Calculator significant digits

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... 

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. 

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. 

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... 

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. 

The above relation holds within the limited precision of our calculations of the singular values, which in the present example is about four significant digits. 

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 ... 

EXAMPLE 2.36. How many significant digits should be retained in the answer when we calculate the number of decimeters in 1.465 m ... 

Calculate the answer to each of the following expressions to the correct number of significant digits. 

Perform the following calculations to the proper number of significant digits. All measurements are in centimeters. 

Estimating distances. For both the worst-case and alternative release scenarios, the source must estimate the distance to where the endpoint is no longer exceeded and estimate the population (rounded to two significant digits) within a circle defined by the distance and centered at the release point. U.S. Census data may be used and it does not have to be updated however, the presence of sensitive populations (e.g. hospitals, schools, etc.) must be noted. In addition, the source must identify and list the types of environmental receptors within the calculated worst-case distance and circle however,... 

The solution has the correct number of significant digits. The units are correct. You know that the reaction was exothermic, because the temperature of the solution increased. The calculated AH is negative, which is correct for an exothermic reaction. 

A balanced chemical equation was used in the calculation. The number of significant digits is correct. The units are also correct. 

A rounded value of 96 500 C/mol is often used in calculations. Note that this rounded value has three significant digits. 

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. 

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