Decimal places

Density is generally measured at 15°C using a hydrometer in accordance with the NF T 60-101 method it is expressed in kg/1 with an error of 0.0002 to 0.0005 according to which category of hydrometer is utilized. However, in practice only three decimal places are usually retained. 

The calculation proceeds as illustrated in Table 2.2, which shows the variation in the coefficients of the atomic orbitals in the lowest-energy wavefunction and the energy for the first four SCF iterations. The energy is converged to six decimal places after six iterations and the charge density matrix after nine iterations. 

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. 

Molecular Identification. In the identification of a compound, the most important information is the molecular weight. The mass spectrometer is able to provide this information, often to four decimal places. One assumes that no ions heavier than the molecular ion form when using electron-impact ionization. The chemical ionization spectrum will often show a cluster around the nominal molecular weight. 

Note that for now we keep enough significant figures to match the number of decimal places to which the signal was measured. The resulting calibration curve is shown in Figure 5.10. 

The standard deviation about the regression, Sr, suggests that the measured signals are precise to only the first decimal place. For this reason, we report the slope and intercept to only a single decimal place. 

A mass spectrometer that can measure mass accurately to several decimal places (rather than just to the nearest integer) can be used to measure such differences. 

A double-focusing mass spectrometer can mea.sure mass accurately to several decimal places, thus enabling the determination of molecular formulae and elemental compositions of fragment ions. 

The schlieren microscope is able to detect refractive index variations to six decimal places. Any small difference in optical path (index difference, film thickness, etc) is very precisely detected by the schlieren microscope, especially in the Dodd modification. It is, in effect, a darkfield method. The specimen is illuminated with light in a portion of the illuminating cone and that direct light is masked in the conjugate back focal plane of the objective (Fig. 3). The only light to pass through this plane is refracted, reflected, or diffracted by the specimen. 

Pressure is defined as force per unit of area. The International System of Units (SI) pressure unit is the pascal (Pa), defined as 1.0 N /m. Conversion factors from non-SI units to pascal are given in Table 1 (see also Units and conversion factors front matter). An asterisk after the sixth decimal place indicates that the conversion factor is exact and all subsequent digits are 2ero. Relationships that are not followed by an asterisk are either the results of physical measurements or are only approximate. The factors are written as numbers greater than 1 and less than 10, with 6 or fewer decimal places (1). 

The conversion factors are presented for ready adaptation to computer readout and electronic data transmission. The factors are written as a number equal to or greater than one and less than 10, with six or fewer decimal places. The number is followed by E (for exponent), a plus or minus symbol, and two digits which indicate the power of 10 by which the number must be multiphed to obtain the correct value. Eor example ... 

An asterisk ( ) after the sixth decimal place indicates that the conversion factor is exact and that all subsequent digits are zero. Where fewer than six decimal places are shown, more precision is not warranted. 

From Hilsenratb et al., N.B.S. Circ. 564, 1955. Some of the above values have been rounded to four decimal places. Values at 10-K increments below 1000 K and at 50 K increments for higher temperatures appear in the original, also for pressures below atmospheric. 

Example In an experiment, six random mimhers (rounded to four decimal places) were observed from the uniform distribiition /(A.) = 1 for 0 < a. < 1 ... 

The Ot) values in Eqs. (13-37) and (13-38) are effective values obtained from Eq. (13-35) or Eq. (13-36). Once these values are available, 0 can be calculated in a straightforward iteration from Eq. (13-38). Since the (ot — 0) difference can be small, 0 should be determined to four decimal places to avoid numerical difficulties. 

The final tolerances the designer would allocate to the component dimensions are also shown in Figure 3.16. The tolerance values are given to three decimal places which, if required for practical use, can be rounded off with minimal effect to the overall assembly tolerance, for example 0.191 mm to 0.190 mm. 

LoglO of Condition Number gives approx, number of decimal % places... 

Importance also applies to what may appear minor decisions in the planning or design phase. If such decisions are incorrect they could result in major problems downstream. If not detected, getting the decimal place wrong or the units of measure wrong can have severe consequences. Audits should verify that the appropriate controls are in place to detect such errors before it is too late. 

This chapter has discussed some of the factors that may affect equipment reliability and necessitate data adjustment. At this time, little documented assistance is available to help develop these data adjustments. It may be necessary to get help from experts in some situations. Lastly, failure rates are often reported to several decimal places, a precision frequently unwarranted by the data. It is suggested that only the failure rate s first significant number and associated exponential power be used. 

We ll look at the main features of this output now. Locate the corresponding section in your file with your editor as you read each comment. Note that the output from your system may vary slightly. Not every numerical value will necessarily agree to the last decimal place, although differences should appear only beyond the fifth decimal place. This subsection concludes the Gaussian tutorial.Once you have completed it, you will be ready to learn about the various types of Gaussian jobs as well as the model chemistries the program offers. 

Prior to 1965, all we had in our armoury were the a and it Hiickel theories, and a very small number of rigorous calculations designated ab initio (to be discussed later). The aims of quantum chemistry in those days were to give total energies and charge distributions for real molecules, and the seventh decimal place in the calculated properties of LiH. Practical chemists wanted things like reliable enthalpy changes for reactions, reaction paths, and so on. It should come as no surprise to learn that the practical chemists therefore treated theoreticians with scepticism. 

Dezimal, n., Dezimale,/. decimal. Dezimal-bruch, m. decimal fraction, -bruch-stelle, -stelle, /, decimal place. -wa(a)ge, /. decimal balance. 

When measured quantities are added or subtracted, the uncertainty in the result is found in a quite different way than when they are multiplied and divided. It is determined by counting the number of decimal places, that is, the number of digits to the right of the decimal point for each measured quantity. When measured quantities are added or subtracted, the number of decimal places in the result is the same as that in the quantity with the greatest uncertainty and hence the smallest number of decimal places. 

Notice that the change in mass doesn t show up until you get to the third decimal place. 

Quantum Mechanics has been the most spectacularly successful theory in the history of science. As is often mentioned the accuracy to which the anomalous magnetic moment of the electron can be calculated is a staggering nine decimal places. Quantum Mechanics has revolutionized the study of radiation and matter since its inception just over one hundred years ago. The impact of the theoiy has been felt in... 

Refractive Index is used to analyse the sample. It can also be used to help determine the percentage of a chemical (such as ethanol) in an aqueous solution. Refractive index (RI) is always reported to four decimal places. An example of the RI scale is shown in Figure 10.4. The correct reading from the RI of the sample would be 1.3764. 

When adding decimals, place the numbers in a column so that the whole numbers are aligned to the left of the decimal and the decimal fractions are aligned to the right of the decimal. 

A note on good practice To avoid overwhelming you with data we usually quote the values of fundamental constants to three decimal places. In actual calculations you should use the more precise values given in tables, including those inside the back cover. 

Addition and subtraction When adding or subtracting, make sure that the number of decimal places in the result is the same as the smallest number of decimal places in the data. For example, 0.10 g + 0.024 g = 0.12 g. 

Benzoic Acid, C6H5COOH.—The treatment of benzoic acid is the same as for nitrobenzene except for the S values. The results for SQ = 4 and Sc (of the carboxyl group) = 0.8 are given in the last column of Table II it is seen that the carboxyl group, like the nitro group, is meta directing, in agreement with experiment. The calculation, however, indicates a rather too small deactivation of the meta position. This effect appears only in the third decimal place of Cmeta, and is consequently not shown in the table. 

It is seen that the convergence is rapid, the final value of A, 6.49903, being trustworthy to within one unit in the last decimal place. 

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