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Significant Figures and Measurement Geneva High School Exact & Measured Numbers • There are 2 different types of numbers – Exact – Measured • Measured number = they are measured with a measuring device so these numbers have ERROR. • Also, we all read measuring instruments slightly differently. How many donuts do you see in this picture? Hopefully you counted four donuts. This is an example of an exact number. An exact number is obtained when you count objects or use a defined relationship. Counted objects are always exact, for example: 2 soccer balls 4 pizzas Exact relationships with predefined values are also exact numbers. For example: 1 foot = 12 inches 1 meter = 100 cm Check Your Learning A. Exact numbers are obtained by 1. using a measuring tool 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool 2. counting 3. definition Check Your Learning A. Exact numbers are obtained by 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool Check Your Learning Classify each of the following as an exact or a measured number. 1 yard = 3 feet The diameter of a red blood cell is 6 x 10-4 cm. There are 6 hats on the shelf. Gold melts at 1064°C. Check Your Learning Classify each of the following as an exact (1) or a measured(2) number. This is a defined relationship. A measuring tool is used to determine length. The number of hats is obtained by counting. A measuring tool is required. Measurements • All scientific measurements have a number and a unit • Ex: 11.2 km 65.304 mg Accuracy • Accuracy is the closeness of a measurement to the actual value of what is being measured. • How close a measured value is to the real value/target. Precision • Precision is a gauge of how exact a measurement is. • How repeatable is a measurement? Accuracy and Precision Significant Figures • Significant digits are all the digits that are known in a measurement, plus the last digit that is estimated. • The more precise a measurement is, the more significant digits the measurement contains. • You cannot record a piece of data that is more precise than the measuring instrument allows. Tomorrow’s Quiz • Scientific Notation • Metric Conversions • Identifying the number of significant figures (yes, you may use your rules sheet I gave you_ For example… • Suppose you measure the mass of a piece of iron to be 34.73 grams on a digital scale. • You then measure the volume to be 4.42 cubic centimeters. • The calculated density of that iron would be: Density = 34.73 g = 7.857466g/cm3 4.42 cm3 This calculated answer can’t be recorded like this, because it contains seven significant figures, while the volume of the sample only had three significant figures in the measurement. You cannot report a calculation or measurement that is more precise than the data used in the calculation. 2.4 Measurement and Significant Figures • • • • Every experimental measurement has a degree of uncertainty. The volume, V, at right is certain in the 10’s place, 10mL<V<20mL The 1’s digit is also certain, 17mL<V<18mL A best guess is needed for the tenths place. Chapter Two 16 What is the Length? 1 • • • • • 2 3 We can see the markings between 1.6-1.7cm We can’t see the markings between the .6-.7 We must guess between .6 & .7 We record 1.67 cm as our measurement The last digit an 7 was our guess...stop there 17 4 cm Learning Check What is the length of the wooden stick? 1) 4.5 cm 2) 4.54 cm 3) 4.547 cm ? 8.00 cm or 3 (2.2/8) 19 Measured Numbers • Do you see why Measured Numbers have error…you have to make that Guess! • All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate. • To indicate the precision of a measurement, the value recorded should use all the digits known with certainty. 20 Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty. 21 Significant Figure Rule 1 • All non-zero digits are significant • Zeros between non-zeros are significant 34.59 1.091 0.3001 All have 4 sig figs Rule 2 • If number ≥ 1 with a decimal point, then… all digits are significant 7.100 10078.00 Rule 3 • If digit ≥ 1 and no decimal point, then all zeros to right of non-zeros are not significant 32,000 890400 290 Note the 4 rules When reading a measured value, all nonzero digits should be counted as significant. There is a set of rules for determining if a zero in a measurement is significant or not. • RULE 1. Zeros in the middle of a number are like any other digit; they are always significant. Thus, 94.072 g has five significant figures. • RULE 2. Zeros at the beginning of a number are not significant; they act only to locate the decimal point. Thus, 0.0834 cm has three significant figures, and 0.029 07 mL has four. Chapter Two 25 • • RULE 3. Zeros at the end of a number and after the decimal point are significant. It is assumed that these zeros would not be shown unless they were significant. 