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Chapter 2 Measurements Cartoon courtesy of Lab-initio.com

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2.1 Units of Measurement Learning Goal Write the names and abbreviations for the metric or SI units used in measurements of length, volume, mass, temperature, and time

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**Chapter 2 Readiness Key Math Skills Identifying Place Values (1.4A)**

© 2014 Pearson Education, Inc.

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**The International System of Units (SI)**

Chemists use the metric system and the International System of Units (SI) for measurement when they measure quantities do experiments solve problems

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**Units of Measurement and Their Abbreviations**

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Metric System Prefixes convert the base units into units that are appropriate for the item being measured. © 2012 Pearson Education, Inc.

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Length Length is measured in units of meters (m) in both the metric and SI systems units of centimeters (cm) by chemists

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Length Useful relationships between units of length include: 1 m = yd 1 m = in. 1 m = 100 cm 2.54 cm = 1 in.

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Volume The most commonly used metric units for volume are the liter (L) and the milliliter (mL). A liter is a cube 1 decimeter (dm) long on each side. A milliliter is a cube 1 centimeter (cm) long on each side. © 2012 Pearson Education, Inc.

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Volume Volume, the space occupied by a substance, is measured using units of m3 in the SI system is commonly measured in liters (L) and milliliters (mL) by chemists

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Volume Useful relationships between units of volume include: 1 m3 = 1000 L 1 L = 1000 mL 1 mL = 1 cm3 1 L = qt 946.3 mL = 1 qt

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Mass The mass of an object, a measure of the quantity of material it contains, is measured on an electronic balance has the SI unit of kilogram (kg) is often measured by chemists in grams (g)

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Mass Useful relationships between units of mass include: 1 kg = 1000 g 1 kg = lb 453.6 g = 1 lb

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Temperature By definition temperature is a measure of the average kinetic energy of the particles in a sample. © 2012 Pearson Education, Inc.

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Temperature In scientific measurements, the Celsius and Kelvin scales are most often used. The Celsius scale is based on the properties of water. 0 C is the freezing point of water. 100 C is the boiling point of water. © 2012 Pearson Education, Inc.

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**Temperature The kelvin is the SI unit of temperature.**

It is based on the properties of gases. There are no negative Kelvin temperatures. K = C © 2012 Pearson Education, Inc.

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Temperature The Fahrenheit scale is not used in scientific measurements. F = 9/5(C) + 32 C = 5/9(F − 32) © 2012 Pearson Education, Inc.

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Time Time is based on an atomic clock and is measured in units of seconds (s) in both the metric and SI systems.

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Learning Check For each of the following, indicate whether the unit describes (1) length, (2) mass, or (3) volume A. A bag of onions has a mass of 2.6 kg. B. A person is 2.0 m tall. C. A medication contains 0.50 g of aspirin. D. A bottle contains 1.5 L of water.

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Solution For each of the following, indicate whether the unit describes (1) length, (2) mass, or (3) volume A. A bag of onions has a mass of 2.6 kg (2) B. A person is 2.0 m tall (1) C. A medication contains 0.50 g of aspirin (2) D. A bottle contains 1.5 L of water (3)

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Learning Check Identify the measurement that has an SI unit. A. John’s height is _____. (1) 1.5 yd (2) 6 ft (3) 2.1 m B. The mass of a lemon is _____. (1) 12 oz (2) kg (3) 0.6 lb C. The temperature is _____. (1) 85 ºC (2) 255 K (3) 45 ºF

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Solution Identify the measurement that has an SI unit. A. John’s height is______. (3) 2.1 m B. The mass of a lemon is _____. (2) kg C. The temperature is _____. (2) 255 K

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Density Learning Goal Calculate the density of a substance; use the density to calculate the mass or volume of a substance.

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**d = m V Derived Units Density is a physical property of a substance.**

It has units (g/mL, for example) that are derived from the units for mass and volume. d = m V © 2012 Pearson Education, Inc.

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**Density Substances that have**

higher densities contain particles that are closely packed together lower densities contain particles that are farther apart Metals such as gold and lead have higher densities because their atoms are packed closely together. © 2014 Pearson Education, Inc.

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Density, Units In the metric system, densities of solids, liquids, and gases are expressed with different units. The density of a solid or liquid is usually given in grams per cubic centimeter (g/cm3) grams per milliliter (g/mL) The density of a gas is usually given in grams per liter (g/L). © 2014 Pearson Education, Inc.

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**Density of Common Substances**

© 2014 Pearson Education, Inc.

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**Guide to Calculating Density**

© 2014 Pearson Education, Inc.

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Learning Check Osmium is a very dense metal. What is its density, in g/cm3, if 50.0 g of osmium has a volume of 2.22 cm3? © 2014 Pearson Education, Inc.

