Significant Figures

1. Significant Figures Part II: Calculations

2. Objectives • When you complete this presentation, you will be able to • determine the number of significant figures in a calculated answer

3. Introduction • We know how to determine the number of significant figures in a measurement. • 3.442 g has 4 significant figures • 0.0025 m has 2 significant figures • 140 s has 2 significant figures • 0.000420 mL has 3 significant figures • Now, we need to learn how to use those measurements in calculations.

4. Rounding • In general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated. • The calculated value must be roundedto make it consistent with the measurements.

5. Rounding • To round, we need to determine how many significant figures the answer needs to have. • This depends on the measurementsand the math processused to determine the answer. • We will cover the details of that later in the presentation. • First, let’s practice rounding numbers to a proper number of significant figures.

6. Rounding • How do we round a numerical value? • We use the same rules we have always used. • If the digit to the immediately to the right of the last significant digit is less than 5, we drop the rest of the digits and the value of the last significant digit remains the same. Round 45.244 to 3 sig figs ➠ 45.244 ➠ 45.2 Round 0.85321 to 2 sig figs ➠ 0.85321 ➠ 0.85

7. Rounding • How do we round a numerical value? • We use the same rules we have always used. • If the digit to the immediately to the right of the last significant digit is 5 or greater we drop the rest of the digits and the value of the last significant digit is increased by 1. Round 62.557 to 3 sig figs ➠ 62.557 ➠ 62.6 Round 0.0545 to 2 sig figs ➠ 0.0545 ➠ 0.055

8. Rounding Example 1: Round each of the following numbers to the indicated number of significant figures: • 54,525.99 m to 3 sig figs • 0.00741554 kg to 4 sig figs • 37.255 s to 1 sig fig • 0.78245 cm to 2 sig figs • 355,000 km to 2 sig figs • 0.0382574925 L to 3 sig figs 54,500 m 0.007146 kg 40 s 0.78 cm 360,000 km 0.0383 L

9. Calculations – addition and subtraction • The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places. • For example: 12.52 m + 349.0 m The 349.0 m measurement has the least number of decimal places. + 8.24 m 369.76 m

10. Calculations – addition and subtraction • The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places. • For example: 12.52 m + 349.0 m That measurement will control the number of decimal places in the answer. + 8.24 m 369.76 m

11. Calculations – addition and subtraction • The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places. • For example: 12.52 m + 349.0 m The reported answer should be 369.8 m + 8.24 m 369.76 m

12. Calculations – addition and subtraction Example 2: Perform each operation. Express the answer in the correct number of sig figs. • 61.2 m + 9.35 m + 8.6 m • 14.2 g + 8.73 g + 0.912 g • 35 s + 72.1 s + 1.876 s • 9.44 kg – 2.11 kg • 1.36 cm + 10.17 cm • 34.61 mL – 17.3 mL 79.2 m 23.8 g 109 s 7.33 kg 11.53 cm 17.3 mL

13. Calculations – multiplication and division • The answer to a multiplication or division calculation should be rounded to the same number of significant digitsas the measurement with the least number of significant digits. • For example: 7.55 m × 0.34 m The 0.34 m measurement has the least number of significant digits (2). 2.567 m2

14. Calculations – multiplication and division • The answer to a multiplication or division calculation should be rounded to the same number of significant digitsas the measurement with the least number of significant digits. • For example: 7.55 m × 0.34 m That measurement will control the number of decimal places in the answer. 2.567 m2

15. Calculations – multiplication and division • The answer to a multiplication or division calculation should be rounded to the same number of significant digitsas the measurement with the least number of significant digits. • For example: 7.55 m × 0.34 m The reported answer should be 2.6 m2 2.567 m2

16. Calculations – multiplication and division Example 3: Perform each operation. Express the answer in the correct number of sig figs. • 2.10 m × 0.70 m • 2.4526 kg ÷ 8.4 • 8.3 m × 2.22 m • 8432 kg ÷ 12.5 • 32 s × 15.125 • 34.61 mL ÷ 17.3 1.5 m2 0.29 kg 18 m2 675 kg 480 s 2.00 mL

17. Summary • A calculated answer cannot be more precise than the least precise measurement from which it was calculated. • The calculated value must be rounded to make it consistent with the measurements.

18. Summary • Rules for rounding: • If the digit to the immediately to the right of the last significant digit is less than 5, we drop the rest of the digits and the value of the last significant digit remains the same. • If the digit to the immediately to the right of the last significant digit is 5 or greater we drop the rest of the digits and the value of the last significant digit is increased by 1.

19. Summary • Significant figures in calculations: • The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places. • The answer to a multiplication or division calculation should be rounded to the same number of significant digits as the measurement with the least number of significant digits.