Significant Figures Exact Numbers

Exact numbers have no uncertainty and thus do not limit the number of significant figures in any calculation. We can regard an exact number as having an unlimited number of significant figures. Exact numbers originate from three sources  

When we use measured quantities in calculations, the results of the calculation must reflect the precision of the measured quantities. We should not lose or gain precision during mathematical operations. Follow these rules when carrying significant figures through calculations. 

In multiplication or division, the result carries the same number of significant figures as the factor with the fewest significant figures. 

A few books recommend a slightly different ronnding procednre for cases In which the last digit is 5. However, the procedure presented here is consistent wItt electronic caicniators and will be nsed throughout this book. 

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An exact number is a small number that can be reproducibly determined by counting or one that is defined to a particular value. Exact numbers have infinite precision and significant figures. Exact numbers are not obtained using measuring devices. 

Exact numbers. Often calculations involve numbers that were not obtained using measuring devices but were determined by counting 10 experiments, 3 apples, 8 molecules. Such numbers are called exact numbers. They can be assumed to have an unlimited number of significant figures. Exact numbers can also arise from definitions. For example, 1 inch is defined as exactly 2.54 centimeters. Thus in the statement 1 in. = 2.54 cm, neither 2.54 nor... 

Exact numbers. Some numbers are exact and have an infinite number of significant figures. Exact numbers occur in simple counting operations when you count 25 dollars, you have exactly 25 dollars. Defined numbers, such as 12 inches in 1 foot, 60 minutes in 1 hour, and 100 centimeters in 1 meter, are also considered to be exact numbers. Exact numbers have no uncertainty. 

Uncertainty in measurement, uncertain (doubtful) digit, significant figures, exact numbers... 

Note Exact conversions 1 m = 100 cm, 1 km = 1000 m. Inexact conversion 1 mile = 1.609 km to 4 significant figures. The number of significant figures in the answer is set by the data (4 sig. figs.) but the answer has extra source of error since the conversion from kilometers to miles is only good to 4 sig. figs. 

One further point about significant figures Certain numbers, such as those obtained when counting objects, are exact and have an effectively infinite number of significant figures. For example, a week has exactly 7 days, not 6.9 or 7.0 or 7.1, and a foot has exactly 12 in., not 11.9 or 12.0 or 12.1. In addition, the power of 10 used in scientific notation is an exact number. That is, the number 103 is exact, but the number 1 X 103 has one significant figure. 

Even though the number 1.9 has two significant figures, we carry the other numbers in Example 2-6 to more significant figures. Then we round at the end to the appropriate number of significant figures. The numbers in the distance conversions are exact numbers. 

Remember that 1 in this case is an exact number by definition and therefore does not limit the number of significant figures (the number 12 is limiting here). 

Exact numbers have an unlimited number of significant figures. Exact mmibers originate from three somces ... 

You report the answer to two significant figures because 3.0 grams has two significant figures. The number 9 is exact and does not detramine the number of significant figures. 

The molar mass of HC9H7O4 is 180.2 g. From this you find that an aspirin tablet contains 0.00180 mol of the acid. Hence, the concentration of the aspirin solution is 0.00180 moFO.500 L, or 0.0036 M. (Retain two significant figures, the number of significant figures in Ka.) Ca/Ka = 0.0036/3.3 X 10 " = 11, which is less than 100, so we must solve the equilibrium equation exactly. 

Exact numbers, such as the stoichiometric coefficients in a chemical formula or reaction, and unit conversion factors, have an infinite number of significant figures. A mole of CaCb, for example, contains exactly two moles of chloride and one mole of calcium. In the equality... 

Significant figures provide an indication of the precision with which a quantity is measured or known. The last digit represents, in a quantitative sense, some degree of doubt. For example, a measurement of 8.12 inches implies tliat Uie actual quantity is somewhere between 8.315 and 8.325 inches. This applies to calculated and measured quantihes quantities tliat are known exactly (e.g., pure integers) have an infinite number of significant figures. 

In applying the rules governing the use of significant figures, you should keep in mind that certain numbers involved in calculations are exact rather than approximate. To illustrate this situation, consider the equation relating Fahrenheit and Celsius temperatures ... 

The numbers 1.8 and 32 are exact Hence they do not limit the number of significant figures in a temperature conversion that limit is determined only by the precision of the thermometer used to measure temperature. 

Stoichiometric coefficients are exact numbers so they do not limit the significant figures of stoichiometric calculations (see Appendix 1C). 

When exact numbers are used, their presence has no effect on the significant figures in the result. For example, we can convert 1.855 hours into seconds. 

In determining the correct number of significant figures, note that the following values are exact 32°F, 1°C/1.8°F, and 1°C/1 K and have an infinite number of significant figures. 

Note that the number of significant figures in the result is determined by the precision of the mass of 202Hg, because the mass of12C is established by definition as an exact number. 

The term (2 mol HC1/1 mol H2) is a mole ratio. We got this mole ratio directly from the balanced chemical equation. The balanced chemical equation has a 2 in front of the HC1, thus we have the same number in front of the mol HC1. The balanced chemical equation has an understood 1 in front of the H2, for this reason the same value belongs in front of the mol H2. The values in the mole ratio are exact numbers, and, as such, do not affect the significant figures. 

Note that there are exactly two equivalents per mole and that the number 2 does not diminish the number of significant figures in the answer because it has an infinite number of significant figures. This is also true in other calculations in this section.)... 

Conversion factors that are exact numbers have an infinite number of significant figures. 

We deal with two types of numbers in chemistry—exact and measured. Exact values are just that—exact, by definition. There is no uncertainty associated with them. There are exactly 12 items in a dozen and 144 in a gross. Measured values, like the ones you deal with in the lab, have uncertainty associated with them because of the limitations of our measuring instruments. When those measured values are used in calculations, the answer must reflect that combined uncertainty by the number of significant figures that are reported in the final answer. The more significant figures reported, the greater the certainty in the answer. 

The answer will be rounded off to 2 significant figures based upon the 2.3 miles, since all the other numbers are exact ... 

Some numerical values are exact to as many significant figures as necessary, by definition. Included in this category are the numerical equivalents of prefixes used in unit definition. For example, 1 cm = 0.01 m by definition, and the units conversion factor, 1.0 x 10-2 m/cm, is exact to an infinite number of significant figures. 

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