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Solution Osmium is a very dense metal. What is its density, in g/cm3, if 50.0 g of osmium has a volume of 2.22 cm3? Step 1 Given g; cm3 Need density, in g/cm3 Step 2 Plan Write the density expression. © 2014 Pearson Education, Inc.

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Solution Osmium is a very dense metal. What is its density, in g/cm3, if 50.0 g of osmium has a volume of 2.22 cm3? Step 3 Express mass in grams and volume in cm3. mass = 50.0 g volume = 22.2 cm3 Step 4 Set up problem, calculate. © 2014 Pearson Education, Inc.

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**Link to the Environment**

The oil pumped out of the ground is called crude oil, or petroleum. It is mostly made of hydrocarbons, or compounds that contain only carbon and hydrogen. In April 2010, an oil rig in the Gulf of Mexico exploded, causing the largest oil spill in U.S. history. It leaked a maximum of 10 million liters a day into the Gulf of Mexico. © 2014 Pearson Education, Inc.

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**Link to the Environment**

Because the density of this oil, 0.8 g/mL, was less than that of water, which is 1.00 g/mL, it floated on the surface, spreading a thin layer of oil over a very large surface. © 2014 Pearson Education, Inc.

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Density of Solids The density of a solid is calculated from its mass and volume. When a solid is submerged, it displaces a volume of water equal to the volume of the solid. © 2014 Pearson Education, Inc.

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Density of Solids In the figure below, the water level rises from 35.5 mL to 45.0 mL after the zinc object is added. © 2014 Pearson Education, Inc.

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Learning Check Scuba divers use lead weights to counteract their buoyancy in water. What is the density of a lead weight that has a mass of 226 g and displaces 20.0 cm3 of water when submerged? © 2014 Pearson Education, Inc.

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Solution What is the density of a lead weight that has a mass of 226 g and displaces 20.0 cm3 of water when submerged? Step 1 Given 226 g lead 20.0 cm3 water displaced Need density (g/cm3) of lead Step 2 Plan. Write the density expression. © 2014 Pearson Education, Inc.

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Solution What is the density of a lead weight that has a mass of 226 g and displaces 20.0 cm3 of water when submerged? Step 3 Express mass in grams and volume in cm3. mass of lead weight = 226 g volume of water displaced = 20.0 cm3 © 2014 Pearson Education, Inc.

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Solution What is the density of a lead weight that has a mass of 226 g and displaces 20.0 cm3 of water when submerged? Step 4 Set up problem, calculate. © 2014 Pearson Education, Inc.

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**Density as a Conversion Factor**

Density can be used as a conversion factor between a substance’s mass and volume. © 2014 Pearson Education, Inc.

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Learning Check If the density of milk is 1.04 g/mL, how many grams of milk are in 0.50 qt? © 2014 Pearson Education, Inc.

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Solution If the density of milk is 1.04 g/mL, how many grams of milk are in 0.50 qt? Step 1 Given qt milk density of milk is 1.04 g/mL Need grams of milk Step 2 Write a plan. © 2014 Pearson Education, Inc.

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Solution If the density of milk is 1.04 g/mL, how many grams of milk are in 0.50 qt? Step 3 Write equalities, conversion factors. 1 L = qt L = 1000 mL 1.04 g = 1 mL © 2014 Pearson Education, Inc.

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Solution If the density of milk is 1.04 g/mL, how many grams of milk are in 0.50 qt? Step 4 Set up problem, calculate. © 2014 Pearson Education, Inc.

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Concept Map © 2014 Pearson Education, Inc.

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**Learning Goal Write a number in scientific notation.**

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Scientific Notation Scientific notation is used to write very large or very small numbers such as the width of a human hair, m, which is also written as 8 × 10−6 m the number of hairs on a human scalp, , which is also written as 1 × 105 hairs © 2014 Pearson Education, Inc.

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**Writing Numbers in Scientific Notation**

A number written in scientific notation contains a coefficient and a power of ten. coefficient power unit of ten × m The coefficient is at least 1 but less than 10. © 2014 Pearson Education, Inc.

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**Scientific Notation In science, we deal with some very LARGE numbers:**

1 mole = In science, we deal with some very SMALL numbers: Mass of an electron = kg

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**Imagine the difficulty of calculating the mass of 1 mole of electrons!**

kg x ???????????????????????????????????

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. 9 8 7 6 5 4 3 2 1 Step #1: Insert an understood decimal point Step #2: Decide where the decimal must end up so that one number is to its left Step #3: Count how many places you bounce the decimal point Step #4: Re-write in the form 2.5 x 10n

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2.5 x 109 The exponent is the number of places we moved the decimal.

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**0.0000579 1 2 3 4 5 Step #2: Decide where the decimal must end**

up so that one number is to its left Step #3: Count how many places you bounce the decimal point Step #4: Re-write in the form 5.79 x 10n

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5.79 x 10-5 The exponent is negative because the number we started with was less than 1.

