• 1.2 Physical Quantities and Units
  • Introduction to Science and the Realm of Physics, Physical Quantities, and Units
  • 1.1 Physics: An Introduction
  • 1.3 Accuracy, Precision, and Significant Figures
  • 1.4 Approximation
  • Section Summary
  • Conceptual Questions
  • Problems & Exercises
  • Introduction to One-Dimensional Kinematics
  • 2.1 Displacement
  • 2.2 Vectors, Scalars, and Coordinate Systems
  • 2.3 Time, Velocity, and Speed
  • 2.4 Acceleration
  • 2.5 Motion Equations for Constant Acceleration in One Dimension
  • 2.6 Problem-Solving Basics for One-Dimensional Kinematics
  • 2.7 Falling Objects
  • 2.8 Graphical Analysis of One-Dimensional Motion
  • Introduction to Two-Dimensional Kinematics
  • 3.1 Kinematics in Two Dimensions: An Introduction
  • 3.2 Vector Addition and Subtraction: Graphical Methods
  • 3.3 Vector Addition and Subtraction: Analytical Methods
  • 3.4 Projectile Motion
  • 3.5 Addition of Velocities
  • Introduction to Dynamics: Newton’s Laws of Motion
  • 4.1 Development of Force Concept
  • 4.2 Newton’s First Law of Motion: Inertia
  • 4.3 Newton’s Second Law of Motion: Concept of a System
  • 4.4 Newton’s Third Law of Motion: Symmetry in Forces
  • 4.5 Normal, Tension, and Other Examples of Forces
  • 4.6 Problem-Solving Strategies
  • 4.7 Further Applications of Newton’s Laws of Motion
  • 4.8 Extended Topic: The Four Basic Forces—An Introduction
  • Introduction: Further Applications of Newton’s Laws
  • 5.1 Friction
  • 5.2 Drag Forces
  • 5.3 Elasticity: Stress and Strain
  • Introduction to Uniform Circular Motion and Gravitation
  • 6.1 Rotation Angle and Angular Velocity
  • 6.2 Centripetal Acceleration
  • 6.3 Centripetal Force
  • 6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force
  • 6.5 Newton’s Universal Law of Gravitation
  • 6.6 Satellites and Kepler’s Laws: An Argument for Simplicity
  • Introduction to Work, Energy, and Energy Resources
  • 7.1 Work: The Scientific Definition
  • 7.2 Kinetic Energy and the Work-Energy Theorem
  • 7.3 Gravitational Potential Energy
  • 7.4 Conservative Forces and Potential Energy
  • 7.5 Nonconservative Forces
  • 7.6 Conservation of Energy
  • 7.8 Work, Energy, and Power in Humans
  • 7.9 World Energy Use
  • Introduction to Linear Momentum and Collisions
  • 8.1 Linear Momentum and Force
  • 8.2 Impulse
  • 8.3 Conservation of Momentum
  • 8.4 Elastic Collisions in One Dimension
  • 8.5 Inelastic Collisions in One Dimension
  • 8.6 Collisions of Point Masses in Two Dimensions
  • 8.7 Introduction to Rocket Propulsion
  • Introduction to Statics and Torque
  • 9.1 The First Condition for Equilibrium
  • 9.2 The Second Condition for Equilibrium
  • 9.3 Stability
  • 9.4 Applications of Statics, Including Problem-Solving Strategies
  • 9.5 Simple Machines
  • 9.6 Forces and Torques in Muscles and Joints
  • Introduction to Rotational Motion and Angular Momentum
  • 10.1 Angular Acceleration
  • 10.2 Kinematics of Rotational Motion
  • 10.3 Dynamics of Rotational Motion: Rotational Inertia
  • 10.4 Rotational Kinetic Energy: Work and Energy Revisited
  • 10.5 Angular Momentum and Its Conservation
  • 10.6 Collisions of Extended Bodies in Two Dimensions
  • 10.7 Gyroscopic Effects: Vector Aspects of Angular Momentum
  • Introduction to Fluid Statics
  • 11.1 What Is a Fluid?
