Appendix 1: REVIEW OF TREATMENT OF SIGNIFICANT FIGURES

Transcription

1 Appendix 1: REVIEW OF TREATMENT OF SIGNIFICANT FIGURES PART I: HOW MANY DIGITS SHOULD YOU RECORD? When you measure an object with a ruler such as Ruler I shown in the figure below, you know for sure that its length is somewhere between 6.2 and 6.3 cm. We always estimate one more digit and we might call it 6.26 cm since it appears to be slightly more than halfway between the two tick marks. If someone else reports it as 6.27 cm, that would be also acceptable. It is understood that the last digit reported always has some uncertainty. We call these three digits significant figures. Significant figures are digits that are of significance. They tell us how finely graduated the measuring device is. The more finely the graduation, the more reproducible the results would be, and therefore the more precise the measurement is. By reporting the length as 6.26 cm (2 decimal places), you are telling people that the measuring device is graduated to 1 decimal place (0.1 cm) and you are estimating to one more decimal place (0.01 cm). Ruler I cm Ruler II cm In comparison, when you measure the same object with Ruler II, which is graduated only to 1 cm, you only know for sure that the length is somewhere between 6 and 7 cm. The next digit you read is an estimation. So, you might read it as 6.2 cm, but you cannot report it as 6.20 or 6.25 cm. By reporting 6.2 cm, you are telling people that the measuring device is graduated to 1 cm and you are estimating to one more place, to 0.1 cm. The general rule is therefore, to read a measurement to one-tenth of the smallest division on the measuring device. For Ruler I, the smallest division is 0.1 cm and so you read measurements to 0.01 cm (two decimal places). For Ruler II, the smallest division is 1 cm and so you read measurements to 0.1 cm. Keep this in mind whenever you make a measurement with equipment (such as rulers, graduated cylinders and burets) that does not give you a digital display (such as an electronic balance or temperature probe). (Practise with Appendix 2, Exercise 1.) PART II: HOW TO RECOGNIZE WHICH DIGITS IN A GIVEN NUMBER ARE CONSIDERED SIGNIFICANT FIGURES? There are various methods to determining which digits in a given number are significant, but ultimately they all point to the same answer. Your textbook may tell you one method, and your instructor may tell you another. You are likely to find that the method shown below is the simplest to remember. A-1

2 A-2 APPENDIX 1: REVIEW OF TREAMTENT OF SIGNIFICANT FIGURES The general rule is as follows: All digits are significant with the exception that: 1. leading zeroes are NEVER significant ( has only one sig. fig.) 2. trailing zeroes in numbers without decimal points are ambiguous. (Zeroes in 700 are ambiguous. Zeroes in are significant.) Such trailing zeroes are generally assumed to be not significant. They must be expressed in scientific notation to remove the ambiguity. For example, [5200] as stated is assumed to have 2 sig. fig. If it were to have 3 sig. fig., it should have been expressed as [5.20 x 10 3 ]. If it were to have 4 sig. fig., it should have been expressed as [5.200 x 10 3 ], or it could have been expressed as [5200.] where the decimal indicates the trailing zeroes to be significant. (Remember trailing zeroes are assumed not significant only when there is no decimal point. Trailing zeroes in numbers with decimal points are significant.) In the following examples, the significant figures are underlined inside the square brackets, [ ]: 30 is assumed to have one sig.fig. [ 30 ]. If you have to report such a number, you MUST express it in scientific notation. 30. has 2 sig. fig. [ 30. ] (The number has a decimal point, so all trailing zeroes are significant.) 30.0 has 3 sig. fig. [ 30.0 ] (Again, the number has a decimal point, so all trailing zeroes are significant.) has 5 sig. fig. [ ] (Leading zeroes are not significant, but the trailing zeroes are significant, because the number has a decimal point.) has 4 sig. fig. [ ] 32.0 x 10 2 has 3 sig. fig. [ 32.0 x 10 2 ] Do not confuse the number of significant figures with the number of decimal places. The number of decimal places refer to the number of digits to the right of the decimal point. Thus [30.0] has three sig. fig. but only one decimal place. (Practise with Appendix 2, Exercise 2.) PART III: WHEN TO USE SCIENTIFIC NOTATION A number should be expressed in scientific notation (with only one nonzero digit to the left of the decimal) under these conditions: 1. A number with ambiguous zeroes (trailing zeroes in a number without a decimal) To remove the ambiguity, it MUST be expressed in sci. not. e.g. It is not clear whether [35000] has 2, 3, 4 or 5 sig.fig. It is not clear whether the three trailing zeroes are significant or not. If you mean [35000] to have 3 sig. fig., then it MUST be expressed as [3.50 x 10 4 ]. 2. A number that is very small (as a rule of thumb, less than 0.01). It is tedious and riskier to copy numbers with a string of avoidable zeroes. e.g. [ ] SHOULD be expressed as [8.3 x 10 7 ] 3. A number that is in exponential form for any reason e.g. [324.3 x 10 8 ] SHOULD be expressed as [3.243 x 10 6 ]

