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Measurement Systems Why do we need a measurement system? Scientific Notation A way to write very large and very small numbers. A number in scientific notation is written in two parts, the coefficient and an exponent of 10. coefficient 5 x 1022 exponent of 10 Scientific Notation Changing standard numbers to scientific notation 1. Numbers greater than 10 a. Move decimal until only ONE number is to the left of the decimal. b. The exponent is the number of places the decimal has moved and it is POSITIVE. Ex. 125 = 1.25 102 15,000,000,000 = 1.5 1010 Scientific Notation Changing standard numbers to scientific notation 2. Numbers less than 1 a. Move decimal until only one number is to the left of the decimal. b. The exponent is the number of places the decimal has moved and it is NEGATIVE. Ex. 0.000189 = 1.89 10-4 0.5476 = 5.476 10-1 Scientific Notation Changing standard numbers to scientific notation 3. To change a number written in incorrect scientific notation: a. Move the decimal until only one number is to the left of the decimal. b. Correct the exponent. (remember: take away, add back) coefficient decreased by 2, so Ex. 504.2 106 = 5.042 108 The the exponent must increase by 2 0.0089 10-2 = 8.9 10-5 The coefficient increased by 3, so the exponent must decrease by 3 Scientific Notation Changing numbers in scientific notation to standard notation 1. 2. If the exponent is (+) move the decimal to the right the same number of places as the exponent. a. 1.65 101 = 16.5 b. 1.65 103 = 1650 If the exponent is (-) move the decimal to the left the same number of places as the exponent. a. 4.6 10-2 = 0.046 b. 1.23 10-3 = 0.00123 Scientific Notation Multiplication and Division in Scientific Notation 1. To multiply numbers in scientific notation a. Multiply the coefficients. b. Add the exponents. c. Convert the answer to correct scientific notation. Ex: (2 109) x (4 103) = 8 x 1012 Scientific Notation Multiplication and Division in Scientific Notation 2. To divide numbers in scientific notation a. Divide the coefficients. b. Subtract the exponents. c. Convert the answer to correct scientific notation. Ex: (8.4 106) (2.1 102) = 4 x 104 Scientific Notation Addition and Subtraction in Scientific Notation 1. Before numbers can be added or subtracted, the exponents must be equal. Ex. (5.4 103) + (6.0 102) = (5.4 103) + (0.6 103) = 6.0 103 Significant Figures Are all the numbers for which actual measurements are made plus one estimated number. 1 2 You would estimate this measurement as 1.5 1 2 You would estimate this measurement as 1.48 Significant Figures Tells the person interpreting your data about the accuracy of the measuring instrument used to obtain the data. Significant Figures Rules for counting sig figs 1. Digits other than zero are always significant. a. 96 = 2 sig figs b. 61.4 = 3 sig figs 2. Zeroes between 2 other sig figs are always significant. a. 5.029 = 4 sig figs b. 306 = 3 sig figs Significant Figures Rules for counting sig figs 3. Leading zeroes are never significant when they are to the left of non-zero numbers. a. 0.0025 = 2 sig figs b. 0.0821 = 3 sig figs 4. Trailing zeroes are only significant if there is a decimal present and they are to the right of nonzero numbers. a. 100 = 1 sig fig b. 100.0 = 4 sig figs c. 0.0820 = 3 sig figs Significant Figures Rules for calculating with sig figs 1. In addition and subtraction, the answer should be rounded off so that it has the same number of decimal places as the quantity having the least number of decimal places. a. 1.1 + 225 = 226.1 = 226 (rounded to no decimal places) b. 2.65 – 1.4 = 1.25 = 1.3 (rounded to 1 decimal place) 2. In multiplication and division, the answer should have the same number of significant figures as the given data value with the least number of significant figures. a. 4.60 45 = 207 = 210 (rounded to 2 sig figs) b. 1.956 3.3 = 0.5927 = 0.59 (rounded to 2 sig figs) Metric System Unit Unit Unit Unit Unit of of of of of length…..meter (m) mass ……gram (g) volume …liter (L) time …….second (s) temperature…degrees Celsius (°C) Metric System The metric system is based on units of 10. Prefix symbol Prefix name Prefix value Fraction or Multiple Power G giga one billion 1,000,000,000 109 M mega one million 1,000,000 106 k kilo one thousand 1000 103 1 10 BASIC UNIT: m, g, L, d deci 1/10 0.1 10-1 c centi 1/100 0.01 10-2 m milli 1/1000 0.001 10-3 µ micro 1/1,000,000 0.000 001 10-6 n nano 1/1,000,000,000 0.000 000 001 10-9 Metric System To convert measurements within the metric system is a simple matter of multiplying or dividing by 10, 100, 1000, etc. Even simpler, it is a matter of moving the decimal point to the left or right. Metric System One way to know where to place the decimal is to draw a "metric line" with the basic unit in the center, marking off six units to the left and six units to the right. To convert from one unit to another simply count the number of places to the left or right, and move the decimal in that direction that many places. Ex. 3 L = 0.003 kL Ex. 3 mg = 3000 µg Two Systems English yard, mile, feet pound, ounce quart, gallon Fahrenheit Metric Meter Gram Liter Celsius Factor-Label T h e m o s t i m p o r t a n t mathematical process for scientists. T r e a t s n u m b e r s a n d units equally. M u l t i p l y w h a t i s g i v e n by fractions equal to one to convert units. Factor-Label What is given A fraction equal to one Factor-Label How many basketballs can be carried by 8 buses? 1 bus = 12 cars 3 cars = 1 truck 1000 basketballs = 1 truck Factor-Label How many basketballs can be carried by 8 buses? 8 buses 1 bus = 12 cars 3 cars = 1 truck 1000 basketballs = 1 truck Factor-Label How many basketballs can be carried by 8 buses? 8 buses 12 cars 1 bus 1 bus = 12 cars 3 cars = 1 truck 1000 basketballs = 1 truck Factor-Label How many basketballs can be carried by 8 buses? 8 buses 12 cars 1 truck 1 bus 3 cars Factor-Label How many basketballs can be carried by 8 buses? 8 buses 12 cars 1 truck 1000 bballs 1 bus 3 cars 1 truck Factor-Label 32000 basketballs can be carried by 8 buses. Factor-Label Convert 5 pounds to kilograms. Factor-Label Convert 5 pounds to kilograms 5 lb 1 kg 2 . 2 0 lb = 2.27 kg Factor-Label Convert 8.3 centimeters to millimeters. Factor-Label Convert 8.3 centimeters to millimeters 8.3 cm 1m 1000 mm 100 cm 1m = 83 mm Factor-Label Method