138.200 m has six significant figures. If the value were known to only four significant figures, we would write 138.2 m. RULE 4. Zeros at the end of a number and before an implied decimal point may or may not be significant. We cannot tell whether they are part of the measurement or whether they act only to locate the unwritten but implied decimal point. Chapter Two 26 Practice Rule #1 Zeros 45.8736 6 •All digits count .000239 3 •Leading 0’s don’t .00023900 5 •Trailing 0’s do 48000. 5 •0’s count in decimal form 48000 2 •0’s don’t count w/o decimal 3.982106 4 1.00040 6 •All digits count •0’s between digits count as well as trailing in decimal form 2.5 Scientific Notation • Scientific notation is a convenient way to write a very small or a very large number. • Numbers are written as a product of a number between 1 and 10, times the number 10 raised to power. • 215 is written in scientific notation as: 215 = 2.15 xChapter 100Two = 2.15 x (10 x 10) = 2.15 x 102 28 Two examples of converting standard notation to scientific notation are shown below. Chapter Two 29 Two examples of converting scientific notation back to standard notation are shown below. Chapter Two 30 • Scientific notation is helpful for indicating how many significant figures are present in a number that has zeros at the end but to the left of a decimal point. • The distance from the Earth to the Sun is 150,000,000 km. Written in standard notation this number could have anywhere from 2 to 9 significant figures. • Scientific notation can indicate how many digits are significant. Writing 150,000,000 as 1.5 x 108 indicates 2 and writing it as 1.500 x 108 indicates 4. • Scientific notation Chapter Twocan make doing 31 arithmetic easier. Rules for doing 2.6 Rounding Off Numbers • Often when doing arithmetic on a pocket calculator, the answer is displayed with more significant figures than are really justified. • How do you decide how many digits to keep? • Simple rules exist to tell you how. Chapter Two 32 • Once you decide how many digits to retain, the rules for rounding off numbers are straightforward: • RULE 1. If the first digit you remove is 4 or less, drop it and all following digits. 2.4271 becomes 2.4 when rounded off to two significant figures because the first dropped digit (a 2) is 4 or less. • RULE 2. If the first digit removed is 5 or greater, round up by adding 1 to the last digit kept. 4.5832 is 4.6 when rounded off Chapter Two to 2 significant figures since the first 33 Practice Rule #2 Rounding Make the following into a 3 Sig Fig number 1.5587 1.56 .0037421 .00374 1367 1370 128,522 129,000 1.6683 106 1.67 106 Your Final number must be of the same value as the number you started with, 129,000 and not 129 Examples of Rounding For example you want a 4 Sig Fig number 0 is dropped, it is <5 4965.03 4965 780,582 780,600 8 is dropped, it is >5; Note you must include the 0’s 1999.5 2000. 5 is dropped it is = 5; note you need a 4 Sig Fig RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers. Chapter Two 36 •RULE 2. In carrying out an addition or subtraction, the answer cannot have more digits after the decimal point than either of the original numbers. Chapter Two 37 Multiplication and division 32.27 1.54 = 49.6958 49.7 3.68 .07925 = 46.4353312 46.4 1.750 .0342000 = 0.05985 .05985 3.2650106 4.858 = 1.586137 107 1.586 107 6.0221023 1.66110-24 = 1.000000 1.000 Addition/Subtraction 25.5 +34.270 12.5 59.770 332.5 59.8 32.72 - 0.0049 320 + 32.7151 32.72 330 Addition and Subtraction .56 __ + .153 ___ = .713 .71 __ 82000 + 5.32 = 82005.32 82000 10.0 - 9.8742 = .12580 .1 10 – 9.8742 = .12580 0 Look for the last important digit Mixed Order of Operation 8.52 + 4.1586 18.73 + 153.2 = = 8.52 + 77.89 + 153.2 = 239.61 = 239.6 (8.52 + 4.1586) (18.73 + 153.2) = = 12.68 171.9 = 2179.692 = 2180. Rule 4 • If number < 1, all zeros to left of non-zeros are not significant 0.098 0.0003 0.90020 Rule 5 • Any number that represents a numerical count or exact definition has an infinite # of sig figs 32 students in the class Sig Figs in Addition/Subtraction • Answer is rounded to same # of decimal places as the measurement with the least # of decimal places. • Ex: 13.1 + 4.25 63.408 This answer would be 80.758 rounded to 80.8 Sig Figs in Multiplication/Division • Round your answer to the measurement with the least # of sig figs. Ex: 4.3 x 2 = 8.6 = 9 18.75 ÷ 3.5 = 5.357 = 5.4 Error Error = experimental value – accepted value Percent Error Percent Error = error / accepted value x 100