Number integers, rounding

We have seen that we must sometimes reduce the number of digits in our calculated result to indicate the accuracy of the measurements that were made. To reduce the number, we round off numbers other than integer digits using the following rules. 

In the problem above, we determined the percentage data from the chemical formula. We can determine the empirical formula if we know the percent compositions of the various elements. The empirical formula tells us what elements are present in the compound and the simplest whole-number ratio of elements. The data may be in terms of percentage, or mass or even moles. However, the procedure is still the same—convert each element to moles, divide each by the smallest, and then use an appropriate multiplier if necessary. We can then determine the empirical formula mass. If we know the actual molecular mass, dividing the molecular formula mass by the empirical formula mass, gives an integer (rounded if needed) that we can multiply each of the subscripts in the empirical formula. This gives the molecular (actual) formula, which tells what elements are in the compound and the actual number of each. 

In contrast to previous examples, the numbers in column (5) are not all integers. The ratio of the numbers of atoms of two elements in a compound must be the ratio of small whole numbers, in order to satisfy one of the postulates of Dalton s atomic theory. Allowing for experimental and calculational uncertainty, we see that the entry of oxygen in column (5), 3.499, is, to within the allowed error, 3.500 or, indeed the ratio of small whole numbers. By rounding off in this way and multiplying each of the entries in column (5) by 2, we arrive at the set of smallest integers that correctly represent the relative numbers of atoms in the compound, as tabulated in column (6). The formula is thus K2Cr20y. 

Table 10.13. Selecting the PARAFAC rank for the VENICE data model. Numbers are rounded to the nearest integer. Component sizes are given as percentage sum of squares relative to the sum of squares of the data...

Round upwards to the nearest integer all fractional turns in the results. Next multiply this result by the desired turns ratio to determine the number of... 

Fractional numbers can pose a problem, but should be rounded to the next integer. 

Figure 3.5. Factorial space. Numbers in circles denote process yields actually measured (initial data set) all other numbers are extrapolated process yields used for planning further experiments (assuming that the repeatability Sy = 0.1 all values are rounded to the nearest integer) the estimated yield of 108% shows that the simple linear model is insufficient.

Intervals were calculated using a method equivalent to rounding numbers to Integer values before assignment to Intervals. 

You should read Technical Support Note TS-230 Dealing with Numeric Representation Error in SAS Applications to learn more about SAS floating-point numbers and storage precision in SAS. Another good resource for rounding issues is Ron Cody s SAS Functions by Example (SAS Press, 2004). In short, whenever you perform comparisons on numbers that are not integers, you should consider using the ROUND function. 

This is still not a whole number ratio, since 1.333 is much too far from an integer to round off. Since 1.333 is about l, multiply both numbers of moles by 3 ... 

Each sector has specific requirements in terms of the number of primary samples that should be taken. Simple empirical rules which have been used in the past to determine the number of samples to be taken from a lot include n = -JN and n = 3 x / N (in each case, N is the total number of items in the lot). In both cases, n is rounded to the nearest integer. 

Nominal Mass Mass of an ion or molecule calculated using the mass of the most abundant isotope of each element rounded to the nearest integer value and equivalent to the sum of the mass numbers of all constituent atoms [1]. 

In real life, other problems involving discrete variables may not be so nicely posed. For example, if cost is a function of the number of discrete pieces of equipment, such as compressors, the optimization procedure cannot ignore the integer character of the cost function because usually only a small number of pieces of equipment are involved. You cannot install 1.54 compressors, and rounding off to 1 or 2 compressors may be quite unsatisfactory. This subject will be discussed in more detail in Chapter 9. 

The random selection in step (iii) is carried out by generating uniform random numbers U e [0, 1], For example, the index of a random particle selected from a set of N particles will be n = intup(//N) where intuP() rounds the argument up to the nearest integer. Note that for constant-density, statistically stationary flow, the effective flow rates will be constant. In this case, steps (i) and (ii) must be completed only once, and the MC simulation is advanced in time by repeating step (iii) and intra-cell processes. For variable-density flow, the mean density field ((p)) must be estimated from the notional particles and passed back to the FV code. In the FV code, the non-uniform density field is held constant when solving for the mean velocity field.15... 

An additional observation for photon counting data there are no fractions of photons and thus the count can only include integer numbers. Thus the measurements in column B are rounded down to the nearest integer. It seems to be reasonable to do the same with the calculated values in column C. However, a test in Excel reveals that such an attempt does not work. The reason is, that the solver s Newton-Gauss algorithm requires the computation of the derivatives of the objective (x2 or ssq) with respect to the parameters. A rounding would destroy the continuity of the function and effectively wipe out the derivatives. 

Different methods can be applied for the split into segments. Mode 111222333 denotes that the first n/s (rounded to integer) objects are put into segment 1, the next n/s objects into segment 2, and so on. Mode 123123123 puts object 1 into segment 1, object 2 into segment 2, and so on. Mode random makes a random split but usually without any user control. We recommend to sort the objects by a user-created random permutation of the numbers 1 to n, and then to apply mode 111222333 —and to repeat this several times. 

Except for the simplest hydrogen atom with a single proton as its entire nucleus, all atoms contain neutrons (particles that are electrically neutral) in addition to protons. For most of the light elements, the numbers of protons and neutrons in the nucleus are nearly equal. Table 3-2 shows the most common nucleus for each element with the atomic weight rounded to the nearest integer. You can see that the rounded-off atomic weights are the sum of the protons and neutrons for each atom. The sum of the protons and neutrons is the mass number of an atom. 

The accessibility of each atom was calculated as discussed in the text. The numbers given are areas of the hydrosphere surface defined as the locus of possible centers of solvent molecules in contact with the given atom or atom group (CHi and CHa). The calculated areas in A2 have been rounded off to the nearest integer. Values between 0 and 0.5 A2 have been indicated by a + and atoms contacting the internal void have been indicated with an Atom designations are standard except that three or four character... 

We solve for J and round the result to an integer to give the rotational quantum number. The vibrational energy is given by... 

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