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9/9/2015 The Problem With The Very Big and The Very Small Believe It Or Not... ...scientific notation makes things easier. It’s just a standardized way of writing very bi and big d very smallll numbers. b Unit 1: Science Is A Verb! Lesson 1.4: Scientific Notation 56 Chemistry Deals With Really Big and Really Small 57 Scientific Notation Always Looks Like This: Examples: (exponent) Coefficient x 10 There are ~ Coefficient- ALWAYS larger than or equal to 1 but smaller than 10 160,000,000,000,000,000,000,000 atoms of carbon in a typical pencil. A typical carbon atom is ~0.00000000015 meters in diameter. exponent- how many powers of 10 you would need to multiply or divide the coefficient by to get back to the unscientifically notated version of the number 58 Big Numbers 59 Big Examples: Back and Forth For numbers larger than 1, the exponent will ALWAYS be a positive number. The Earth is 4 600 000 000 years old. This number in scientific notation is: 4.6 x109 1) The coefficient is 4.6 2) After the 4, there are 9 places back to the “ones” place = exponent f h h 9 l b k h “ ” l of X 109 There are 33 400 000 000 000 000 000 molecules in one milliliter of water. This number in scientific notation is: 3.34 x1019 1) The coefficient is 3.34 2) After the first 3, there are 19 places back to the “ones” place = exponent of X 1019 60 61 1 9/9/2015 Small Numbers Small Examples: Back and Forth For numbers smaller than 1, the exponent will ALWAYS be a negative number. An atom of carbon weighs 0.000 000 000 000 000 000 000 1993 g. This number in scientific notation is: 1.993 x10‐22 1) The first integer in the coefficient is 1, followed by a decimal point, followed by the rest of the digits = 1.993 2)) There are 22 places in front of the first integer = exponent of p g p X 10‐22 The distance between H atoms in a molecule of H2 gas is 0.000 000 0370 mm. This number in scientific notation is: 3.70 x10‐8 1) The first integer in the coefficient is 3, followed by a decimal point, followed by the rest of the digits = 3.70 2) There are 8 places in front of the first integer back to the ones place = exponent of X 10‐8 62 63 But Wait! That’s Not All! Practice Exponents let us do some interesting things with math: 1. Circle the larger value in the pairs below: a) 2.0 x 105 or 3.0 x 104 b) 5.7 x 10 5 7 x 1012 or 1.3 x 10 or 1 3 x 107 c) 3.3 x 10‐6 or 9.7 x 10‐2 d) 6.5 x 10‐2 or 2.8 x 100 Multiplying p y g exponents p = Add them! Dividing exponents = Subtract them! 64 Multiplication and Division (always, always, always) Let’s try two! Multiply 1.25 x 1016 by 4.00 x 1030 1. Do the math that involves your coefficients first • 65 This includes significant figure rounding, of course 2. Then deal with the exponential calculations Divide 6.25 x 104 by 1.25 x 1010 66 67 2 9/9/2015 Addition and Subtraction (always, always always) Let’s Try Two More! Add 4.02 x 1010 + 9.23 x 109 1. Adjust the exponents first. 2. Then do the math that involves the (adjusted) coefficients. • This includes significant figure rounding, of course (more on this in our next lesson). Subtract 1.4 x 105 from 8.2 x 104 68 69 Scientific Notation and your Calculator Calculator Tip Most calculators have a short‐cut to deal with Powers of 10 exponents. This is usually signified with a button that says “E” or “EE”. If you’re feeling saucy, you can write scientific notation this way in general: Ex. 1.356 x 1037 = 1.356 E37 The scientific notation key 70 Scientific Notation and your Calculator 3