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**Writing Numbers in Scientific Notation**

The number of spaces moved to obtain a coefficient between 1 and 10 is shown as a power of ten. = × 104 move decimal 4 spaces left = × 10−3 move decimal 3 spaces right © 2014 Pearson Education, Inc.

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Some Powers of Ten © 2014 Pearson Education, Inc.

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**Some Measurements in Scientific Notation**

© 2014 Pearson Education, Inc.

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**Comparing Numbers in Standard and Scientific Notation**

Standard Format Scientific Notation Diameter of the Earth m × 107 m Mass of a human 68 kg × 101 kg Diameter of a virus cm 3 × 10−7 cm © 2014 Pearson Education, Inc.

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**Scientific Notation and Calculators**

You can enter a number written in scientific notation on many calculators using the EE or EXP key. © 2014 Pearson Education, Inc.

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**Scientific Notation and Calculators**

When a calculator display appears in scientific notation, it is shown as a number between 1 and 10, followed by a space and the power (exponent). © 2014 Pearson Education, Inc.

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**Scientific Notation and Calculators**

On many scientific calculators, a number is converted to scientific notation, using the appropriate keys. nd or 3rd function key SCI Key Key = 5.2 −04 or 5.2−04 = 5.2 × 10−4 Calculator display © 2014 Pearson Education, Inc.

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**Guide to Writing a Number in Scientific Notation**

© 2014 Pearson Education, Inc.

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Learning Check Write the following number in the correct scientific notation, g. © 2014 Pearson Education, Inc.

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**Move the decimal 5 places to the right, to give a coefficient of 5.8.**

Solution Write the following number in the correct scientific notation, g. Step 1 Move the decimal point to obtain a coefficient that is at least 1 but less than Move the decimal 5 places to the right, to give a coefficient of 5.8. © 2014 Pearson Education, Inc.

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**Step 2 Express the number of places moved as a power of 10. **

Solution Write the following number in the correct scientific notation, g. Step 2 Express the number of places moved as a power of 10. Moving the decimal 5 places to the right gives a power of −5. © 2014 Pearson Education, Inc.

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Solution Write the following number in the correct scientific notation, g. Step 3 Write the product of the coefficient multiplied by the power of 10 with the unit. 5.8 × 10−5 g © 2014 Pearson Education, Inc.

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**Select the correct scientific notation for each. A. 0.000 008 **

Learning Check Select the correct scientific notation for each. A (1) 8 × 106 (2) 8 × 10−6 (3) 0.8 × 10−5 B (1) 7.2 × 104 (2) 72 × 103 (3) 7.2 × 10−4 © 2014 Pearson Education, Inc.

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**Select the correct scientific notation for each. A. 0.000 008 **

Solution Select the correct scientific notation for each. A (Move the decimal 6 places to right.) (2) 8 × 10−6 B (Move the decimal 4 places to the left.) (1) 7.2 × 104 © 2014 Pearson Education, Inc.

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**Write each as a standard number. A. 2.0 × 10−2 **

Learning Check Write each as a standard number. A. 2.0 × 10−2 (1) 200 (2) (3) 0.020 B. 1.8 × 105 (1) (2) (3) © 2014 Pearson Education, Inc.

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**Write each as a standard number. A. 2.0 × 10−2 (3) 0.020 B. 1.8 × 105 **

Solution Write each as a standard number. A. 2.0 × 10−2 (3) 0.020 B. 1.8 × 105 (1) © 2014 Pearson Education, Inc.

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Dimensional Analysis We use dimensional analysis to convert one quantity to another. Most commonly, dimensional analysis utilizes conversion factors (e.g., 1 in. = 2.54 cm) 1 in. 2.54 cm or © 2012 Pearson Education, Inc.

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Dimensional Analysis Use the form of the conversion factor that puts the sought-for unit in the numerator: Given unit desired unit desired unit given unit Conversion factor © 2012 Pearson Education, Inc.

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**Dimensional Analysis For example, to convert 8.00 m to inches,**

convert m to cm convert cm to in. 8.00 m 100 cm 1 m 1 in. 2.54 cm 315 in. © 2012 Pearson Education, Inc.

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Uncertainty in Data © 2012 Pearson Education, Inc.

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**Uncertainty in Measurements**

Different measuring devices have different uses and different degrees of accuracy. © 2012 Pearson Education, Inc.

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**Accuracy versus Precision**

Accuracy refers to the proximity of a measurement to the true value of a quantity. Precision refers to the proximity of several measurements to each other. © 2012 Pearson Education, Inc.

