Significant Figure Primer

Significant Figure Primer The following guidelines/rules apply to determining the number of significant figures in a MEASURED quantity: 1. All nonzero digits are significant. For example, 457 cm has three significant figures whereas 0.25 cm has two significant figures. 2. Zeros between nonzero digits are significant. For example, 1005 kg has four significant numbers and 1.03 in has three significant figures. 3. Zeros to the left of the first nonzero digit in a value are not significant; they merely indicate the position of the decimal point. For example, 0.02 g has one significant figure and 0.026 has two significant figures. 4. Zeros that fall both at the end of a number and to the right of the decimal point are significant. For example, 0.0200 oz has three significant figures as does 30.0 in. 5. When a number ends in zeros that are not to the right of a decimal point, the zeros are not necessarily significant. Under most situations if a decimal is not given after the zero value it is considered no significant. For example 130 cm only has two significant figures given this strict interpretation. 6. To remove any ambiguity as to where or not the zero at the end of a number are significant or not it is best to use exponential or scientific notation (rule #5). For example if the zero value is significant in the number 130 cm it should be written as follows: 1.30 x 102 cm has three significant figures 1.300 x 102 cm has four significant figures In the above numbers, all zeros to the right of the decimal point are significant (Rule #2 and #4). 7. Exact numbers can be treated as if they have an infinite number of significant figures. An example of an exact number is that there are 12 eggs in a dozen or there are 100 pennies in one dollar. 8. This rule also applies to conversion factors. For example, there are 12 inches in 1 foot. The number 12 is exact, there is only and always only 12 inches not 12.1 or 11.9 inches in one foot. Mathematical Operations and Significant Figures Reporting Multiplication and Division In multiplication and division the reporting of significant figures in the FINAL answer is determined by the MEASURED value used in the calculation that had the least number of significant digits (See the example below) Example problem: A chemical substance has a mass of 3.02 grams. Its volume was measured to be 4.5002 milliliters. Determine the density of the substance. The measured value 3.02 gram has three significant figures and the value 4.5002 milliliters has five significant figures. The final answer to the problem has to be reported to three significant figures. 3.02 grams / 4.5002 mL = 0.671 g/mL (three significant figures) Note: All measurements have two parts - a numerical value and a unit. When rounding off numbers if the value is less than five you need to round down and if it is greater than five round up. In the above example, a value of 0.67108 is obtained. The value 0 is less than 5 so the reported value is 0.671 g/mL. Addition and Subtraction In addition and subtraction the reporting of significant figures for the FINAL answer is determined by the MEASURED values used in the calculation. Again the least significant dictates the number of significant figures to be reported in the answer (See example below). Example problem: A solution contains 99.6 grams of water, 31.62 grams of NaCl and 9.765 grams of KCl. Determine the total mass of the solution. 99.6 g 31.62 g 9.765 g 140.985 g ------ > 141.0 grams is the final answer Logarithms A logarithm consists of two parts: the characteristic and the mantissa. The characteristic is the digits/numbers to the left of the decimal. The mantissa is the digits to the right of the decimal. For example, the log of 2578 = 3.4113, where the 3 is the characteristic and .4113 is the mantissa. The general rule governing the reporting of significant figures for logarithms is that the mantissa should have the same number of significant figures as the original number. For example: Log 2 = 0. 3 One significant figure for the original value of 2 has one significant figure. Log 2.0 = 0. 30 Two significant figures since the original value has two significant digits. The same line of reasoning applies when determining an antilog. The number of significant figures for the final answer should have the same number of significant digits as does the mantissa. For example: Antilog 0.48 = 3.0 Two significant digits in the mantissa, so the answer is reported to two significant figures Antilog 3.4771 = 3.000 X 103 Four significant digits in the mantissa, so the answer must be reported to have four significant digits. To determine the number of significant digits to report when series of multiple mathematical operations is needed is calculate a final answer depends on the order of each mathematical operation. See the examples below. Given: [(21.87)(190)] + [(30.10/0.7010] = X [4200] + [42.9] = X X = 4243 Given: {[5.040 + 0.0078)/0.8009] - 31.6} = X {[(5.048)/0.8009] – 31.6} = X {6.303 – 31.6} = X X = - 25.3