Review of Significant Figures

1. Review ofSignificant Figures Dr. C. Yau Spring 2015 (Loosely based on Sec 1.5) from Jespersen 7th edition)

2. Accuracy vs. Precision • Accuracy is how close a measurement is to the correct value (commonly accepted value). • Precisionis how close several measurements are to each other. • When calibrated instruments are used properly, the greater the number of significant figures, the greater is the degree of precision for a given measurement.

3. Uncertainty in Measurement When making a measurement, we record all the digits that we are certain of and then estimate one more digit. mL mL What is the volume of the liquid in the graduated cylinder to the left? to the right? Ans. 21.3 mL? 21.4 mL? Ans. 21.33 mL? 21.32 mL? There is always uncertainty in the last digit we record.

4. Uncertainty in Measurement Another way of looking at this is to say that we record to 1/10 th of the smallest increment of the measuring device. Smallest increment of the graduated cylinder on the right is _____ mL. 1/10 th of that would be ___ mL. Its reading should therefore be recorded to what decimal place? Ans. ...to 2 decimal places(21.32 mL) 4

5. Uncertainty in Measurement Smallest increment of the graduated cylinder on the left is _____ mL. 1/10 th of that would be ___ mL. Its reading should therefore be recorded to what decimal place? Ans. ...to 1 decimal place(21.3 mL) 5

6. (Graduated cylinder on the left) 21.3 mL has 3 significant figures (sig. fig.) and 1 decimal place. (Graduated cylinder on the right) 21.32 mL has 4 sig.fig. and 2 decimal places. Know the difference between the number of sig. fig. and decimal places!

7. Errors Arise From A Number Of Sources Including: • Errors - inherent error due to the equipment or procedure (instrumental or method errors) • Changing volume due to thermal expansion or contraction (temperature changes) • Improperly calibrated equipment • procedural design allows variable measurements • Mistakes-blunders that you know that you have made. Do not use these data • Spillage • Incomplete procedures • Reading scales incorrectly • Using the measuring device incorrectly

8. Reducing Error: • Errors can often be detected by making repeated measurements • Error can be reduced by calibrating equipment • The average or meanreduces data variations: it helps find a central value

9. How many sig.fig. are in a number? All digits are significant EXCEPT: • Zeros to the left of the first nonzero digit (leading zeros) are never counted as significant. 0.00045 has only 2 sig. fig. • Zeros at the end of a number without a decimal point are assumed not to be significant (Trailing without a decimal place) • 5900 is assumed to have 2 sig. fig. • We don’t really know for sure. • The 2 zeroes are ambiguous, and the number should be expressed in scientific notation. • 30.0 on the other hand has 3 sig.fig. • The zeroes are not ambiguous. Why not?

10. Learning Check: How Many Significant Figures Are There In the Following? 2.33 3 500.0 4 Ambiguous, assumed to be 1 1000 3 .0500 How many sig. figs. are there in the number 010.010? 5 sig. fig.

11. Numbers with trailing zeroes & without a decimal are said to be ambiguous in terms of sig. fig. and must be expressed in scientific notation: 3400 in 3 sig. fig. ______________ 1530 in 2 sig. fig. ______________ 1400 x 103 in 3 sig. fig. _______________ 3000 x 10-2 in 2 sig. fig._______________

12. Exact Numbers • Numbers that come from definitions are exact and have no uncertainty. • They can be assumed to contain an infinite number of significant figures. Example: 25 mph means 25 miles/hour 25 has 2 sig. fig. Per hour means exactly 1 hour They do not affect the sig. fig. in a calculation. (1.000000000…. hour)

13. Measurements Limit The Precision Of Calculated Results Rules for combining measurements depend on the type of operation performed: • Multiplication and division • The # of sig. fig. in the answer should be the same as the # of sig. figs of the least precise measurement.

14. How many sig. figs. result from the following: 12.33 x 0.00002? Ans. ONLY ONE! 0.0002466 = 0.0002 = 2 x 10-4

15. Addition and Subtraction The answer should have the same # of decimal places as the number that is least precise, with the fewest # of decimal places. 3.247 ← 3 decimal places 41.36 ← 2 decimal places +125.2 ← 1 decimal place 169.807 ← ans rounded to 1 decimal place Ans. 169.8 How many sig. figs. result from the following: 10.33 - 0.0344? Tricky! Do this carefully before blurting out the answer! 10.2956 = 10.30 Ans. Four!

16. Ambiguous zeroes Addition and Subtraction What if the numbers do not have a decimal points? e.g. 7200 + 132 = ? Mathematically, the answer is 7332, but should we keep all 4 digits? 7200 is precise only to the hundred place. It will limit the precision of the answer to the hundred place. NO 7200132 7332 = 7300 = 7.3x103

17. Think Carefully! How many sig. figs. result from the following? • 2 • 3 • 4 • 5 • none of these 10.0 x 10.88 = 108.8 108.8 – 12.2 = 96.6 = 44 Ans. 2 sig. fig.

18. Scientific Notation Let’s pinpoint exactly what is meant by “scientific notation.” It is to express a number so that the decimal is behind the first nonzero digit. e.g. 427.1 x 104 = 0.0051 = 1302 x 10-5 = 4.271 x 106 5.1 x 10-3 1.302 x 10-2 Expressing a number in sci. notation should not change the # of sig. fig.

19. Exponential vs. Scientific Notation How is “exponential notation” different from “scientific notation”? Technically, “exponential notation” means expressing a number to include x10n without the decimal being behind the first nonzero digit necessarily. 0.25 in exponential notation could be written as 25x10-2 or 2.5x10-1 or 0.0025x102. 0.25 in scientific notation MUST be written as 2.5x10-1.

20. Do NOT use scientific notation indiscriminately! Scientific notation is needed ONLY.... 1) ...if the number has ambiguous zeroes. e.g. 150 x 2.0 = 2) ...if the number is very small (smaller than 0.01) e.g. 0.003 = 3) ...if the number is already in exponential notation (but not scientific notation) e.g. 24.831 x 1056 = 300 = 3.0 x 102 3 x 10-3 2.4831 x 1057

21. What is wrong with using scientific notation when it is not necessary? It shows your ignorance on the use of scientific notation. Would it make sense for someone to tell you to measure out 3.5x101 mL, when all he/she needs to say is ? Would it make sense for someone to tell you to weigh out 4.5 x 10-1 g when all he/she needs to say is ? 35 mL 0.45 g

22. Significant Figures When Calculating Averages What is the average of 35.7 and 35.8? You probably can come up with an answer without a calculator or doing mental arithmetic. In math, it would be the midpoint, 35.75. In science, however, if these were measured numbers you might consider the operation: What is wrong with this answer? The average cannot have more precision than the numbers from which it is derived! Ans. 35.8

23. One last question on sig. fig...... Think before you blurt out the answer! What is the answer to the correct sig. fig. for this calculation? 3.42 x 10-32 + 2.1 x 10-38 Ans. 3.42 x 10-32not 3.4 x 10-32 3.42 x 10-32 + 2.1 x 10-38 5.52 x 10-?? NO!! 3.42 x 10-32 + 0.0000021 x 10-32 3.42 x 10-32 3 sig. fig.! More practice problems are available on my HomePage: Click here.