Repeating decimal values

Fractions all have decimal values, but some of these decimals terminate (come to an end) and some repeat (never end). As long as the denominator of the fraction is the product of 2s and 5s and nothing else, then the decimal equivalent of the fraction will terminate. To find this terminating decimal, you divide the denominator (bottom) of the fraction into the numerator (top) and keep dividing until there s no remainder. You may have to keep adding 0s in the divisor for a while, but the division will end. 

Terminating decimals are just dandy, but they re in no way the only type of decimal value out there. Repeating decimals occur when you change a fraction to a decimal and the denominator of the fraction has some factor other than 2 or 5. It only takes one such factor to create the repeating situation. For... 

When you re using repeating decimals in a problem, you decide how many decimal places you want and then round the number to that approximate value. Rounded decimals aren t exactly the same value as the repeating decimals, but you can make them pretty accurate in an application by using enough digits in the decimal equivalent. 

The middle step in changing a fraction to a percent is finding the decimal equivalent (or, in the case of a repeating decimal, the approximate). Table 6-1 shows you some fractions, their decimal value, and then the percent that you get by moving the decimal point two places to the right. 

What about fractions that are repeating decimals We rounded those decimals to the nearest thousandth, and we can convert that rounded value to a percent. 

The process of achieving agreement between an observed value and the repeatability (2.3.7) of the analytical procedure. The maximum rounding off interval is equal to the largest decimal unit determined to be smaller than half... 

Ordinarily X does not have to be known to very fine precision, but a rough ballpark, say one place behind the decimal is sufficient. If greater precision in the estimation of X is needed, once a value of X on the interval — 2, 2 is identified, e.g., 0.5, then the grid can be refined and the process repeated, e.g., 0.4-0.6 by... 

Revise your solution to Problem 8.52 to apply the pib basis functions to the onedimensional quartic oscillator with V = cx. See Problem 8.55 for hints. Take the box to extend from x, = —3.5 to 3.5, where x, is as found in Problem 4.32. Increase the number of pib basis functions until the lowest three energy values remain stable to three decimal places Compare the lowest three energies with those found by the Numerov method in Problem 452. Check the appearance of the lowest three variational functions. Now repeat for the box going from X, = -4.5 to 4.5. For which box length do we get faster convergence to the true energies ... 

Figure 19-6 shows that, in the first attempt, we obtained 0.9573 md and 1.911 cp in the second and third attempts, we have 1.059 md and 2.131 cp, and 1.016 md and 2.033 cp, respectively. These values compare favorably with the assumed 1 md and 2 cp shown in Figure 19-5. The disagreement arises because only four decimal places of information are used from Figure 19-5. Again, sensitivity studies must be performed to show that known values of formation properties remain stable to slight errors in input mudcake assumptions. When performing time lapse analysis in the presence of mudcake, significant differences between mudcake and formation mobility heighten this sensitivity. Only when the two are comparable, for example, as in the case where mudcake builds on likewise low permeability rock, can such predictions prove robust, repeatable, and accurate.

Flush and dry the sample tube as described in 9.2.1 and allow the display to reach a steady reading. If the display does not exhibit the correct density for air at the temperature of test, repeat the cleaning procedure or adjust the value of constant B commencing widi the last decimal place until the correct density is displayed. 

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