  • 11.2 Density
  • 11.3 Pressure
  • 11.4 Variation of Pressure with Depth in a Fluid
  • 11.5 Pascal’s Principle
  • 11.6 Gauge Pressure, Absolute Pressure, and Pressure Measurement
  • 11.7 Archimedes’ Principle
  • 11.8 Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action
  • 11.9 Pressures in the Body
  • Introduction to Fluid Dynamics and Its Biological and Medical Applications
  • 12.1 Flow Rate and Its Relation to Velocity
  • 12.2 Bernoulli’s Equation
  • 12.3 The Most General Applications of Bernoulli’s Equation
  • 12.4 Viscosity and Laminar Flow; Poiseuille’s Law
  • 12.5 The Onset of Turbulence
  • 12.6 Motion of an Object in a Viscous Fluid
  • 12.7 Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes
  • Introduction to Temperature, Kinetic Theory, and the Gas Laws
  • 13.1 Temperature
  • 13.2 Thermal Expansion of Solids and Liquids
  • 13.3 The Ideal Gas Law
  • 13.4 Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature
  • 13.5 Phase Changes
  • 13.6 Humidity, Evaporation, and Boiling
  • Introduction to Heat and Heat Transfer Methods
  • 14.2 Temperature Change and Heat Capacity
  • 14.3 Phase Change and Latent Heat
  • 14.4 Heat Transfer Methods
  • 14.5 Conduction
  • 14.6 Convection
  • 14.7 Radiation
  • Introduction to Thermodynamics
  • 15.1 The First Law of Thermodynamics
  • 15.2 The First Law of Thermodynamics and Some Simple Processes
  • 15.3 Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
  • 15.4 Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
  • 15.5 Applications of Thermodynamics: Heat Pumps and Refrigerators
  • 15.6 Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
  • 15.7 Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation
  • Introduction to Oscillatory Motion and Waves
  • 16.1 Hooke’s Law: Stress and Strain Revisited
  • 16.2 Period and Frequency in Oscillations
  • 16.3 Simple Harmonic Motion: A Special Periodic Motion
  • 16.4 The Simple Pendulum
  • 16.5 Energy and the Simple Harmonic Oscillator
  • 16.6 Uniform Circular Motion and Simple Harmonic Motion
  • 16.7 Damped Harmonic Motion
  • 16.8 Forced Oscillations and Resonance
  • 16.10 Superposition and Interference
  • 16.11 Energy in Waves: Intensity
  • Introduction to the Physics of Hearing
  • 17.2 Speed of Sound, Frequency, and Wavelength
  • 17.3 Sound Intensity and Sound Level
  • 17.4 Doppler Effect and Sonic Booms
  • 17.5 Sound Interference and Resonance: Standing Waves in Air Columns
  • 17.6 Hearing
  • 17.7 Ultrasound
  • Introduction to Electric Charge and Electric Field
  • 18.1 Static Electricity and Charge: Conservation of Charge
  • 18.2 Conductors and Insulators
  • 18.3 Coulomb’s Law
  • 18.4 Electric Field: Concept of a Field Revisited
  • 18.5 Electric Field Lines: Multiple Charges
  • 18.6 Electric Forces in Biology
  • 18.7 Conductors and Electric Fields in Static Equilibrium
  • 18.8 Applications of Electrostatics
  • Introduction to Electric Potential and Electric Energy
  • 19.1 Electric Potential Energy: Potential Difference
  • 19.2 Electric Potential in a Uniform Electric Field
  • 19.3 Electrical Potential Due to a Point Charge
  • 19.4 Equipotential Lines
  • 19.5 Capacitors and Dielectrics
  • 19.6 Capacitors in Series and Parallel
  • 19.7 Energy Stored in Capacitors
  • Introduction to Electric Current, Resistance, and Ohm's Law
  • 20.1 Current
  • 20.2 Ohm’s Law: Resistance and Simple Circuits
  • 20.3 Resistance and Resistivity
  • 20.4 Electric Power and Energy
  • 20.5 Alternating Current versus Direct Current
  • 20.6 Electric Hazards and the Human Body
  • 20.7 Nerve Conduction–Electrocardiograms
  • Introduction to Circuits and DC Instruments
  • 21.1 Resistors in Series and Parallel
  • 21.2 Electromotive Force: Terminal Voltage
  • 21.3 Kirchhoff’s Rules
  • 21.4 DC Voltmeters and Ammeters
  • 21.5 Null Measurements
  • 21.6 DC Circuits Containing Resistors and Capacitors
  • Introduction to Magnetism
  • 22.1 Magnets
  • 22.2 Ferromagnets and Electromagnets
  • 22.3 Magnetic Fields and Magnetic Field Lines
  • 22.4 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
  • 22.5 Force on a Moving Charge in a Magnetic Field: Examples and Applications
