Significant digit

In general, all operations are carried out in full, and only the final results are rounded, in order to avoid the loss of information from repeated rounding. For this reason, several additional digits are carried in all calculations until the final selected set of data is developed, and only then are data rounded. 

The uncertainty of a value basically defines the number of significant digits a value should be given. 

The Gibbs energy minimisation and optimisation program, NONLINT-SIT 

In this Appendix, no attempt will be made to present details of all the equations involved in calcnlations we restrict onrselves to a very general description of the for-mnlation of Gibbs-energy minimisation problem and the optimisation procedure, illustrated by an example. Readers interested in the details of the programs and the detailed equations involved shonld consnlt [1995FEL] and the very detailed nser mannal of INSIGHT, [1997STE/FEL]. 

The program can be nsed to analyse different types of experimental data (e.g., solvent extraction, ion-exchange, potentiometric, solubility of pure phases, solubility of solid solutions) involving aqneons, solid, and gaseous phases at different temperatnres. 

For example, consider the rectangular block pictured in Fig. 2-1. The ruler at the top of the block is divided into centimeters. You can estimate the length of a block to the nearest tenth of a centimeter (millimeter). 

EXAMPLE 2.20. Which of the two rulers shown in Eig. 2-1 was used to make each of the following measurements  

The measurements reported in (a), (c), and (e) can easily be seen to have two decimal places. Since they are reported to the nearest hundredth of a centimeter, they must have been made by the more accurate ruler, the millimeter ruler. The measurements reported in (b) and (d) were made with the centimeter ruler at the top. In part (e), the 0 at the end shows that this measurement was made with the more accurate ruler. Here the distance was measured as more nearly 0.90 cm than 0.89 or 0.91 cm. Thus, the results are estimated to the nearest hundredth of a centimeter, but that value just happens to have a 0 as the estimated digit. 

Suppose that we want to report the measurement 4.95 cm in terms of meters. Is our measurement any more or less precise No, changing to another set of units does not increase or decrease the precision of the measurement. Therefore, we must use the same number of significant digits to report the result. How do we change a number of centimeters to meters  

The zeros in 0.0495 m do not indicate anything about the precision with which the measurement was made they are not significant. (They are important, however.) In a properly reported number, all nonzero digits are significant. Zeros are significant only when they help to indicate the precision of the measurement. The following rules are used to determine when zeros are significant in a properly reported number  

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Note that the answers have been rounded to three significant digits. Since the even-tempered formula is only an approximation, this does not introduce any significant additional error. 

Rule II. The characteristic of a number less than 1 is found by subtracting from 9 the number of ciphers between the decimal point and the first significant digit, and writing —10 after the result. 

When connecting numbers to logarithms, use as many decimal places in the mantissa as there are significant digits in the number. 

When finding the antilogarithm, keep as many significant digits as there are decimal places in the mantissa. 

Previously published and 1994 data aie rounded off by the U.S. Bureau of Mines to three significant digits and may not add to totals shown. Table includes data available through July 5, 1995. 

Sig nifica.ntDig lts. Any digit that is necessary to define the specific value or quantity is said to be significant. A problem arises, however, when a value of, eg, 4 in. is given. This may be intended to represent 4, 4.0, 4.00, 4.000 or even more accuracy with a corresponding increase in significant digits (equivalent to 102, 101.6, 101.60, and 101.600 mm, respectively). 

Numerical values of the constants that follow are approximate to the number of significant digits given. 

Example Assume three-decimal floating arithmetic (i.e., only the three most significant digits of any niimher are retained), and solve the following system hy Gauss ehmination ... 

The form of the equations here is given to provide good accuracy when many terms are used and to provide the variances of the parameters. Another form of the equations for a and b is simpler, but is sometimes inaccurate unless many significant digits are kept in the calculations. The minimization ofx" when Gj is the same for all i gives the following equations for a and b... 

Convergence to the final solution is rapid with the TG method for narrowboiling feeds but may be slow for wide-boiling feeds. Generally, at least five column iterations are required. Convergence is obtained when successive sets of tear variables are identical to approximately four significant digits. This is accompaniedby 0 = 1.0, a. = normahzed x, and nearly identical successive values of ft as well as Q,.. 

Only on-site releases of the toxic chemical to the environment for the calendar year are to be reported in this section of the form. The total releases from your facility do not include transfers or shipments of the chemical from your facility for sale or distribution in commerce, or of wastes to other facilities fortreatment ordisposal (see Pari III, Section 6). Both routine releases, such as fugitive air emissions, and accidental or nonroutine releases, such as chemical spills, must be included in your estimate of the quantity released. EPA requires no more than two significant digits when reporting releases (e.g., 7521 pounds would be reported as 7500 pounds). 

The analyst can then calculate the total probability of failure (Ft) by summing the probability of all failure paths (Fi-s). The probability of a specific path is calculated by multiplying the probabilities of each success and failure limb in that path. Note The probabilities of success and failure sum to 1.0 for each branch point. For example, the probability of Error B is 0.025 and the probability of Success b is 0.975.) Table 5.2 summarizes the calculations of the HRA results, which are normally rounded to one significant digit after the intermediate calculations are completed. 

The significant digits of a number are tlie digits from tlie first nonzero digit on tlie left to eitlier (a) the last digit (whetlier it is a nonzero or zero) on tlie right if tliere is a decimal point, or (b) tlie last nonzero digit of the number if tliere is no decimal point. For example ... 

Total mass 5.136(7) x 10 tonnes H2O 0.017(1) x 10 tonnes dry atmosphere 5.119(8) x 10 tonnes. Figures in parentheses denote estimated uncertainty in last significant digit. 

Double precision—Vdilue stored as two words, rather than one, representing a real number, but allowing for approximately double the number of significant digits. 

If the digits to be discarded are less than- -500. . . .leave the last digit unchanged. Masses of 23.315 g and 23.487 g both round off to 23 g if only two significant digits are required. 

Data are presented in their fuU precision, although often no more than the first four or five significant digits are used figures in parentheses represent the standard deviation uncertainty in the least significant digits. 

Case Amount c Subtracted Digits Typed in Number of Significant Digits... 

As indicated in Section 1.7.2, the standard deviations determined for the small sets of observations typical for analytical chemistry are trustworthy only to one or two significant digits. 

The second significant digit in (underlined) corresponds to the third decimal place of Xmean- In reporting this result, one should round as follows ... 

Notice that a result of this type, in order to be interpretable, must comprise three numbers the mean, the (relative) standard deviation, and the number of measurements that went into the calculation. All calculations are done using the full precision available, and only the final result is rounded to an appropriate precision. The calculator must be able to handle >4 significant digits in the standard deviation. (See file SYS SUITAB.xls.)... 

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... 

Sres is easy to calculate and, since the relevant information resides in the first significant digits, its calculation places no particular demands on the soft- or hardware (cf. Section 3.3) if the definition of r,- in Table 2.1 and Eqs. (1.3a)-(1.3d) is used. 

Two numbers of s significant digits each, when multiplied, require 2 s digits for the result. 

The division of two numbers of s significant digits each can yield results that require nearly any number of digits (only a few if one number is a simple multiple of the other, many if one or both numbers are irrational) usually the number of bytes the computer displays or uses for calculation... 

W weight solution to five significant digits, in grams... 

Example 48 The result is thus CL(/4) = 7.390 0.028 mM/g, and should be either left as given or rounded to one significant digit in C1 7.39 0.03 The %-variance contributions are given in parentheses (Eq. (4.24)). Note that the analytical method with the best precision (titrimetry), because of the particular numerical constellation, here gives rise to the largest contribution (77%). 

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