Significant Figures: Rules, Rounding, and Calculations

SIGNIFICANT FIGURES Number1 1. All digits (1-9) are significant Significant Figures 1.589 m 0.897 m 3600 m 4 3 22 2. Significant Zeroes a. All sandwiched zeroes 13.02 m 1.0002 m 10.5 m 4 5 3 b. All trailing zeroes preceeded by a digit 0.0200 m 15.0 m 3 3 3. Nonsignificant Zeroes a. Leading zeroes 0.0200 (2.00 x 10-2) m 3 b. Trailing zeroes to left of decimal in a number without decmial part 360 (3.6 x 102) m 56,000 (5.6 x 104) m NOTES: 1 2 22 22 Write numbers in exponential notation if you have any doubt; any zeros used to indicate the power of 10 (order of magnitude) are not significant. Note that nearly all numbers have units associated with them. These must always be written with the number. For these types of numbers, the zero could be counted as a significant digit if the number results from a measurement or a calculation. See rules on the next page (K = oC + 273 = 37 + 273 = 4.50 x 102 K, for 3 significant figures). Numbers in science usually result from measurements and calculations in the laboratory. ROUNDING OFF 1. If the last digit to be retained in a number is followed by a number <5, ROUND DOWN. Round to 3 significant figures: 2. If the last digit to be retained in a number is followed by a number >5, ROUND UP. Round to 2 significant figures: 3. 28.23 g rounds to 28.2 g 578.1 g rounds to 578 g 5.998 cm rounds to 6.0 cm 0.00258 cm rounds to 0.0026 cm 3.6502 cm rounds to 3.7 cm If the last digit to be retained in a number is followed by 5 (0000000...implied), ROUND UP if the digit to be retained is ODD (1,3,5,7,9) but ROUND DOWN if the digit to be retained is EVEN (0,2,4,6,8) Round to 2 significant figures: 1.75 g rounds to 1.8 g (7 is odd) 1.050 g rounds to 1.0 g (0 is even) 1.45 g rounds to 1.4 g (4 is even) Round to 4 significant figures: 67.835 g rounds to 67.84 g (3 is odd) 67.885 g rounds to 67.88 g (8 is even) UNCERTAINTY AND SIGNIFICANT FIGURES THE LEAST ACCURATE NUMBER (LAN) DETERMINES THE NUMBER OF DIGITS TO WHICH THE ANSWER IS EXPRESSED MULTIPLICATION AND DIVISION a) ADDITION AND SUBTRACTION a) The LAN is the number with the least number of significant figures. The LAN is the number one with the least number of decimal places. The answer (SUM or DIFFERENCE) can have not more decimal places than the LAN. b) The answer (PRODUCT OR QUOTIENT) can have no more significant figures than the LAN. EXAMPLE: EXAMPLE: 212 g water (LAN) 2.1 g salt 1.88 g sugar Calculate the volume of a rectangular solid that has a length of 4.16 cm, a width of 2.2 cm, and a height of 2.00 cm. 215.98 g (incorrect) Volume = length x width x height LAN = (4.16 cm)(2.2 cm)(2.00 cm) = 18.304 cm3 (incorrect) 216 b) g (correct) If a number has no decimal places, the LAN is the number with the fewer digits and with the last digit in the highest place. = 18 cm3 (correct) EXAMPLE: c) 3600 111 55 mL water (LAN) mL ethanol mL isopropyl alcohol 3766 mL (incorrect) 3800 mL (correct) Zeros may be counted as significant digits if they are the result of a calculation in which the calculated zero is in a place that is known by the numbers used in the addition or subtraction. EXAMPLE: Convert 37 oC to K K = oC + 273 = 37 + 273 (both known in ones place) = 4.50 x 102 K = 450. K (notation less commonly used) PRACTICE PROBLEMS 1. Give the number of significant figures: a. b. c. d. 9.9099 m 0.9090 m 0.0909 m 9 x 10-3 m 2. Express your answer to the proper number of significant figures: a. b. c. d. e. f. 99.099 g + 1.91 g = 3.33 x 10-2 cm - 1.45 x 10-3 cm = 15.15 cm x 2.10 cm = 0.516 cm3 /1.0301 g/cm3 = 83.15 m /(9.0 x 102 m) = K = 177 oC + 273 =