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1.2 Measurement & Scientific Notation

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## 1.2 Measurement & Scientific Notation

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**Measurement**You make a measurement every time you • Measure your height. • Read your watch. • Take your temperature. • Weigh a cantaloupe.**Measurement in Science**In science and allied health we • Measure quantities. • Do experiments. • Calculate results. • Use numbers to report measurements. • Compare results to standards.**Standards of Measurement**• When we measure, we use a measuring tool to compare some dimension of an object to a standard. • Calipers are used to measure the thickness of the skin fold at the waist.**Stating a Measurement**• In every measurement, a number is followed by aunit. • Observe the following examples of measurements: number + unit 35 m 0.25 L 225 lb 3.4 hr**The Metric System (SI)**The metric system is • A decimal system based on 10. • Used in most of the world. • Used by scientists and in hospitals.**Units in the Metric System**In the metric and SI systems, a basic unit identifies each type of measurement:**Length Measurement**• In the metric system, length is measured in meters using a meter stick. • The metric unit for length is the meter (m).**Volume Measurement**• Volume is the space occupied by a substance. • The metric unit of volume is the liter (L). • The liter is slightly bigger than a quart. • A graduated cylinder is used to measure the volume of a liquid.**Mass Measurement**• The mass of an object is the quantity of material it contains. • A balance is used to measure mass. • The metric unit for mass is the gram (g).**Temperature Measurement**• The temperature of a substances indicates how hot or cold it is. • In the metric system, temperature is measured on the Celsius scale. • On this thermometer, the temperature is 19ºC or 66ºF.**Scientific Notation**• A number in scientific notation contains a coefficient and a power of 10. coefficient power of ten coefficient power of ten 1.5 x 102 7.35 x 10-4 • Place the decimal point after the first digit. Indicate the spaces moved as a power of ten. 52 000 = 5.2 x 104 0.00378 = 3.78 x 10-3 4 spaces left 3 spaces right**Measured Numbers**• You use a measuring tool to determine a quantity such as your height or the mass of an object. • The numbers you obtain are called measured numbers.**Reading a Meter Stick**. l2. . . . l . . . . l3 . . . . l . . . . l4. . cm • To measure the length of the blue line, we read the markings on the meter stick. The first digit 2 plus the second digit 2.7 • Estimatingthe third digit between 2.7–2.8 gives a final length reported as 2.75 cm or 2.76 cm**Known + Estimated Digits**• In the length measurement of 2.76 cm, • the digits 2 and 7 are certain (known). • the third digit 5(or 6) is estimated (uncertain). • all three digits (2.76) are significant including the estimated digit.**Zero as a Measured Number**. l3. . . . l . . . . l4. . . . l . . . . l5. . cm • The first and second digits are 4.5. • In this example, the line ends on a mark. • Then the estimated digit for the hundredths place is 0. • We would report this measurement as 4.50cm.**Exact Numbers**• An exact number is obtained when you count objects or use a defined relationship. Counting objects 2 soccer balls 4 pizzas Defined relationships 1 foot = 12 inches 1 meter = 100 cm • An exact number is not obtained with a measuring tool.**Significant Figures in Measurement**• The numbers reported in a measurement depend on the measuring tool. • Measurements are not exact; they have uncertainty. • The significant figures for a measurement include all of the known digits plus one estimated digit.**Counting Significant Figures**• All non-zero numbers in a measured number are significant. Measurement Number of Significant Figures 38.15 cm 4 5.6 ft 2 65.6 lb 3 122.55 m 5**Leading Zeros**• Leading zeros precede non-zero digits in a decimal number. • Leading zeros in decimal numbers are not significant. • Measurement Number of Significant Figures • 0.008 mm 1 • 0.0156 oz 3 • 0.0042 lb 2 • 0.000262 mL 3**Sandwiched Zeros**• Sandwiched zeros occur between nonzero numbers. • Sandwiched zeros are significant. • Measurement Number of Significant Figures • 50.8 mm 3 • 2001 min 4 • 0.0702 lb 3 • 0.40505 m 5**Trailing Zeros**• In numbers without decimal points, trailing zeros follow non-zero numbers. • Trailing zeros are usually place holders and not significant. • Measurement Number of Significant Figures • 25 000 cm 2 • 200 kg 1 • 48 600 mL 3 • 25 005 000 g 5**Significant Figures in Scientific Notation**• All digits including zeros that appear in the coefficient of a number written in scientific notation are significant. Scientific Notation Number of Significant Figures 8 x 104 m 1 8.0 x 104 m 2 8.00 x 104 m 3**Significant Numbers in Calculations**• A calculated answer must relate to the measured values used in the calculation. • In calculations involving addition or subtraction, the number of decimal places are counted. • In calculations involving multiplication or division, significant figures are counted to determine final answers.