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Determining Significant Figures A4 Science Learning Center University of Michigan, Dearborn Directions This module consists of a written explanation and examples problems. As you read through the material, be sure to complete all example problems. When you have completed reading the module and you feel you fully understand the material, ask for a posttest. If you pass the posttest, make sure your name is recorded in the SLC database. If you do not pass it, you may review the module and retake the test as many times as needed. Objectives Know what significant figures are. Know which numbers are significant. Be able to express the result of a measurement with the correct number of significant figures. Distinguish between significant and non-significant zeros. Be aware that the number of figures in scientific notation indicates precision. Be able to express the result of a calculation with the correct number of significant figures. What are Significant Figures? Significant Figures and Measurements A significant figure is one that has some significance but does not necessarily denote a certainty. Whenever you estimate any kind of measurement, for example the length or weight of an object, there is always a limit to the number of digits you can read. The number of significant figures in a measurement is the number of digits that are known with certainty plus the last one that is not absolutely certain. What are Significant Figures? Significant Figures and Measurements As a general rule you should attempt to read any scale to one tenth of its smallest division by visual interpolation. In the case below, you would read to + 0.01cm. This estimated figure will always be your last significant figure with the implied accuracy of + 1. Therefore, the measurement is written as 4.63 + 0.01cm. Generally, read any scale to 1/10 of smallest division. 4.63 + 0.01 cm What are Significant Figures? Significant Figures and Measurements A length measurement of 5.63 cm contains three significant figures. The first two, the 5 and 6, are certain. The last digit, the 3, is uncertain. The uncertainty in the last significant figure is usually + 1. The result is 5.63 + 0.01cm. An Analytical Balance is precise to four decimal places with an uncertainty of +1 in the last significant figure. Therefore, the measurement 13.7654g is written as 13.7654 + 0.0001 g and has six significant figures. Precision of Two Instruments Ruler 5.63 + 0.01 cm 3 significant figures Analytical Balance 13.7654 + 0.0001 g 6 significant figures What are Significant Figures? Significant Figures and Measurements Here we see another kind of measurement, the reading of the position of a buret meniscus, (the curved liquid surface in a buret). The liquid level is always read at the bottom of the meniscus for transparent liquids. The reading in this buret is 12.75. Four significant figures are implied. The last significant figure, 5, is obtained by visual interpolation between the 0.1 milliliter divisions. All observers should agree with the first three significant figures but not necessarily with the last figure recorded here. Disagreements of +1 in the last digit are expected with visual interpolations. The measurement is, therefore, written as 12.75 + 0.01ml and has four significant figures. Measurement with a Buret Read the bottom of meniscus at eye level 12.75 + 0.01 ml 4 significant figures What are Significant Figures? Significant Figures and Measurements When reading a measurement from a meter, you should also read to one digit past the smallest division on the meter. On this meter, the reading should be 1.27. The 1 and the 2 are certain. The 7 must be estimated visually. The measurement is written as 1.27 + 0.01g. 1.00 2.00 0.00 grams Reading: 1.27 + 0.01 g The 7 is estimated and, therefore, uncertain. Writing Significant Figures During any calculation—addition, subtraction, multiplication, or division— your answer could be expressed with too few or too many significant figures. These numeric values may imply a precision that does not exist in the experiment being evaluated. If you round off incorrectly, your answer will have an incorrect number of significant figures and will lose precision. EXAMPLE 1.024 x 1.2 = 1.2288 Too many numerals Too precise 1.024 x 1.2 = 1 Too few numerals Not precise enough We, therefore, have developed rules for determining the correct number of significant figures in a number and apply these rules to calculations. Writing Significant Figures First we need to learn how to evaluate the number of significant figures a given number contains. This is necessary for calculations such as addition, subtraction, multiplication and division. Any written number that is not a zero is significant. In this table the significant figures are underlined: 3 significant figures 4 significant figures 5 significant figures 23.4 2.34 0.234 345.6 3.456 0.03456 678.90 Note: The zeros in 0.234 and 0.03456 are not significant, but the zero in 678.90 is a significant figure. Zeros have special rules as we will discuss in the next several slides. Writing Significant Figures Standard Exponential Notation and Zeros Zeros appearing between nonzero numbers are significant. Examples 40.7 L 87,009 km 3 significant figures 5 significant figures Writing Significant Figures Standard