2.4 Significant Figures in Measurement The significant figures in a measurement include all the digits that are known precisely plus one last digit that is estimated. Example: With a thermometer that has 1° intervals, you may determine that the temperature is between 24°C and 25°C and estimate it to be 24.3°C. You know the first two digits (2 and 4) with certainty, and the third digit (3) is a “best guest” By estimating the last digit, you get additional information

Rules 1. Every nonzero digit in a recorded measurement is significant. - Example: 24.7 m, 0.743 m, and 714 m all have three sig. figs. 2. Zeros appearing between nonzero digits are significant. - Example: 7003 m, 40.79 m, and 1.503 m all have 4 sig. figs.

Rules 3. Zeros appearing in front of all nonzero digits are not significant; they act as placeholders and cannot arbitrarily be dropped (you can get rid of them by writing the number in scientific notation). - Example: 0.0071 m has two sig. figs. And can be written as 7.1 x 10-3 4. Zeros at the end of the number and to the right of a decimal point are always significant. - Example: 43.00 m, 1.010 m, and 9.000 all have 4 sig. figs.

Rules 5. Zeros at the end of a measurement and to the left of the decimal point are not significant unless they are measured values (then they are significant). Numbers can be written in scientific notation to remove ambiguity. - Example: 7000 m has 1 sig. fig.; if those zeros were measured it could be written as 7.000 x 103

Rules 6. Measurements have an unlimited number of significant figures when they are counted or if they are exactly defined quantities. - Example: 23 people or 60 minutes = 1 hour * You must recognize exact values to round of answers correctly in calculations involving measurements.

Significant Figures – Example 1 How many significant figures are in each of the following measurements? a. 123 m b. 0.123 cm c. 40506 mm d. 9.8000 x 104 m e. 4.5600 m f. 22 meter sticks g. 0.07080 m h. 98000 m 3 (rule 1) 3 (rule 3) 5 (rule 2) 5 (rules 4 and 5) 5 (rule 4) Unlimited (rule 6) 4 (rules 2, 3, and 4) 2 (rule 5)

2.4 Concept Practice 7. Write each measurement in scientific notation and determine the number of significant figures in each. a. 0.05730 m b. 8765 dm c. 0.00073 mm d. 12 basketball players e. 0.010 km f. 507 thumbtacks 5.730 x 10-2 m, 4 8.765 x 103 dm, 4 7.3 x 10-4 mm, 2 1.2 x 101 BB players, unlimited 1.0 x 10-2 km, 2 5.07 x 102 thumbtacks, unlimited

Significant Figures in Calculations The number of significant figures in a measurement refers to the precision of a measurement; an answer cannot be more precise than the least precise measurement from which it was calculated. Example: The area of a room that measures 7.7 m (2 sig. figs.) by 5.4 m (2 sig. figs.) is calculated to be 41.58 m2 (4 sig. figs.) – you must round the answer to 42 m2

Rounding – The Rule of 5 If the digit to the right of the last sig. fig is less than 5, all the digits after the last sig. fig. are dropped. Example: 56.212 m rounds to 56.21 m (for 4 sig. figs.) If the digit to the right is 5 or greater, the value of the last sig. fig. is increased by 1. Example: 56.216 m rounds to 56.22 m (for 4 sig. figs.)

Rounding – Example 2

Addition and Subtraction The answer to an addition or subtraction problem should be rounded to have the same number of decimal places as the measurement with the least number of decimal places.

Multiplication and Division In calculations involving multiplication and division, the answer is rounded off to the number of significant figures in the least precise term (least number of sig. figs.) in the calculations