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Appendix 1 SIGNIFICANT FIGURES

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Part I: Reading Measuring Devices 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. To figure what digit should come after the 2, you visually divide up the space in ten parts and note the approximate location of the right edge of the object. Because the right edge appears to be about 6/10th of the way between the 0.2 and the 0.3 marks, we would say that the length is 6.26 cm. If someone else reports the measurement as 6.27 cm, that would also be 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 are all the accurately known digits plus the first uncertain digit in a measurement. 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 someone that the smallest divisions on the ruler are 0.1 cm apart and that the last digit is uncertain. Ruler I

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 estimate. 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 someone that the smallest divisions on the ruler are 1 cm apart and that the last digit is your best estimate of reading between the 6 cm and 7 cm marks. The general rule is therefore, to read a measurement to one-tenth of the smallest division on the measuring device. That is, you should add one more digit than can be read directly from the calibration marks. For Ruler I, the smallest division is 0.1 cm and so you read measurements to two decimal places. For Ruler II, the smallest division is 1 cm and so you read measurements to one decimal place. 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). When using a digital display, you must record all the digits displayed—they are all significant. Learning to read measuring devices properly is very important for laboratory work. To check your understanding, do the following practice exercise.

cm Ruler II

cm

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Practice Exercise 1 Write your answers on the blanks, then check them against answers provided at the end of this appendix. 1.1) Record the measurements to the correct significant figures. Don’t forget your units! A B C D

A = __________ B = __________ C = __________ D = __________ 1.2) E F G H

E = __________ F = __________ G = __________ H = __________ 1.3) I J K L

I = __________ J = __________ K = __________ L = __________ Part II: Identifying Significant Figures in Numbers 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. The general rule is as follows:

All digits in a measurement are significant with the exception that: 1. leading zeroes are NEVER significant (0.0005 has only one sig. fig.) 2. tailing zeroes in numbers without decimal points are ambiguous. (Zeroes in

700 are ambiguous. Zeroes in 700.0 are significant.) • Such tailing zeroes are generally assumed to be not significant. • They can be expressed in scientific notation to remove the ambiguity.

| | | | cm

| | | | cm

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For example, 5200 as stated is assumed to have 2 sig. figs. If it were to have 3 sig. figs., it should have been expressed as 5.20 x 103. If it were to have 4 sig. figs., it should have been expressed as 5.200 x 103, or it could have been expressed as 5200. with the decimal point after the last zero. This indicates that the tailing zeroes are significant. (Remember tailing zeroes are assumed not significant only when there is no decimal point. Tailing zeroes in numbers with decimal points are significant.)

In the following examples, the significant figures are underlined.

30 is assumed to have one sig. fig. 30. has 2 sig. figs. (The number has a decimal point, so all tailing zeroes are significant.) 30.0 has 3 sig. figs. (Again, the number has a decimal point, so all tailing zeroes are

significant.) 0.0050200 has 5 sig. figs. (Leading zeroes are not significant, but the tailing zeroes are

significant, because the number has a decimal point.) 12.00 has 4 sig. figs. 3.20 x 102 has 3 sig. figs.

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. figs. but only one decimal place. The practice exercise below provides opportunity for you to distinguish between the number of significant figures and the number of decimal places in a number. It also provides the opportunity to distinguish between zeroes that are significant and those that are not significant. Practice Exercise 2 2.1) Give the number of sig. figs. and the number of decimal places in each of the number

below. # sig. figs. # decimal places # sig. figs. # decimal places

12.92 _______ ______________ 8,000 _______ ______________ 30.009 _______ ______________ 8,000. _______ ______________ 0.005 _______ ______________ 8,000.00 _______ ______________ 0.00260 _______ ______________

2.2) These numbers have ambiguous zeroes. Express them in scientific notation to remove

the ambiguity. 35000 in 2 sig. figs. ______________ 1800 in 3 sig. figs. ______________ 35000 in 3 sig. figs. ______________ 680,000 in 4 sig. figs. ______________ 35000 in 4 sig. figs. ______________ 2700 x 10-8 in 3 sig. figs. ______________

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Part III: Using 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 (tailing zeroes in a number without a decimal) To remove the ambiguity, it can be expressed in scientific notation. e.g. It is not clear whether 35000 has 2, 3, 4 or 5 sig. figs. It is not clear whether the three “tailing zeroes” are significant or not. Suppose you mean 35000 to have 3 sig. figs., then it should be expressed as 3.50 x 104.

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. 0.000 000 83 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 Some students indiscriminately express all their numbers in scientific notation. 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.

20.0 x 5.0 = 100 This should be expressed as 2 sig. figs. Because the tailing zeros are ambiguous, the number should be expressed in scientific notation. Correct answer = 1.0 x 102

0.004 ÷ 800 = 0.000005 This should be expressed as 1 sig. fig. Because there are so many leading zeros, the number can be expressed in exponential notation. Correct answer = 5 x 10−6

(42 x 10 3) x 2 = 84 x 103 This should be in 1 sig. fig. and being a very large number, needs to be in scientific notation.

Correct answer = 8 x 104 22 x 2.0 = 44

This should be in 2 sig. figs. There is nothing wrong with the way it is stated. Correct answer = 44

The following practice exercise provides an opportunity for you to check your understanding of when to use scientific notation.

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Practice Exercise 3 3.1) Which of the following numbers require scientific notation? For any that require it,

give the correct way of expressing it. A. 350 cm B. 0.13 g C. 38 mL D. 0.00032871 E. 235.2 x 104

3.2) Convert the following numbers from scientific notation to standard notation.

A. 1.5 x 104 B. 4.59 x 10-7

Part IV: Rounding-off Numbers In correcting a number to express the proper number of sig. figs., 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.

