VIEWS: 119 PAGES: 28 CATEGORY: Technology POSTED ON: 9/6/2010
Scientific Notation, Exponents and Significant Figures Scientific Notation Scientific Notation – It is notation used to express very large or very small numbers using powers of 10. It is written as a number multiplied by 10x Example: 1000 = 1 x 103 (10 x 10 x 10) Here the 1000 is in standard notation 1 x 103 is scientific notation Scientific Notation How do we express these terms? For Scientific Notation: The number is expressed as >1 and <10 and then multiplied by a power of 10. Example: 523,000 = 5.23 x 105 Scientific Notation Prefix Symbol Standard notation Exponent giga- G 1 000 000 000 109 mega- M 1 000 000 106 kilo- k 1000 103 deci- d 0.1 10-1 centi- c 0.01 10-2 milli- m 0.001 10-3 micro- µ 0.000 001 10-6 nano- n 0.000 000 001 10-9 pico- p 0.000 000 000 001 10-12 Scientific Notation So, how do we change from standard notation to scientific notation? Move the decimal point to create number that is between 1 and 10 Example: 7,231,967 = 7.231967 x 106 0.00003433 = 3.433 x 10-5 Scientific Notation Rules: The decimal place should end up at to the right of the first nonzero digit. The total number of spaces moved becomes the exponent of 10 in the scientific notation. If the given number is greater than 1, the exponent is positive. If the given number is less than 1 (but >0), the exponent is negative. Scientific Notation Practice writing these in scientific notation. 17 mL 153 kg 24883.5 km 2000 miles 0.4502 g 0.00063401 m Scientific Notation You can also use this information to write a number in standard notation. Example: 2.3445 x 103 g= 2344.5 g 2.21 x 10-7 m = 0.000000221 m Scientific Notation Practice: Write the follow ing in standard notation: 6.423 x 103 g 6423 g 1.002 x 10-6 m 0.000001002 m 5.0023 x 1010 m 50,023,000,000 m 3.3 x 10-9 sec 0.0000000033 sec Rules of Exponents Remember that exponents, especially with powers of 10, help count zeros. It is easier to see keep track of zeros with and exponent like 106 than with the standard notation of 1,000,000. Using the rules of exponents, you can multiply and divide exponents easily. Exponents Rules of Exponents Example (10m)(10n) = 10m+n (102)(103 100,000 100*1000)== 105 (10m)n = 10m*n (103)2 = 106 (1000)2 = 1,000,000 10m/10n = 10m-n 106/102 = 10410,000 1,000,000 = 100 10-m = 1/10m 10-8 = 1/108 1 = 0.00000001 100,000,000 100 = 1 = 1 x 10-8 Scientific Notation & Exponents Prefix Symbol Standard notation Exponent giga- G 1 000 000 000 109 mega- M 1 000 000 106 kilo- k 1000 103 deci- d 0.1 10-1 centi- c 0.01 10-2 milli- m 0.001 10-3 micro- µ 0.000 001 10-6 nano- n 0.000 000 001 10-9 pico- p 0.000 000 000 001 10-12 Scientific Notation & Exponents Practice: Convert numbers or exponents to prefix. 1000 g 1 x 103 g or 1 kg 5.3 x 103 m 5.3 km 4.5 x 10-6 m 4.5 µm 1.7 x 10-3g 1.7 mg 22000 seconds 22 kiloseconds Scientific Notation & Exponents 2.4 mg 2.4 x 10-3 g 2 km 2 x 103 m 1.6 Mm (megameter) 1.6 x 106 m 15 msec (milliseconds) (1.5 x101) x 10-3 sec or 1.5 x 10-2 sec 253 km (2.53 x 102) x 103 m Or 2.53 x 105 m Exponents How many milligrams are in a kilogram? 1 kg = 1000 g = 103 g x 1 mg = 10-3g = 106 mg Exponents How many picograms in a microgram? 1 µg = (1 x 10-6 g)(1 pg )= 1x10-12 g 1/(1 x 10-6) pg = 106 pg = 1,000,000 pg Significant Figures With scientific measurements, you want to know accuracy, precision and certainty. Accuracy – How close a measurement is to an accepted value Precision – How close a measurement is to other measurements of the same thing. Certainty – Degree of confidence of a measurement. The last digit to the right is usually an uncertain digit. Significant Figures So for any measured value, we’ll record all of the certain digits plus an uncertain digit. All together, they are the significant figures of the measurement. Significant Figure RULES 1. All non zero digits (1,2,3,4,5,6,7,8, and 9) are significant. 2. Final zeros to the right of the decimal point are significant. 3. Zeros between two significant digits are significant. 4. Zeros used for spacing the decimal point are not significant. 5. For numbers in scientific notation, all of the digits before the “x 10x” are significant. How Many Significant Figures? Measurement # of Sig Figs 135.3 4 sig figs 4.6025 5 sig figs 200,035 6 sig figs 0.0000300 3 sig figs 2.0000300 8 sig figs 0.002 1 sig fig 4.44 x 103 3 sig figs 2.0 x 10-2 2 sig figs 10.00 4 sig figs 10 1 sig fig 102,000 3 sig figs Significant Figures Multiplying and Dividing with Sig Figs When multiplying or dividing measurements, the answer must have the same number of sig figs as the measurement with the fewest sig figs. Example: 22 feet x 9 feet = 198 square feet…but Since 9 feet only has 1 sig fig the correct answer is 200 ft2 Calculate the Area in square blocks 3 2 1 0 1 2 3 4 5 Significant Figures Calculation Calc’d Answer Ans w/sig figs 2.86 m x 1.824 m 5.21664 m2 5.22 m2 460 miles/ 8 hours 57.5 mi/hr 60 mi/hr 98.50 in x 1.82 in 179.27 in2 179 in2 Significant Figures Calculation Calc’d Answer Ans w/sig figs 2.100 m x 0.0030 m 0.0063 m2 0.0063 m2 10.00 g / 5.000 L 2 g/L 2.000 g/L 4.610 ft x 1.7 ft 7.837 ft2 7.8 ft2 Significant Figures Defined numbers – part of a definition and is not measured. So, defined numbers (unit conversion factors) do not limit the sig figs in an answer. Also, counting numbers do not limit sig figs. Example: You cut a 24 ft piece of wood into 4 pieces. Each is 24 ft/4 = 6.0 ft/piece. Significant Figures Addition and Subtraction The sig figs with addition and subtraction are handled differently than with x and /. The answer cannot have more certainty than the least certain measurement. This means the answer must have the same number of sig figs to the right of the decimal as the measurement with the fewest sig figs to the right of the decimal place. Significant Figures Example: 4.271 g (3 sig figs to right of decimal) 2 g (0 sig figs to right of decimal) 10.0 g (1 sig fig to right of decimal) 16.271 g is calculated answer but… since 2 g has no sig figs to right of decimal the final answer is 16 g.