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Practical Estimation Using Scientific Notation Created by Dick Heckathorn Can you quickly approximate an answer to the following problem? How many seconds are there in one year? 365 days 24 hours 3600 sec 1 year x x x 1 year 1 day 1 hour Can you quickly determine which is larger? 3684 x 27.36 x 416.3 ? or 63.72 x 273 x 4175 ? Can you quickly approximate an answer to the following problem? 3746 x 21.4 x 81.74 0.0975 A review of place holders 57,246.9318 5 7 2 4 6 ten-thousands thousands hundreds tens ones 9 3 1 8 tenths hundredths thousandths ten-thousandths A review of what happens to a number when it is multiplied by ten (10). 2 2 x 10 = 20 2 x 10 x10 = 200 2 x 10 x 10 x 10 = 2000 2 / 10 = 0.2 2 / 10 / 10 = 0.02 2 / 10 / 10 / 10 = 0.002 A review of numbers that are different by a power of 10. Number How number is formed 1 10 100 1000 10000 100000 1000000 = = = = = = = 1 1 x 10 1 x 10 x 10 1 x 10 x 10 x 10 1 x 10 x 10 x 10 x 10 1 x 10 x 10 x 10 x 10 x 10 1 x 10 x 10 x 10 x 10 x 10 x 10 A review of how numbers are changed to power of ten format. Power of 10 Number How number is formed 1 10 100 1,000 10,000 100,000 1,000,000 1 x no tens 1 x 10 1 x 10 x 10 1 x 10 x 10 x 10 1 x 10 x 10 x 10 x 10 1 x 10 x 10 x 10 x 10 x 10 1 x 10 x 10 x 10 x 10 x 10 x 10 100 101 102 103 104 105 106 A review of how numbers are changed to power of ten format. Power of 10 Number How number is formed 100 10-1 10-2 10-3 10-4 10-5 10-6 1 1 / 10 1 / 10 / 10 1 / 10 / 10 / 10 1 / 10 / 10 / 10 / 10 1 / 10 / 10 / 10 / 10 / 10 1 / 10 / 10 / 10 / 10 / 10 x 10 1 0.1 0.01 0.001 0.0001 0.00001 0.000001 Scientific Notation Is a system in which the numbers are expressed as the product of the COEFFICIENT which is a number that is equal to or greater than one but less than ten and the appropriate POWER OF TEN Here is an example. The Number 63847 63847 Same number in scientific notation 6.3847 x 104 Coefficient Power of Ten Let us now examine the steps to change a number to scientific notation format. Number to be Changed 76348 76348 First move the decimal point so that one non-zero digit is to the left of the decimal point. 7.6348 Next determine what you did with the decimal point. 76348 7.6348 It was moved 4 places to the left. How does this number compare to the original number? 76348 7.6348 The number is 10,000 (104) times smaller. 76348 7.6348 To make the number in scientific notation the same as the original number, what must we do to the powers of ten? 76348 7.6348 We must multiply the number by ten to the fourth power (4). 7.6348 x 104 Number to be Changed 0.000385 0.0003857 First move the decimal point so that one non-zero digit is to the left of the decimal point. 3.857 Next determine what you did with the decimal point. 0.0003857 3.857 It was moved 4 places to the right. How does this number compare to the original number? 0.0003857 3.857 The number is 10,000 (104) times larger. 0.0003857 3.857 To make the number in scientific notation the same as the original number, what must we do to the powers of ten? 0.0003857 3.857 We thus multiply 7.6348 by 10 to the minus four power (10-4). 3.857 x 10-4 Change the following number to scientific notation format. 724000 Check your answer on the next slide when finished. 724000 The answer is 7.24 x 104 Change the following number to scientific notation format. 0.0000517 Check your answer on the next slide when finished. 0.0000517 The answer is 5.17 x 10-5 Let us now solve a problem where the numbers are written in scientific notation. The Problem (3.1 x 102) x (2.0 x 104) (3.1 x 2 10 ) x (2.0 x 4 10 ) First separate the coefficients & powers of ten. (3.1 x 2.0) x (102 x 104) (3.1 x 2 10 ) x (2.0 x 4 10 ) (3.1 x 2.0) x (102 x 104) Next show how the powers of ten are combined. (3.1 x 2.0) x (2 + 4) 10 (3.1 x 2 10 ) x (2.0 x 4 10 ) (3.1 x 2.0) x (102 x 104) (3.1 x 2.0) x 10(2 + 4) Finally finish the calculation. 6.2 x 6 10 Let us solve another problem where the numbers are written in scientific notation. The Problem 3 6.4 x 10 -5 3.2 x 10 3 6.4 x 10 -5 3.2 x 10 First separate the coefficients & powers of ten. 3 6.4 10 x -5 3.2 10 3 6.4 x 10 -5 3.2 x 10 3 6.4 10 x -5 3.2 10 Next show how the powers of ten are combined. 6.4 3-(-5) x 10 3.2 3 6.4 x 10 -5 3.2 x 10 3 6.4 10 x -5 3.2 10 6.4 3-(-5) x 10 3.2 Finally finish the calculation. 2.0 x 10 8 Let us now solve a problem by first changing the number to scientific notation format and then determine the answer. The Problem 6749 x 263 6749 x 263 First write the number in scientific notation. (6.749 x 103) x (2.63 x 102) 6749 x 263 (6.749 x 103) x (2.63 x 102) Next combine the coefficients and powers of ten. (6.749 x 2.63 ) x 103 + 2 6749 x 263 (6.749 x 103) x (2.63 x 102) (6.749 x 2.63 ) x 103 + 2 Next combine the coefficients and the powers of ten. 17.75 x 105 6749 x 263 (6.749 x 2.63 ) x 103 + 2 17.75 x 105 Finally write in proper scientific notation. 1.775 x 106 Set up the following problem on a piece of paper showing each step. 0.00463 x 6734 x 27 When finished check your answer on the next slide. The answer is: 8.42 x 102 Fermi Questions By Dick Heckahtorn To solve Fermi questions such as how may blades of grass are there inside the fence that surrounds the soccer field, one needs to learn how to round a number to its nearest order of magnitude. For example: 6.35 x 105 In the desired format is: 106 For example: 2.35 x 105 In the desired format is: 105 But what about 4.35 x 5? 10 4.35 x 5? 10 The answer is: 6 10 How many of you said 105 ? How can this be? Since we are dealing only with orders of magnitude, we need to determine the half-way point between one order of magnitude and the next. For example between 101 and 102. Would you agree that the half way point between 1 2 1.5 10 and 10 is 10 ? 1.5 10 is I hope you agree because the midpoint between 1 2 10 and 10 ? Our task then is quite simple. What is the value of 101.5 ? Did you say 50? If you did, you have responded with an answer that most students give. But 50 is an incorrect answer. You see, we are not dealing with an ordinary number scale. We are dealing with an exponential scale. This is true because each order of magnitude is 10 times larger than the previous. Without going into a lot of mathematical explanation, perform the following steps on your calculator. 1. Press the ‘2nd’ key 2. Press the ‘LOG’ key 3. Enter 1.5 4. Press the ‘ENTER’ key The answer is …….. Did you get 31.6227766? You should if you have the calculator set on float. Otherwise you will get a rounded value. An Example Examine the following: 100 - order of magnitude = 0 101 - order of magnitude = 1 105 - order of magnitude = 5 10-1 - order of magnitude = -1 -3 10 - order of magnitude = -3 -5 10 - order of magnitude = -5