Scientific Notation Often used to express very large or very small numbers. Also used to maintain correct number of significant figures. Form: (# from 1 to 9.999) x 10exponent 800 = 8 x 10 x 10 = 8 x 102 2531 = 2.531 x 10 x 10 x 10 = 2.531 x 103 0.0014 = 1.4 / 10 / 10 / 10 = 1.4 x 10-3 Change to standard form. 000000187000000 . . 1.87 x 10–5 = 0.0000187 3.7 x 108 = 370,000,000 7.88 x 101 = 78.8 2.164 x 10–2 = 0.02164 Change to scientific notation. 12,340 = 0.369 = 0.008 = 1,000,000,000 = 1.234 x 104 3.69 x 10–1 8 x 10–3 1 x 109 Using the Exponent Key on a Calculator EE EXP EE or EXP means “times 10 to the…” How How to to type type out out 6.02 6.02 xx10 102323:: 6 0 . 2 EE 2 3 Don’t do it like this… 6 WRONG! 0 . yx 2 2 3 …or like this… 6 . 0 WRONG! 2 x 1 …or like this: 6 . 0 0 EE 2 3 TOO MUCH WORK. 2 x 1 0 yx 2 3 Also, know when to hit your (–) sign… …before the number, …after the number, …or either one. 1.2 x 105 Example: 2.8 x 1013 Type this calculation in like this: 1 . 2 EE 5 2 . 8 EE 1 3 = Calculator gives… 4.2857143 –09 or… 4.2857143 E–09 This is NOT written… 4.3–9 But instead is written… 4.3 x 10–9 or 4.3 E –9 7.5 x 10-6 - 8.7 x 10-4 = -6.525 x 10-9 report -6.5 x 10-9 (2 sig. figs.) 4.35 x 106 1.23 x 10-3 = 5.3505 x 103 or 5350.5 report 5.35 x 103 (3 sig. figs.) 5.76 x 10-16 9.86 x 10-4 = 5.84178499 x 10-13 report 5.84 x 10-13 (3 sig. figs.) 8.8 x 10 3.3 x 10 11 11 = 2.904 x 1023 report 2.9 x 1023 (2 sig. figs.) 6.022 x 1023 - 5.1 x 10-8 = -3.07122 x 1016 report -3.1 x 1016 (2 sig. figs.) Scientific Notation • Scientific Notation • Converting Numbers to Scientific Notation • How to Use a Scientific Calculator Scientific Notation We often use very small and very large numbers in chemistry. Scientific notation is a method to express these numbers in a manageable fashion. Thus 0.000 000 1 cm can be written 1 x 10-7 cm. Lets see why… Scientific notation expresses a number as the product of two factors, the first falling between 1 and 10 and the second being a power of 10. Method to express really big or small numbers. Format is Mantissa Decimal part of original number 6.02 x Base Power x Decimal you moved 23 10 We just move the decimal point around. 602000000000000000000000 Scientific Notation Numbers are written in the form M x 10n, where the factor M is a number greater than or equal to 1 but less than 10 and n is a whole number. 5000 = 5 x 103 or 5 x (10 x 10 x 10) 5 x 1000 5000 5 Numbers > one have a positive exponent. Numbers < one have a negative exponent. xEE 10n 3 Converting Numbers to Scientific Notation 0.00002205 1 2 3 4 2.205 x -5 10 5 In scientific notation, a number is separated into two parts. The first part is a number between 1 and 10. The second part is a power of ten. How to Use a Scientific Calculator Divide: (5.44 x 107) .. (8.1 x 104) 07 04 671.604938 5.44 8.100 54400000. How to enter this on a calculator: 5.44 EE 7 .. 8.1 EE 4 ENTER 8.1 EXP 4 = OR 5.44 EXP 7 .. 671.6049383 rounded to 6.7 x 102 Davis, Metcalfe, Williams, Castka, Modern Chemistry, 1999, page 52 Rule for Multiplication Calculating with Numbers Written in Scientific Notation When multiplying numbers in scientific notation, multiply the first factors and add the exponents. Sample Problem: Multiply 3.2 x 10-7 by 2.1 x 105 (3.2) x (2.1) = 6.72 6.72 x 10-2 (-7) + (5) = -2 or 10-2 Exercise: Multiply 14.6 x 107 by 1.5 x 104 2.19 x 1012 Rule for Division Calculating with Numbers Written in Scientific Notation When dividing numbers in scientific notation, divide the first factor in the numerator by the first factor in the denominator. Then subtract the exponent in the denominator from the exponent in the numerator. Sample Problem: Divide 6.4 x 106 by 1.7 x 102 . (6.4) . (1.7) = 3.76 3.76 x 104 (6) - (2) = 4 or 104 Exercise: Divide 2.4 x 10-7 by 3.1 x 1014 7.74 x 10-22 Rule for Addition and Subtraction Calculating with Numbers Written in Scientific Notation In order to add or subtract numbers written in scientific notation, you must express them with the same power of 10. Sample Problem: Add 5.8 x 103 and 2.16 x 104 27.4 x 103 2.74 x 104 Exercise: Add 8.32 x 10-7 and 1.2 x 10-5 1.28 x 10-5 (5.8 x 103) + (21.6 x 103) = Using Scientific Notation for Expressing the Correct Number of Significant Figures Measurement Number of significant figures it contains 25 g 2 0.030 kg 2 1.240560 x 106 mg 7 6 x 104 sec 1 246.31 g 5 20.06 cm 4 1.050 m 4 Measurement Number of significant figures it contains 0.12 kg 2 1240560. cm 7 6000000 kg 1 6.00 x 106 kg 3 409 cm 3 29.200 cm 5 0.02500 g 4

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