3 APPENDIX 1: REVIEW OF TREATMENT OF SIGNIFICANT FIGURES A-3 Some students indiscriminately express all their numbers in scientific notation because they are too lazy to figure out when it is necessary. Although it is not wrong to do so, you should learn when it is appropriate. For example, it would not be appropriate to tell someone to weigh out 2.5 x 10 grams of salt when 25 grams of salt would do equally well x 5.0 = 100 This should be expressed as 2 sig.fig.. As stated, it has ambiguous zeroes. Correct answer = 1.0 x = This should be in 1 sig. fig.. As stated there are too many leading zeroes. Correct answer = 5 x 10 6 (42 x 10 3 ) x 2 = 84 x 10 3 This should be in 1 sig. fig. and being a very large number, needs to be in scientific notation. Correct answer = 8 x x 2.0 = 44 This should be in 2 sig. fig.. There is nothing wrong with the way it is stated. Correct answer = 44 (Practise with Appendix 2, Exercise 3.) PART IV: ROUNDING OFF NUMBERS In correcting a number to express the proper number of sig. fig., we often have to drop off unwanted digits. The rules for rounding off numbers are explained in your textbook and/or lab manual. Here is a summary: Rules for rounding off numbers: If the digit immediately to the right of the last sig. fig. is equal or greater than 5, you round up. If the digit immediately to the right of the last sig. fig. is less than 5, you round down in 3 sig. fig. is in 3 sig. fig. is in 2 sig. fig. is and to remove ambiguity, answer is 3.0 x ,528 in 4 sig. fig. is and to remove ambiguity, answer is x 10 4 (Practise with Appendix 2, Exercise 4.)

4 A-4 APPENDIX 1: REVIEW OF TREAMTENT OF SIGNIFICANT FIGURES PART V: KEEPING TRACK OF SIGNIFICANT FIGURES DURING CALCULATIONS Rule 1: During Addition or Subtraction, the answer has the same number of decimal places as that with the least. e.g decimal places decimal places (should have only 2 decimal places) = 5.02 Rule 2: During Multiplication or Division, the answer has the same number of sig. fig. as that with the least. e.g. 3.5 x 2.78 = 9.73 = 9.7 (2 sig.fig.) (3 sig. fig.) (should have 2 sig. fig.) e.g x 3.0 = 6.0 (2 sig. fig.) 2.00 Rule 3: When Addition, Subtraction, Multiplication, and Division are mixed together, apply rules 1 and 2 one step at a time. This is very tricky, so think through this very carefully. e.g = 0.05 = = Count 1 sig. fig. for the division ( ) involves 2 decimal places, thus the answer has 2 decimal places (0.05). In the next step, 0.05 is divided by Here we need to consider # sig. fig. instead. There is only 1 sig. fig. in 0.05 and therefore the answer needs to be rounded to 1 sig. fig. (0.03). Rule 4: When there are several steps before you get to the final answer, carry one extra digit and round off properly at the end. Often we keep track of the extra digit by writing a line through it. e.g =? = = = 5.2 (limiting ans to one decimal place in the addition)

5 APPENDIX 1: REVIEW OF TREATMENT OF SIGNIFICANT FIGURES A-5 Rule 5: Keep in mind that you cannot get more precision just by doing a calculation such as finding the average of several numbers. The average must have the same number of decimal places as the individual numbers themselves. e.g. The average of 37 and 38 mathematically comes out to 37.5, but as written, the average would have more digits than 37 and 38. The correct answer is 38 (37.5 rounded off.) (Practise with Appendix 2, Exercise 5.) PART VI: EXACT NUMBERS Certain types of numbers are considered exact. For example, there are exactly 16 ounces in one pound. The number 16 would have as many significant figures as needed. So one pound has ounces. Calculations involving this number should not be limited by the significant figures shown in 16 oz/lb. For example, if we want to calculate how many ounces are in 2.0 lb, we would set up the problem thus: x oz = 2.00 lb x 16 oz = 32.0 oz 1 lb The answer has 3 sig. fig. even though 16 appears as 2 sig. fig.. The answer is limited by 2.00 lb (3 sig. fig.) and not by 16 because 16 here is an exact number. In the same way, the answer is not limited by 1 in 1 lb because that, too, is an exact number. Which types of numbers are considered exact? Below are the general rules. 1. Conversions between units within the English System are exact. e.g. 12 in = 1 ft or 12 in/1 ft (12 and 1 are both exact.) 2. Conversions between units within the Metric System are exact. e.g. 1 m = 100 cm or 1 m/100 cm (1 and 100 are both exact.) 3. Conversions between English and Metric system are generally not exact. Exceptions will be pointed out to you. Example of an exception: 1 in = 2.54 cm exactly (Both 1 and 2.54 are exact.) Example of general rule: 454 g = 1 lb or 454 g/1 lb (454 has 3 sig. fig., but 1 is exact.) 4. Per means out of exactly one. e.g. 45 miles per hour means 45 mi =1 hr or 45 mi/1 hr. (45 has 2 sig. fig. but 1 is exactly one.) 5. Percent means out of exactly one hundred. e.g. 25.9% means 25.9 out of exactly 100 or 25.9/100. (25.9 has 3 sig. fig., but 100 is exact.) 6. Counting numbers are exact. Sometimes it is hard to decide whether a number is a counting number or not. In most cases it would be obvious. Ask when in doubt. e.g. There are 5 students in the room. (5 would be an exact number because you cannot have a fraction of a student in the room.) e.g. Find the average of 3.27 and (To find the average, you add the two numbers together and divide by 2. 2 is an exact number. Do not round you average to 1 sig. fig.)

6 A-6 APPENDIX 1: REVIEW OF TREAMTENT OF SIGNIFICANT FIGURES