Exponential Notation and Zeros Zeros appearing in front of nonzero numbers are not significant. Examples 0.095987 m 0.0009 kg 5 significant figures 1 significant figure Writing Significant Figures Standard Exponential Notation and Zeros Zeros appearing at the end of a number and to the right of a decimal point are significant. Examples 850.00 g 9.500000000 mm 5 significant figures 10 significant figures Writing Significant Figures Standard Exponential Notation and Zeros Without a decimal, large numbers containing zeros, such as 45,600 grams, pose a special problem. As the number is written, we cannot tell whether the two zeros indicate the precision of the measurement or whether the zeros merely locate the decimal point. If the zeros indicate precision, they are significant and the implied uncertainty is + 1. This means that the measurement lies between 45,599 and 45,601. If, however, the zeros merely locate the decimal point and are not significant, the implied uncertainty is + 100. Then we know the measurement lies between 45,500 and 45,700. Thus a number written in this form is ambiguous. Example 45,600 grams Zeros Significant 45,600 grams + 1 gram 45,599 45,601 5 significant figures Zeros Not Significant 45,600 grams + 100 grams 45,500 45,700 3 significant figures **For the purposes of this module and its posttest, assume the zeros are not significant for ambiguous numbers.** Writing Significant Figures Standard Exponential Notation and Zeros Zeros appearing after a nonzero number which are not followed by another significant figure or a decimal point are not significant. Examples 85,000 85,000,000 85,000,000.0 85,000,000. 2 significant figures 2 significant figures 9 significant figures 8 significant figures Note: The decimal point after the zeros indicates that all the numbers are significant. Writing Significant Figures Summary: Rules for Zeros Zeros appearing between two nonzero numbers are significant. Examples: The number 2.035 has 4 significant figures The number 3,0007 has 5 significant figures. Zeros appearing in front of nonzero numbers are not significant. Example: The number 0.0567 has only 3 significant figures. Zeros appearing at the end of a number and to the right of a decimal point indicate precision and are significant. Example: 4.700 has 4 significant figures. Zeros appearing after a nonzero number which are not followed by another significant figure or a decimal point are not significant Example: The number 1200 has 2 significant figures, the number 1200. has 4 significant figures, and the number 1200.0 has 5 significant figures. Writing Significant Figures Standard Exponential Notation and Zeros Ambiguity about the precision and number of significant figures for a particular number may be avoided by expressing the number in standard exponential notation or scientific notation. If the number 45,600.0 contains 5 significant figures, the number would be expressed as 4.5600 x 104. This notation implies 5 significant figures. If only 3 numbers are significant, the number would be expressed as 4.56 x 104. Normal notation 45,600 Scientific notation 4.5600 x 104 4,5600 4.56 x 104 Writing Significant Figures Standard Exponential Notation and Zeros Scientific notation is also useful for clearly expressing very large and very small numbers with the correct precision and number of significant figures. The number 0.001230 can be expressed as 1.230 x 10-3 which contains 4 significant figures. The number 900.00 can be expressed as 9.0000 x 102 which contains 5 significant figures. Finally, the number 2000. has a decimal on the end indicating precision and that all of the zeros are significant. Thus, 2000. can be expressed as 2.000 x 103 which contains 4 significant figures. Normal notation Scientific notation 0.001230 1.230 x 10-3 900.00 9.0000 x 102 2000. 2.000 x 103 Practice Problems 1 How many significant figures are in each of the following numbers? a) b) c) d) e) f) g) h) 1.234 1.2340 1.234 x 10-3 1.2340 x 10-3 1234 12340 0.012340 10234 Solutions to Practice Problems 1 a) 1.234: 4 b) 1.2340: 5 c) 1.234 x 10-3: 4 d) 1.2340 x 10-3: 5 e) 1234: 4 f) 12340: 4 g) 0.012340: 5 h) 10234: 5 Calculations You have learned how to determine how many significant figures are in a number. Now you will learn how many significant figures should be expressed in the result of a calculation. Calculations Adding and Subtracting When adding or subtracting quantities, the rule is to determine which number in the calculation has the least number of digits to the right of the decimal point. Your result will have that same number of digits to the right of the decimal point. Calculations Adding and Subtracting Example 26.46 + 4.123 30.583 Least number of digits to the right of the decimal = 2 26.46 - 4.123 22.337 30.58 Round to 2 decimals 22.34 For example, when adding 26.46 to 4.123, the calculated sum is 30.583. The original number 26.46 has the least number of digits to the right of the decimal point, two, so the calculated sum is rounded to 30.58. In subtracting 4.123 from 26.46, the calculated difference is 22.337. The original number 26.46 has the least number of digits to the right of the decimal point, two, so the calculated difference is rounded to 22.34. Calculations Adding and Subtracting Example 2.634 + 0.02 2.654 Least digits to the right = 2 2.634 - 0.02 2.614 2.65 Round to 2 decimals 2.61 Another example of addition is 2.634 plus 0.02. The calculated sum, 2.654, will be rounded off to 2.65. **The number 0.02 has the least number of digits to the right of the decimal point, two, regardless of the fact that only one of the digits is significant.