**Rules for Rounding Off Calculated Answers**• To obtain the correct number of significant figures, an answer may be rounded off. • When digits of 4 or less are dropped, the rest of the numbers are the same. For example, rounding 45.832 to 3 significant figures gives 45.832 rounds to 45.8 (3 SF) • When digits of 5 or greater dropped, the last retained digit is increased by 1. For example, rounding 2.4884 to 2 significant figures gives 2.4884 rounds to 2.5 (2 SF)**Adding Significant Zeros**• When a calculated answer requires more significant digits, zeros are added. Calculated answer Zeros added to give 3 significant figures 4 4.00 1.5 1.50 0.2 0.200**Adding and Subtracting**• An answer obtained by adding or subtracting has the same number of decimal places as the measurement with the fewest decimal places. • Proper rules of rounding are used to adjust the number of digits in the answer. 25.2 one decimal place + 1.34two decimal places 26.54 calculated answer 26.5 answer withone decimal place**Multiplying and Dividing**• An answer obtained by multiplying or dividing has the same number of significant figures as the measurement with the fewest significant figures. • Use rounding to limit the number of digits in the answer. 110.5 x 0.048 = 5.304 (calculator) 4 SF 2 SF • The final answer is rounded off to give 2 significant figures = 5.3(2 SF)**Prefixes**• A prefix in front of a unit increases or decreases the size of that unit. • The new units are larger or smaller that the initial unit by one or more factors of 10. • A prefix indicates a numerical value. prefix=value 1 kilometer = 1000 meters 1 kilogram = 1000 grams**Metric Equalities**• An equality states the same measurement in two different units. • Equalities are written using the relationships between two metric units. • For example, 1 meter can be expressed as 100 cm or as 1000 mm. 1 m = 100 cm 1 m = 1000 mm**Metric Equalities for Mass**• Several equalities can be written for mass in the metric system 1 kg = 1000 g 1 g = 1000 mg 1 mg = 0.001 g 1 mg = 1000 µg**Equalities**• The quantities in an equality use two different units to describe the same measured amount. • Equalities are written for relationships between units of the metric system, U.S. units or between metric and U.S. units. For example, 1 m = 1000 mm 1 lb = 16 oz 2.20 lb = 1 kg**Exact and Measured Numbers in Equalities**• Equalities written between units of the same system are definitions; they are exact numbers. • Equalities written between metric-U.S. units, which are in different systems, represent measured numbers and must be counted as significant figures.**Equalities on Food Labels**• The contents of packaged foods in the U.S. are listed as both metric and U.S. units. • The content values indicate the same amount of substance in two different units.**Conversion Factors**• A conversion factor is a fraction in which the quantities in an equality are written as the numerator and denominator. Equality: 1 in. = 2.54 cm • Each unit can be written as the numerator or denominator. Thus, two conversion factors are possible for every equality. 1 in. and 2.54 cm 2.54 cm 1 in.**Conversion Factors in a Problem**A word problem may contain information that can be used to write conversion factors. Example 1: At the store, the price of one pound of red peppers is $2.39. 1 lb red peppers$2.39 $2.39 1 lb red peppers Example 2: At the gas station, one gallon of gas is $1.34. 1 gallon of gas$1.34 $1.34 1 gallon of gas**Initial and Final Units**• To start solving a problem, it is important to identify the initial and final units. A person has a height of 2.0 meters. What is that height in inches? • The initial unit is the unit of the given height. The final unit is the unit needed for the answer. Initial unit = meters (m) Final unit = inches (in.)**Problem Setup**• In working a problem, start with the initial unit. • Write a unit plan that converts the initial unit to the final unit. Unit 1 Unit 2 • Select conversion factors that cancel the initial unit and give the final unit. Initial x Conversion = Final unit factor unit Unit 1 x Unit 2 = Unit 2 Unit 1**Setting up a Problem**How many minutes are 2.5 hours? Solution: Initial unit = 2.5 hr Final unit =? min Unit Plan= hr min Setup problem to cancel hours (hr). InitalConversion Final unit factor unit 2.5 hr x 60 min = 150 min (2 SF) 1 hr**Using Two or More Factors**• Often, two or more conversion factors are required to obtain the unit of the answer. Unit 1 Unit 2 Unit 3 • Additional conversion factors are placed in the setup to cancel the preceding unit Initial unit x factor 1 x factor 2 = Final unit Unit 1 x Unit 2 x Unit 3 = Unit 3 Unit 1 Unit 2**Example: Problem Solving**How many minutes are in 1.4 days? Initial unit: 1.4 days Unit plan: days hr min Set up problem: 1.4 days x 24 hr x 60 min = 2.0 x 103 min 1 day 1 hr 2 SF Exact Exact = 2 SF**Check the Unit Cancellation**• Be sure to check your unit cancellation in the setup. What is wrong with the following setup?1.4 day x 1 day x 1 hr 24 hr 60 min Units = day2/minis Not the final unit needed Units don’t cancel properly. The units in the conversion factors must cancel to give the correct unit for the answer.