The numerical value of every observed measurement is an approximation. No physical measurement, such as mass, length, time, volume, velocity, is ever absolutely correct. The accuracy (reliability) of every measurement is limited by the reliability of the measuring instrument, which is never absolutely reliable.
Consider that the length of an object is recorded as 15.7 cm. By convention, this means that the length was measured to the nearest tenth of a centimeter and that its exact value lies between 15.65 and 15.75 cm. If this measurement were exact to the nearest hundredth of a centimeter, it would have been recorded as 15.70 cm. The value 15.7 cm represents three significant figures (1, 5, 7), while 15.70 cm represents four significant figures (1, 5, 7, 0). A significant figure is one which is known to be reasonably reliable.
Similarly, a recorded mass of 3.406 2 g, observed with an analytical balance, means that the mass of the object was determined to the nearest tenth of a milligram and represents five significant figures (3, 4, 0, 6, 2), the last figure (2) being reasonably correct and guaranteeing the certainty of the preceding four figures.
A 50-mL buret has markings 0.1 ml apart, and the hundredths of a milliliter are estimated. A recorded volume of 41.83 cm3 represents four significant figures. The last figure (3), being estimated, may be in error by one or two digits in either direction. The preceding three figures (4, 1, 8) are completely certain.
In elementary measurements in chemistry and physics, the last digit is an estimated figure and is considered as a significant figure.
A recorded volume of 28 mL represents two significant figures (2, 8). If this same volume were written as 0.028 L, it would still contain only two significant figures. Zeros appearing as the first figures of a number are not significant, since they merely locate the decimal point. However, the values 0.028 0 L and 0.280 L represent three significant figures (2, 8, and the last zero); the value 1.028 L represents four significant figures (1, 0, 2, 8); and the value 1.028 0 L represents five significant figures (1, 0, 2, 8, 0). Similarly, the value 19.00 for the atomic mass of fluorine contains four significant figures.
The statement that a body of ore weighs 9 800 lb does not indicate definitely the accuracy of the weighing. The last two zeros may have been used merely to locate the decimal point. If it was weighed to the nearest hundred pounds, the weight contains only two significant figures and may be written exponentially as 9.8 × 103 lb. If weighed to the nearest ten pounds it may be written as 9.80 × 103 lb, which indicates that the value is accurate to three significant figures. Since the zero in this case is not needed to locate the decimal point, it must be a significant figure. If the object was weighed to the nearest pound, the weight could be written as 9.800 × 103 lb (four significant figures). Likewise, the statement that the velocity of light is 186 000 mi/s is accurate to three significant figures, since this value is accurate only to the nearest thousand miles per second; to avoid confusion, it may be written as 1.86 × 105 mi/s. (Normally the decimal point is placed after the first significant figure.)
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 × 10–2 m/cm, is exact to an infinite number of significant figures.
Other numerical values are exact by definition. For example, the atomic mass scale was established by fixing the mass of one atom of 12C as 12.000 0 u. As many more zeros could be added as desired. Other examples include the definition of the inch (1 in = 2.5400 cm) and the calorie (1 cal = 4.184 00 J).
A number is rounded off to the desired number of significant figures by dropping one or more digits to the right. When the first digit dropped is less than 5, the last digit retained should remain unchanged; when it is greater than 5, 1 is added to the last digit retained. When it is exactly 5, 1 is added to the last digit retained if that digit is odd. Thus the quantity 51.75 g may be rounded off to 51.8 g; 51.65 g to 51.6 g; 51.85 g to 51.8 g. When more than one digit is to be dropped, rounding off should be done in a block, not one digit at a time.
The answer should be rounded off after adding or subtracting, so as to retain digits only as far as the first column containing estimated figures. (Remember that the last significant figure is estimated.)
EXAMPLES Add the following quantities expressed in grams.
An alternative procedure is to round off the individual numbers before performing the arithmetic operation, retaining only as many columns to the right of the decimal as would give a digit in every item to be added or subtracted. Examples (2), (3), and (4) above would be done as follows:
Note that the answer to (4) differs by one in the last place from the previous answer. The last place, however, is known to have some uncertainty in it.
The answer should be rounded off to contain only as many significant figures as are contained in the least exact factor. For example, when multiplying 7.485 × 8.61, or when dividing 0.1642 ÷ 1.52, the answer should be given in three significant figures.
This rule is an approximation to a more exact statement that the fractional or percentage error of a product or quotient cannot be any less than the fractional or percentage error of any one factor. For this reason, numbers whose first significant figure is 1 (or occasionally 2) must contain an additional significant figure to have a given fractional error in comparison with a number beginning with 8 or 9.
By the approximate rule, the answer should be 1.1 (two significant figures). However, a difference of 1 in the last place of 9.3 (9.3 ± 0.1) results in an error of about 1 percent, while a difference of 1 in the last place of 1.1 (1.1 ± 0.1) yields an error of roughly 10 percent. Thus the answer 1.1 is of much lower percentage accuracy than 9.3. Hence in this case the answer should be 1.06, since a difference of 1 in the last place of the least exact factor used in the calculation (9.3) yields a percentage of error about the same (about 1 percent) as a difference of 1 in the last place of 1.06 (1.06 ± 0.01). Similarly, 0.92 × 1.13 = 1.04.
In nearly all practical chemical calculations, a precision of only two to four significant figures is required. Therefore the student need not perform multiplications and divisions manually. Even if an electronic calculator is not available, an inexpensive 10-in slide rule is accurate to three significant figures, and a table of 4-place logarithms is accurate to four significant figures.
Since not all electronic calculators are alike, detailed instructions cannot be given here. Read your instruction manual. You should purchase a calculator which, in addition to +, –, ×, and ÷ functions, provides at least the following: scientific notation (powers of ten); logarithms and antilogarithins (inverse logarithms) both natural and common (base ten); and exponentials (yx). If it has these functions, it will probably have reciprocals (1/x), squares, square roots, and trigonometric functions as well.
The use of the arithmetic functions is fairly obvious, but you should use powers of ten except in trivial cases. To enter “96 500” for instance, consider it 9.65 × 104 and enter 9.65 EE4. (On most calculators “EE4” means × 104). The calculator keeps track of the decimal point and provides an answer between one and ten times the appropriate power of ten. It will usually display many more figures than are significant, and you will have to round off the final result. If at least one factor was entered as a power of ten, the power-of-ten style will prevail in the display, and you need not fear running “off scale,” nor will any significant figures disappear off scale.