and imtiooal numbers are represented by a finite number of digits. In both cases, then, approximations are made and die errors introduced in the result depend on the number of significant figures carried by the computer - the machine precision. In die case of irrational numbers such errors cannot be avoided. 

We must report the results of our calculations to the proper number of significant digits. We almost always use our measurements to calculate other quantities and the results of the calculations must indicate to the reader the limit of accuracy with which the actual measurements were made. The rules for significant digits as the result of additions or subtractions with measured quantities are as follows ... 

When you take measurements and use them to calculate other quantities, you must be careful to keep track of which digits in your calculations and results are significant. Why Your results should not imply more certainty than your measured quantities justify. This is especially important when you use a calculator. Calculators usually report results with far more digits than your data warrant. Always remember that calculators do not make decisions about certainty. You do. Follow the rules given below to report significant digits in a calculated answer. 

The value with the fewest number of significant digits, going into a calculation, determines the number of significant digits that you should report in your answer. 

Examine each group s data and calculated value for density. Note how the number of significant digits in each value for density compares with the number of significant digits in the measured quantities. 

If we just leave the answer the way our electronic calculator gives it to us, anyone could assume that the measurement had been carried out with a precision of 1 part in 86,387, which is not true. We must reduce the number of significant digits in the answer to two because the factor with fewer significant digits has two. Thus, we change the answer to 86 cm. ... 

The reciprocal of a number is 1 divided by the number. The reciprocal has the same number of significant digits as the number itself. For example, the reciprocal of 2.00 is 0.500. A number times its reciprocal is equal to 1. The reciprocal key is especially useful if there is a calculated value in the display that is to be used as a denominator. For example, to calculate a b + c) with the value of + c in the display, divide by a, then take the reciprocal to get the answer. Alternatively, with the value of + c in the display, take its reciprocal and then multiply that value by a. 

With the calculator, determine the value of each expression to the proper number of significant digits ... 

The same information is obtained if consecutive decisions are reached between choices with different values of m, either as alternatives or as multiple choice questions (Fig. 3.3-11). This allows to calculate how much information a certain analytical procedure may supply or how much information is needed to solve a particular analytical problem. An analytical measure, such as a melting point or a refractive index with 3 significant digits, may supply Id 999 = 9.96 = 10 bit. In order to identify one compound in a spectral collection of N different samples, Id N bit are required to identify one out of 100 000 spectra, at least 16.61 bit are needed. 

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