For example, 72.49 in 3 sig. figs. is 72.5 45.52 in 3 sig. figs. is 45.5 299 000 in 2 sig. figs. is 300000 and to remove ambiguity, the answer is 3.0 x 105 92528 in 4 sig. figs. is 92530 and to remove ambiguity, answer is 9.253 x 104

The practice exercise below provides an opportunity to check your understanding of rounding. Practice Exercise 4 4.1) Round the following numbers to the specified significant figures:

A. 26000 to one sig. fig. Ans. _____________ B. 3510 to two sig. figs. Ans. _____________ C. 0.00375 to two sig. figs. Ans. _____________ D. 0.002787 x 103 to three sig. figs. Ans. _____________

4.2) A student was given the numbers in column A and asked to round them off to three sig. figs. The student’s answers are in column B. Indicate whether the student’s answers are correct or incorrect. If incorrect, give the correct answer. Column A Column B Correct or Incorrect? 4925 493 0.0006399 0.0006400 535.456 535.000

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Part V: Handling Significant Figures in Calculations Rule 1: During Addition or Subtraction, the answer has the same number of decimal places as the measurement with the least number of decimal places.

e.g. 3.255 3 decimal places

+ 1.76 2 decimal places 5.015 (should have only 2 decimal places) = 5.02 Rule 2: During Multiplication or Division, the answer has the same number of sig. figs. as the measurement with the least number of sig. figs. e.g. 3.5 x 2.78 = 9.73 = 9.7 (2 sig. figs.) (3 sig. figs.) (should have 2 sig. figs.) e.g. 4.00 x 3.0 = (2 sig. figs.) 2.00 Rule 3: When Addition, Subtraction, Multiplication, or 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. 4.05 − 4.00 = 0.05 = 0.025 = 0.03 2.00 2.00 Count 1 sig. fig. for the division 2 decimal places for the subtraction 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. You can keep track of where the last digit should be by placing a line under the digit in that position. Or, we can keep track of the extra digit by writing a line through it. e.g. 78.2 + 5.23 = ? 21.3 3.4 = 3.671 + 1.54 = 5.211 = 5.2 (limiting answer to one decimal place in the addition)

6.0

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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.) Practice Exercise 5 Provide answers for the following computations. 5.1) 69.76 – 65.2 Ans. _____________ 5.2) 9.21 + 7.242 Ans. _____________ 5.3) 21 x 3 = Ans. _____________

5.4) 5.0 + 3.0

= 2.00

Ans. _____________

5.5) 33.9 - 32.1

= 2.00

Ans. _____________

5.6) Find the average of 73.2, 73.8 and 74.2. Ans. _____________ 5.7) Find the average of 82.3 and 82.4. Ans. _____________ Part VI: Calculating with Exact Numbers

Certain types of numbers are considered “exact.” For example, there are exactly 16 ounces in one pound. The number 1 and the number 16 would have an unlimited number of significant figures. So one pound (1.00000000000…), for example, has 16.000000000000.... ounces. Calculations involving these number should not be limited by the significant figures shown in “16 oz/lb.” If we want to calculate how many ounces are in 2.00 lb, for example, we would set up the problem thus:

2.00 lb x 16 oz

= 32.0 oz 1 lb

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The answer has 3 sig. figs. even though 1 appears as 1 sig. fig. and 16 appears as 2 sig. figs. The answer is limited by 2.00 lb (3 sig. figs.) and not by 1 or 16 because they are “exact” numbers. 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 3.87. (To find the average, you add the two numbers

together and divide by 2. “2” is an exact number. Do not round your average to 1 sig. fig.)

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Answers to Practice Exercises Practice Exercise 1

A = 1.64 cm B = 3.04 cm C = 5.00 cm D = 8.97 cm E = 0.2 cm F = 2.9 cm G = 6.0 cm H = 8.3 cm I = 0.602 in. J = 0.696 in. K = 0.794 in. L = 0.822 in. Because the last digit in any measurement should be an estimate, your measurements can be different in the last digit.

Practice Exercise 2 2.1) sig. fig. # decimal places # sig. fig. # decimal places

12.92 ___4___ _____2________ 8000 assume 1 _____none_____ 30.009 ___5___ _____3________ 8000. ___4___ _____none_____ 0.005 ___1___ _____3________ 8000.00 ___6___ _______2______ 0.00260 ___3___ _____5________

2.2) These numbers have ambiguous zeroes. Express them in scientific notation to remove

the ambiguity. 35000 in 2 sig. fig. ___3.5 x 104____ 1800 in 3 sig. fig. __1.80 x 103____ 35000 in 3 sig. fig. ___3.50 x 104___ 680,000 in 4 sig. fig. __6.800 x 105___ 35000 in 4 sig. fig. ___3.500 x 104__ 2700 x 10-8 in 3 sig. fig. __2.70 x 10−5___ Practice Exercise 3 3.1) A = 3.5 x 102 cm D = 3.2871 x 10−4 E = 2.352 x 106

B and C do not require scientific notation. 3.2) A. 1.5 x 104 = 15000 (2 sig. figs) B. 4.59 x 10-7 = 0.000000459 Practice Exercise 4 4.1) A = 3 x 104 B = 3.5 x 103 C = 3.8 x 10−3 D = 2.79 4.2) Column A Column B Correct or Incorrect? 4925 493 Incorrect, 4.93 x 103 0.0006399 0.0006400 Incorrect, 6.40 x 10-4

535.456 535.000 Incorrect, 535 Practice Exercise 5

5.1) 4.6 5.2) 16.45 5.3) 6 x 101 5.4) 4.0 5.5) 0.90 5.6) 73.7 5.7) 82.4

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