** The result will have two digits to the right of the decimal point. When subtracting 2.634 minus 0.02, the calculated difference, 2.614, is rounded off to 2.61. Practice Problems 2 Complete the following arithmetic operations and express the answer with the correct number of significant figures: a) b) c) d) e) f) g) h) 1.421+ 0.4372 = 0.0241 + 0.11 = 0.14 + 1.2243 = 760.0 + 0.011 = 1.0123 - 0.002 = 123.69 - 20.1 = 463.231 - 14.0 = 47.2 - 0.01 = Solutions to Practice Problems 2 a) 1.421+ 0.4372 = 1.858 b) 0.0241 + 0.11 = 0.13 c) 0.14 + 1.2243 = 1.36 d) 760.0 + 0.011 = 760.0 e) 1.0123 - 0.002 = 1.010 f) 123.69 - 20.1 = 103.6 g) 463.231 - 14.0 = 449.2 h) 47.2 - 0.01 = 47.2 Calculations Multiplying and Dividing For the multiplication and division of numbers, there is a different rule for determining the number of significant figures. When multiplying or dividing, determine which number entering the calculation has the smallest total number of significant figures regardless of the position of the decimal point. Your calculated result will have that same number of significant figures. Calculations Multiplying and Dividing Example Least number of significant figures 2.61 x 1.2 = 3.132 2.61 / 1.2 = 2.175 Round to: 3.1 Round to: 2.2 In the example above, 2.61 multiplied by 1.2, the result is rounded off to 3.1. The number 1.2 has the least number of significant figures, two. So the calculated answer will also have two significant figures. When 2.61 is divided by 1.2, the result is also rounded off to two significant figures. The calculated answer, 2.175, is rounded off to 2.2. Practice Problems 3 Perform the indicated operations. Express your answers with the correct number of significant figures: a) b) c) d) 42.3 x 2.61 = 0.61 x 42.1 = 46.1 / 1.21 = 23.2 / 4.1 = Solutions to Practice Problems 3 a) 42.3 x 2.61 = 110. b) 0.61 x 42.1 = 26 c) 46.1 / 1.21 = 38.1 d) 23.2 / 4.1 = 5.7 Review of Rules for Calculations Addition/ Determine which number in the calculations subtraction has the least number of digits to the right of the decimal point. Your result will also have the same number of digits to the right of the decimal point. 234.7 + 1.623 236.323 Result: 236.3 Multiplicati Determine which number in the calculations on / has the least total number of significant Division figures (regardless of the decimal point’s position). Your result will also have that same number of significant figures. 44.2 x 2.662 117.6604 Result: 117 Additional Principles As you begin to apply the principles of significant figures to actual problems or laboratory experiments, three additional principles should be presented. Principle 1: If you are using exact constants, they do not affect the number of significant figures in your answer. For example, you might need to calculate how many feet equal 26.1 yards. The conversion factor you would need to use, 3 ft/yard, is an exact constant and does not affect the number of significant figures in your answer. 26.1 yards multiplied by 3 feet per yard equals 78.3 feet which has 3 significant figures. Example EXACT CONSTANT: 3 ft = 1 yd 26.1 yd 3 significant figures X 3 ft / 1 yd EXACT = 78.3 ft 3 significant figures Additional Principles Principle 2: If you are using constants which are not exact (such as pi = 3.14 or 3.142 or 3.14159) select the constant that has at least one or more significant figures than the smallest number of significant figures in your original data. This way the number of significant figures in the constant will not affect the number of significant figures in your answer. For example, if you multiply 4.136 ft., which has four significant figures, times pi, you should use 3.1416 which has 5 significant figures for pi and your answer will have 4 significant figures. EXAMPLE 4.316 ft. x π = 4.316 ft. x 3.1415 = 13.56 4 significant figures 5 significant figures for π answer in 4 significant figures Additional Principles Principle 3: When you are doing several calculations, carry out all of the calculations to at least one more significant figure than you need. When you get the final result, then round off. For example, you would like to know how many meters per second equals 55 miles per hour. The conversion factors you would use are: 1 mile = 1.61 x 103 meters and 1 hour = 3600 seconds Your answer should have two significant figures. Your result would be 88.55 divided by 3600 which equals 24.59 m/sec. This rounds off to 25 m/sec. By carrying this calculation out to at least one extra significant figure, we were able to round off and give the correct answer of 25 m/sec rather than 24 m/sec. EXAMPLE How many meters per second is 55 miles per hour? 1 mile = 1.61 x 103 m (not exact constant; 3 sig. figs.) 1 hour = 3600 seconds (exact constant; 4 sig. figs.) 55 miles / hour = 55 miles / 1 hr. x 1.61 x 103 m / 1 mile x 1 hour / 3600 sec = 24.597 m / 1 sec = 25 m/s Take the Posttest! You are now finished with this module. If you haven't already done the practice problems, we recommend you try them. When you're done, obtain a posttest from the Science Learning Center personnel and complete it.