Time Arithmetic

If you enter 10 AM in cell A1 of a worksheet, and =A1 +3 in cell B1, you may at first be confused when cell B1 displays 10 00 AM. But remember that you ve added three days, not three hours if you apply the date format m/d/yy to the cell, you ll see that you ve calculated a date three days from the current date. If you change the formula to =A1+3 00, you get an Error in formula message. That s because 3 00 is not a numerical value. To obtain the desired result, 1 00 PM, use the formula =A1+ 03 00 . Excel recognizes that 03 00 is text and evaluates it just as it would if you d typed it into a cell. 

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If you keep in mind that Excel stores dates and times as date serial numbers, performing date or time arithmetic is simple. For example, in a kinetics experiment you may have a table of times at which data points were recorded at irregular intervals (Figure 3-15). To analyze the data you need the elapsed time from finitial- Subtracting the time values from the initial value yields numbers that are decimal fractions of a day and are converted into minutes by multiplying by 24 X 60. The formula in cell B2 is =(A2- A 2) 1440. 

Note the use of date and time arithmetic to calculate the elapsed time between readings. The formula in cell B7 is ... 

To estimate the computational time required in a Gaussian elimination procedure we need to evaluate the number of arithmetic operations during the forward reduction and back substitution processes. Obviously multiplication and division take much longer time than addition and subtraction and hence the total time required for the latter operations, especially in large systems of equations, is relatively small and can be ignored. Let us consider a system of simultaneous algebraic equations, the representative calculation for forward reduction at stage is expressed as... 

By the time j ou get to a ten step synthesis, the arithmetic demon ensures that the yield is down to a miserly 35%. And this with 90% yield in eaeh step So clearly a short synthesis is a good one. But we ean cheat the arithmetic demon by making our five steps convergent rather than linear. The convergent version is this ... 

Operations such as the above are carried out very rapidly by the computer through voltage switching, each switch lasting only a few nanoseconds. Therefore, although it is clumsier to represent numbers in binary for the human mind, and instead we use ten symbols (0, 1,. .., 9) to help us with complicated arithmetic, the speed with which we can do this arithmetic is nothing like the speed of the computer. Computer addition seems instantaneous, whereas human response to addition takes a finite time. 

U. Single water drop in air, liquid side coefficient / jy l/2 ki = 2 ), short contact times / J 1 lcontact times dp [T] Use arithmetic concentration difference. Penetration theory, t = contact time of drop. Gives plot for k a also. Air-water system. [lll]p.. 389... 

The voltage must approach a sinusoidal waveform and should be balanced. If at the time of conducting the tests the voltage is almost btit not completely balaticeil. arithmetical average of the phase volttige must be used for calculating the machine s performance. 

In general, air quality data are classified as a function of time, location, and magnitude. Several statistical parameters may be used to characterize a group of air pollution concentrations, including the arithmetic mean, the median, and the geometric mean. These parameters may be determined over averaging times of up to 1 year. In addition to these three parameters, a measure of the variability of a data set, such as the standard deviation... 

Fig. 19-13. Three-parameter averaging-time model fitted through the arithmetic mean and the second highest 3-hr and 24-hr SOj concentrations measured in 1972 a few miles from a coal-burning power plant. Source From Larsen (21).

Balancing the electrons is the next step, but this time it is a little more difficult. We see that if we multiply equation (90) by 5 and equation (97) by 2, there will be 10 electrons in each case. Arithmetic teachers call this finding the least common multiple ... 

In this chapter we described Euler s method for solving sets of ordinary differential equations. The method is extremely simple from a conceptual and programming viewpoint. It is computationally inefficient in the sense that a great many arithmetic operations are necessary to produce accurate solutions. More efficient techniques should be used when the same set of equations is to be solved many times, as in optimization studies. One such technique, fourth-order Runge-Kutta, has proved very popular and can be generally recommended for all but very stiff sets of first-order ordinary differential equations. The set of equations to be solved is... 

In such matters some progress can be achieved by combinations of the decomposition method and the method of separation of variables. For example, this can be done using the method of separation of variables for the reduced system (6) upon eliminating the unknown vectors with odd subscripts j. This trick allows one to solve problem (2) here the expenditures of time are Q 2nin2 og N2 arithmetic operation, half as much than required before in the method of separation of variables. 

Visco-elastic substances can be described with the spring/shock-absorber model of Kevin and Voigt, and have phase displacements of 0° to 90°. In analogy to other time dependent processes in physics, the oscillation tests are evaluated with complex arithmetics. Obtained are the complex quantities ... 

The specification of a value like d = 235 pm means the distance d amounts to 235 times 1 picometer. Arithmetically, this is a multiplication of the numerical value with... 

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