Decimal value, significance

The possibility to have well resolved peaks allows the measurement of the accurate mass of an ion, i.e. m/z values significant to four decimals, and to calculate its elemental composition. 

Express each of the following measurements as a decimal value. State how many significant digits are in each result. Could we tell just from looking at the results without knowing the original valnes ... 

The steps for developing X-bar and R control charts follow. AU averages that are calculated should be rounded to one more decimal place (significant figure) than the values being averaged. 

The digits that appear before (to the left of) the decimal point specify the value of n, that is, the power of 10 involved in the expression. In that sense, they are not experimentally significant. In contrast, the digits that appear after (to the right of) the decimal point specify the value of the logarithm of C the number of such digits reflects the uncertainty in C. Thus,... 

When evaluating exponents of e, the number of significant figures generally equals the number of decimal places in the In value, hi this case, whereas = 5.62 x 10, ... 

The result has three decimal places because the ° value is accurate to three decimal places, and the correction is a subtraction. The low concentration of H3 in pure water reduces this cell potential nearly to zero, but the reaction is still spontaneous, so permanganate can oxidize water. Solutions of potassium permanganate slowly deteriorate and cannot be stored for long times. The reaction is slow enough, however, that significant oxidation does not occur over days or weeks. 

Note Keep all the significant figures and round at the end. Remember the number of decimal places in pH or pOH values are set by the number of significant figures in the [H+] or [OH-] this is a result of working with logarithms. 

Solving the quadratic equation gives a value of x, correct to one decimal place. As a result, [H2] can have only one significant figure. The calculated value of is equal to the given value, within the error introduced by rounding. 

The value of is reasonable for a weak organic base. The final answer has two significant digits, consistent with the two decimal places in the given pH. 

For logarithmic values, only the digits to the right of the decimal point count as significant digits. The digit to the left of the decimal point fixes the location of the decimal point of the original value. 

For many mole conversions, you need to look up atomic masses on the periodic table (see Chapter 4). The atomic masses you see in different periodic tables may vary slightly, so for consistency, we ve rounded all atomic mass values to two decimal places before plugging them into the equations. We round answers according to significant figure rules (see Chapter 1 for details). 

The position of the decimal point in a measured value has nothing to do with the number of significant figures. The diameter of a dime may be given as 1.794 cm or as 17.94 mm. In either case, four significant figures are used. 

Final zeros after the decimal point are significant figures and are used to indicate the decimal place to which the measurements are reliable. Thus 1.0 cm indicates a length reliably known to tenths of a centimeter but not to hundredths of a centimeter, whereas 1.000 cm indicates a length reliably known to thousandths of a centimeter. A very common mistake is leaving out these zeros when the measured quantity has an integral value. 

The procedure is very similar to the one set out in Toolbox 10.1. The only difference is that the acid or base is now an ion added as a salt. Although we shall calculate pH and pfC values to the number of significant figures appropriate to the data, the answers are often considerably less reliable than that. For instance, we might calculate the pH of a solution as 8.82, but in practice the answer is unlikely to be reliable to more than one decimal place (pH = 8.8). One reason for this poor reliability is that we are ignoring interactions between the ions in solution when we use concentrations in place of activities. 

Zeros at the end of a number and after the decimal point are always significant. The assumption is that these zeros would not be shown unless they were significant. Thus, 55.220 K has five significant figures. (If the value were known to only four significant figures, we would write 55.22 K.)... 

Example 12000 Pa would be considered to have five significant digits by many scientists, but in the context, "The pressure rose from 11000 Pa to 12000 Pa," it almost certainly only has two. "12 thousand pascal" only has two significant figures, but 12000. Pa has five because of the decimal point. The value should be represented as 1.2X104 Pa (or 1.2000 X104 Pa). The best alternative would be to use 12 kPa or 12.000 kPa. 

This value must be rounded off to three significant digits after the decimal point because this is the lowest precision of the added values. The four is an exact number. This means rounding downwards to 3.460 g, eliminating choices C and D. The volume of the collected stones is found from the increase in the level read off the cylinder ... 

For summarizing the practical results of the error analysis, let us assume that the solution of the system of equation started by setting x.i°= 1. The error in the elements of Xi° in that case is of the order of 1CT where t is the number of significant decimal digits carried by the computer. The accuracy of the solution (x/ jX 1 ) depends mainly on the absolute value of the norm j s Til Let us say that this value is 10. If k t, then the solution will be quite accurate. For k >%t, the solution will be very inaccurate and some elements of x2 probably be completely... 

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