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Sig Figs
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## Sig Figs

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1. Sig Figs Easy as…

2. IDing Sig Figs

3. Significant Figures • All the digits used to report a measurement including the uncertain digit (your guess). • The only digits that are NOT considered significant are zeros that are present simply as placeholders.

4. Rules to help decide which digits are significant • All non-zero numbers in a measurement are significant. (Ex: 1, 2, 3 ect) • Zeros are significant only if they: • are surrounded by other significant figures. Sandwiched in between (ex. 120,001m) • appear at the end of a decimal number. (ex. 3.940) • have a bar over them. (ex. 6,32Ō KJ)

5. Examples • How many significant figures are in 23,457.12km? • Rule 1 – All non-zero numbers are significant. Therefore there are 7 significant figures. • How many significant figures are in 98,001g? • Use Rule 1 and 2a – Zeros are significant if they are surrounded by other significant figures. Therefore there are 5 significant figures.

6. More Examples • How many significant figures are in 2.3100s? • Use Rules 1, 2a, and 2b – Zeros at the end of a decimal number are significant. There are 5 significant figures. • How many significant figures are in 57ŌŌ00 nm? • Use rules 1, 2a, 2c – Zeros with a bar over them are significant. Therefore the last 2 zeros are NOT significant. There are 4 significant figures

7. One More Example How many significant figures are in 0.0030160L? • Use rules 1, 2a, and 2b. The first 3 zeros do NOT fit any rules, therefore there are 5 significant figures.

8. Figuring Significance • Figuring Significance • Indicate the number of significant figures in each of the following numbers.

9. Significant Measurements 6,000 L • Look at the scale to the right and record the measurement in the space below. A. 5,880 L • How many significant figures does the measurement have? Explain your answer. A. 3, The last zero is not significant • Assume there is another arrow pointing exactly at the 6,000 L mark, how would you report the measurement so that you have the correct number of significant figures? A. 6,0 0 0 L 5,000 L

10. Exact or Counted Numbers • Significant figures only apply to measurements when there is an uncertain digit. • Counted numbers are perfectly exact and do not have uncertainty. They have an infinite number of significant figures. • Example: 12 eggs could be 12.0000000…eggs (a dozen always means 12 no matter what your talking about)

11. Rounding and Rule of 5

12. Rounding Rules • Always use the number to the right of the number you are rounding to. • 1, 2, 3, 4 always round down: So 3.3 rounds to 3 • 6,7,8, and 9 always round up : So 4.7 rounds to 5

13. Rule of 5’s • 5 will round a number up • Examples • Round 4.85 to 2 sig figs • 4.9 • Round 7.335 to 3 sig figs • 7.34

14. Let’s Review • 4.5 rounds to 5 • 3.5 rounds to 4 • 2.57 rounds to 3 • 2.5000001 rounds to 3

15. Measurements used in calculations • When measurements are used in calculations the answer cannot be any more accurate than the original measurements. • Example: Rectangle with a width of 1.8 cm and a length of 4.3256 cm. What is the area? • Multiplying gives you 7.79608 cm. • You can only have 2 significant figures. Therefore the answer is 7.8 cm.

16. Practice • Round the following numbers to four significant figures. 23,005.1 g 2.78320 g 0.0089345 L 201,455 m • Round the following numbers to five significant figures. 872,345 g 0.00435671 s 200.0651 cm 0.0987335 g

17. Multiplication and Division

18. Multiplication and Division • The answer should contain the same number of significant figures as the measurement with the least number of significant figures • 3 Sig Figs 5 sig figs • Example: (4.56 mm) (3.4624mm) =  3 sig figs 3 sig figs 2 sig figs Example: (3.45m) (6.24m) (2.0m) = 

19. Adding and Subtracting

20. Addition and Subtraction • The answer must be rounded to the last decimal place of the least accurate number. • Example: 3.23m • + 154.2 m • 157.43 m • The correct answer should be… 157.4m

21. Examples • Example: 17,000m • - 6,430m • 10,570m • The correct answer should be… 11,000 since the least accurate measurement is 17,000 with uncertainty in the thousands place.

22. Adding to Your Significance • Solve the following math problems, reporting your answers to the correct number of significant figures. Indicate the uncertain figure in each number by underlining it. Be sure to also show the pre- and post rounded numbers. • Initial Answer Rounded with correct sig figs • 34.8 nm + 22.45 nm =  • 3.67 kg – 3.62 kg =  • 7.39m + 3.467 m • + 1.0 m – 8 m = 

23. Mixed Problems

24. Mixed Problems • Follow the Order of Operations • PEMDAS • Please Excuse My Dear Aunt Sally • Parentheses • Exponents and Roots • Multiplication and Division • Addition and Subtraction 2 4 2 3 • EX: (23 cm+1.435 cm) (450 cm) / (16.3 cm) = • SHOW YOUR WORK!!! (24.435 cm x 450 cm) / (16. 3 cm) = 674.58588957 = 670 cm

25. Significant Operations • Solve the following math problems, reporting your answers to the correct number of significant figures. For addition or subtraction, indicate the uncertain figure in each number by underlying it. For multiplication and division, indicate the number of significant figures in each number by placing a small number above each measurement. Be sure to also show the pre- and post rounded numbers. 3 3 2 • (176 m)(325 m)(1.2 m) = =68,640m3 (2 sig fig) =69,000m3 2 4 1 3 2. (0.0000023 cm)(1.435 cm)(7,000 cm) / (122 cm) = = 0.000189373cm2 (1 sig fig) 0.0002cm2 (Continued on Next Page)

26. Significant Operations cont 9 6 • (1.33490001 m)(16,000.0 m) = = 21358.40016m2 (6 sig fig) = 21,358.4m2 4 3 • (1.948m) / (2.43s) = = 0.801646091 m/s (3 sig fig) = 0.802 m/s 3 4 2 5 5. (0.663 kg)(4.391 m) / [(3.2 s)(9.1000 s)] = 0.009997366 kg m/s2 (2 sig fig) =0.010 kg-m/s2