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Significant Figures and Scientific Notation

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Significant Figures ► When using our calculators we must determine the correct answer; our calculators are mindless drones and don’t know the correct answer. ► There are 2 different types of numbers –Exact –Measured ► Measured number = they are measured with a measuring device so these numbers have ERROR. ► When you use your calculator your answer can only be as accurate as your worst measurement Chapter Two 2

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3 Exact Numbers An exact number is obtained when you count objects or use a defined relationship. Counting objects are always exact 2 soccer balls 4 pizzas Exact relationships, predefined values, not measured 1 foot = 12 inches 1 meter = 100 cm For instance is 1 foot = 12.000000000001 inches? No 1 ft is EXACTLY 12 inches.

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4 Learning Check 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.

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5 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. Solution

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Measurement and Significant Figures ► Every experimental measurement has a degree of uncertainty. ► The volume 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 6

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7 What is the Length? ► We can see the markings between 1.6-1.7cm ► We must guess between.6 &.7 ► We record 1.67 cm as our measurement

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Learning Check What is the length of the wooden stick? A. 4.5 cm B. 4.54 cm C. 4.547 cm

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Chapter Two 9 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.

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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. 10 Chapter Two

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► 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 11

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Practice 45.8736.000239.00023900 48000. 48000 3.982 10 6 1.00040 6 3 5 5 2 4 6 All digits count Leading 0’s don’t Trailing 0’s do 0’s count in decimal form 0’s don’t count w/o decimal All digits count 0’s between digits count as well as trailing in decimal form

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Examples of Rounding For example you want a 4 Sig Fig number 4965.03 780,582 1999.5 0 is dropped, it is <5 8 is dropped, it is >5; Note you must include the 0’s 5 is dropped it is = 5; note you need a 4 Sig Fig 4965 780,600 2000.

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Practice Rule #2 Rounding Make the following into a 3 Sig Fig number 1.5587.0037421 1367 128,522 1.6683 10 6 1.56.00374 1370 129,000 1.67 10 6 Your Final number must be of the same value as the number you started with, 129,000 and not 129

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RULE 1. RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers. Chapter Two 15

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Chapter Two Multiplication and division 32.27 1.54 = 49.6958 3.68 .07925 = 46.4353312 1.750 .0342000 = 0.05985 3.2650 10 6 4.858 = 1.586137 10 7 6.022 10 23 1.661 10 -24 = 1.000000 49.7 46.4.05985 1.586 10 7 1.000

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► RULE 2. In carrying out an addition or subtraction, the answer cannot have more significant digits BEFORE or AFTER the DECIMAL point than either of the original numbers. Chapter Two 17

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Addition/Subtraction 25.5 32.72 320 +34.270 ‑ 0.0049 + 12.5 59.770 32.7151 332.5 59.8 32.72 330

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__ ___ __ Addition and Subtraction.56 +.153 =.713 82000 + 5.32 = 82005.32 10.0 - 9.8742 =.12580 10 – 9.8742 =.12580.71 82000.1 0 Look for the last important digit

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Chapter Two Mixed Order of Operation 8.52 + 4.1586 18.73 + 153.2 = (8.52 + 4.1586) (18.73 + 153.2) = 239.6 2180. = 8.52 + 77.89 + 153.2 = 239.61 = = 12.68 171.9 = 2179.692 =

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How wide is our universe? 210,000,000,000,000,000,000,000 miles (22 zeros) This number is written in decimal notation. When numbers get this large, it is easier to write them in scientific notation.

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Scientific Notation A number is expressed in scientific notation when it is in the form a x 10 n where a is between 1 and 10 and n is an integer

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Write the width of the universe in scientific notation. 210,000,000,000,000,000,000,000 miles Where is the decimal point now? After the last zero. Where would you put the decimal to make this number be between 1 and 10? Between the 2 and the 1

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2. 10,000,000,000,000,000,000,000. How many decimal places did you move the decimal? 23 When the original number is more than 1, the exponent is positive. The answer in scientific notation is 2.1 x 10 23

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Express 0.0000000902 in scientific notation. Where would the decimal go to make the number be between 1 and 10? 9.02 The decimal was moved how many places? 8 When the original number is less than 1, the exponent is negative. 9.02 x 10 -8

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Write 28750.9 in scientific notation. A. 2.87509 x 10 -5 B. 2.87509 x 10 -4 C. 2.87509 x 10 4 D. 2.87509 x 10 5

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Express 1.8 x 10 -4 in decimal notation. 0.00018 Express 4.58 x 10 6 in decimal notation. 4,580,000 On the calculator, scientific notation is done with the button. 4.58 x 10 6 is typed 4.58 6

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Use a calculator to evaluate: 4.5 x 10 -5 1.6 x 10 -2 Type 4.5 -5 1.6 -2 You must include parentheses if you don’t use those buttons!! (4.5 x 10 -5) (1.6 x 10 -2) 0.0028125 Write in scientific notation. 2.8 x 10 -3

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Use a calculator to evaluate: 7.2 x 10 -9 1.2 x 10 2 On the calculator, the answer is: 6.E -11 The answer in scientific notation is 6.0 x 10 -11 The answer in decimal notation is 0.000000000060

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Write (2.8 x 10 3 )(5.1 x 10 -7 ) in scientific notation. A. 14.28 x 10 -4 B. 1.4 x 10 -3 C. 14.28 x 10 10 D. 1.428 x 10 -3

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Write in PROPER scientific notation. (Notice the number is not between 1 and 10) 234.6 x 10 9 2.346 x 10 11 0.0642 x 10 4 6.42 x 10 2

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Write 531.42 x 10 5 in scientific notation. A..53142 x 10 2 B.5.3142 x 10 3 C. 53.142 x 10 4 D. 531.42 x 10 5 E. 53.142 x 10 6 F. 5.3142 x 10 7 G..53142 x 10 8

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