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**Accuracy and Precision**

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**Accuracy and Precision**

Low accuracy, high precision High accuracy, low precision High accuracy, high precision

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**Accuracy and Precision Accepted mass of the object is 10.91 g**

Trails Mass (g) 1 11.50 2 11.20 3 10.50 4 10.60 5 10.30 Poor precision, but good accuracy

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**Accuracy and Precision mass of object has actual mass of 25.11 g**

Data Set 1 Data Set 2 Data Set 3 Data Set 4 24.06 g 25.12 g 23.76 g 26.51 g 28.09 25.09 g 23.80 g 25.08 g 29.56 g 25.14 g 23.78 g 23.63 g Data set 1 neither accurate or precise Data set 2 accurate and precise Data set 3 precise, not accurate Data set 4 accurate (based on average), but not precise

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Significant Figures The term significant figures refers to digits that were measured. When rounding calculated numbers, we pay attention to significant figures so we do not overstate the accuracy of our answers. © 2012 Pearson Education, Inc.

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**Significant Figures A number is a significant figure if it is**

Not a zero One or more zeros between nonzero digits One or more zeros at the end of a decimal number In the coefficient of a number written in scientific notation 2. A zero is not significant if it is At the beginning of a decimal number written in scientific notation Used as a placeholder in a large number without a decimal point © 2012 Pearson Education, Inc.

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**Rules for Counting Significant Figures**

55.32 has 4 significant figures

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has 5 significant figures

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has 1 significant figures

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has 6 significant figures

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**Sig Fig In Class Practice**

How many significant figures in each of the following? 1 sig figs 2 sig figs 3 sig figs 501 4 sig figs 1 sig figs 4 sig figs

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**Sig Fig In Class Practice**

How many significant figures in each of the following? 6 sig figs 1 sig figs 4 sig figs 5 sig figs 3 sig figs 2 sig figs

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**Sig Fig In Class Practice**

How many significant figures in each of the following? 4 sig figs 4 sig figs 3 sig figs 5 sig figs 4 sig figs 154 3 sig figs

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**Sig Fig In Class Practice**

How many significant figures in each of the following? 3 sig figs 5 sig figs

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Significant Figures When addition or subtraction is performed, answers are rounded to the least significant decimal place. When multiplication or division is performed, answers are rounded to the number of digits that corresponds to the least number of significant figures in any of the numbers used in the calculation. © 2012 Pearson Education, Inc.

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**calculated answers are usually rounded off **

Rounding Off In calculations calculated answers are usually rounded off rounding rules are used to obtain the correct number of significant figures © 2014 Pearson Education, Inc.

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Rules for Rounding Off 1. If the first digit to be dropped is 4 or less, then it and all following digits are simply dropped from the number. 2. If the first digit to be dropped is 5 or greater, then the last retained digit is increased by 1. © 2014 Pearson Education, Inc.

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**Rounding Off and Significant Figures**

© 2014 Pearson Education, Inc.

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Learning Check Adjust the following calculated answers to give answers with three significant figures: A cm B g 8.2 L © 2014 Pearson Education, Inc.

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Solution Adjust the following calculated answers to give answers with three significant figures: A cm 825 cm B g g 8.2 L L © 2014 Pearson Education, Inc.

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**Multiplication and Division with Measured Numbers**

In multiplication and division, the final answer is written to have the same number of significant figures (SFs) as the measurement with the fewest SFs. For example, 24.65 × = 17 4 SF SF Calculator Final answer © 2014 Pearson Education, Inc.

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**= 4 4.00 Adding Significant Zeros**

When the calculator answer is a small whole number and more significant figures are needed, we can add one or more zeros. For example, = 3 SF Calculator Final answer © 2014 Pearson Education, Inc.

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**Addition and Subtraction with Measured Numbers**

In addition or subtraction, the final answer is written so that it has the same number of decimal places as the measurement having the fewest decimal places. For example, Thousandths place Tenths place Calculator display Answer, rounded off to tenths place © 2014 Pearson Education, Inc.

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Learning Check Give an answer for each with the correct number of significant figures. 2.19 × 4.2 = (1) 9 (2) 9.2 (3) 9.198 B × = × 0.060 (1) 11.3 (2) (3) © 2014 Pearson Education, Inc.

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Solution Give an answer for each with the correct number of significant figures. A × = (2) 9.2 B × = (2) 11 × 0.060 © 2014 Pearson Education, Inc.

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Learning Check For each calculation, round the answer to give the correct number of digits. A = (1) 257 (2) (3) B – = (1) (2) (3) 40.7 © 2014 Pearson Education, Inc.

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**Solution A. 235.05 Hundredths place +19.6 Tenths place + 2 Ones place**

rounds to answer (1) B Thousandths place – Tenths place rounds to answer (3) © 2014 Pearson Education, Inc.

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3.1 Measurements and Their Uncertainty

3.1 Measurements and Their Uncertainty

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