  • 22.6 The Hall Effect
  • 22.7 Magnetic Force on a Current-Carrying Conductor
  • 22.8 Torque on a Current Loop: Motors and Meters
  • 22.9 Magnetic Fields Produced by Currents: Ampere’s Law
  • 22.10 Magnetic Force between Two Parallel Conductors
  • 22.11 More Applications of Magnetism
  • Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies
  • 23.1 Induced Emf and Magnetic Flux
  • 23.2 Faraday’s Law of Induction: Lenz’s Law
  • 23.3 Motional Emf
  • 23.4 Eddy Currents and Magnetic Damping
  • 23.5 Electric Generators
  • 23.6 Back Emf
  • 23.7 Transformers
  • 23.8 Electrical Safety: Systems and Devices
  • 23.9 Inductance
  • 23.10 RL Circuits
  • 23.11 Reactance, Inductive and Capacitive
  • 23.12 RLC Series AC Circuits
  • Introduction to Electromagnetic Waves
  • 24.1 Maxwell’s Equations: Electromagnetic Waves Predicted and Observed
  • 24.2 Production of Electromagnetic Waves
  • 24.3 The Electromagnetic Spectrum
  • 24.4 Energy in Electromagnetic Waves
  • Introduction to Geometric Optics
  • 25.1 The Ray Aspect of Light
  • 25.2 The Law of Reflection
  • 25.3 The Law of Refraction
  • 25.4 Total Internal Reflection
  • 25.5 Dispersion: The Rainbow and Prisms
  • 25.6 Image Formation by Lenses
  • 25.7 Image Formation by Mirrors
  • Introduction to Vision and Optical Instruments
  • 26.1 Physics of the Eye
  • 26.2 Vision Correction
  • 26.3 Color and Color Vision
  • 26.4 Microscopes
  • 26.5 Telescopes
  • 26.6 Aberrations
  • Introduction to Wave Optics
  • 27.1 The Wave Aspect of Light: Interference
  • 27.2 Huygens's Principle: Diffraction
  • 27.3 Young’s Double Slit Experiment
  • 27.4 Multiple Slit Diffraction
  • 27.5 Single Slit Diffraction
  • 27.6 Limits of Resolution: The Rayleigh Criterion
  • 27.7 Thin Film Interference
  • 27.8 Polarization
  • 27.9 *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light
  • Introduction to Special Relativity
  • 28.1 Einstein’s Postulates
  • 28.2 Simultaneity And Time Dilation
  • 28.3 Length Contraction
  • 28.4 Relativistic Addition of Velocities
  • 28.5 Relativistic Momentum
  • 28.6 Relativistic Energy
  • Introduction to Quantum Physics
  • 29.1 Quantization of Energy
  • 29.2 The Photoelectric Effect
  • 29.3 Photon Energies and the Electromagnetic Spectrum
  • 29.4 Photon Momentum
  • 29.5 The Particle-Wave Duality
  • 29.6 The Wave Nature of Matter
  • 29.7 Probability: The Heisenberg Uncertainty Principle
  • 29.8 The Particle-Wave Duality Reviewed
  • Introduction to Atomic Physics
  • 30.1 Discovery of the Atom
  • 30.2 Discovery of the Parts of the Atom: Electrons and Nuclei
  • 30.3 Bohr’s Theory of the Hydrogen Atom
  • 30.4 X Rays: Atomic Origins and Applications
  • 30.5 Applications of Atomic Excitations and De-Excitations
  • 30.6 The Wave Nature of Matter Causes Quantization
  • 30.7 Patterns in Spectra Reveal More Quantization
  • 30.8 Quantum Numbers and Rules
  • 30.9 The Pauli Exclusion Principle
  • Introduction to Radioactivity and Nuclear Physics
  • 31.1 Nuclear Radioactivity
  • 31.2 Radiation Detection and Detectors
  • 31.3 Substructure of the Nucleus
  • 31.4 Nuclear Decay and Conservation Laws
  • 31.5 Half-Life and Activity
  • 31.6 Binding Energy
  • 31.7 Tunneling
  • Introduction to Applications of Nuclear Physics
  • 32.1 Diagnostics and Medical Imaging
  • 32.2 Biological Effects of Ionizing Radiation
  • 32.3 Therapeutic Uses of Ionizing Radiation
  • 32.4 Food Irradiation
  • 32.5 Fusion
  • 32.6 Fission
  • 32.7 Nuclear Weapons
  • Introduction to Particle Physics
  • 33.1 The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited
  • 33.2 The Four Basic Forces
  • 33.3 Accelerators Create Matter from Energy
  • 33.4 Particles, Patterns, and Conservation Laws
  • 33.5 Quarks: Is That All There Is?
  • 33.6 GUTs: The Unification of Forces
  • Introduction to Frontiers of Physics
  • 34.1 Cosmology and Particle Physics
  • 34.2 General Relativity and Quantum Gravity
  • 34.3 Superstrings
  • 34.4 Dark Matter and Closure
  • 34.5 Complexity and Chaos
  • 34.6 High-temperature Superconductors
  • 34.7 Some Questions We Know to Ask
  • A | Atomic Masses
  • B | Selected Radioactive Isotopes
  • C | Useful Information
  • D | Glossary of Key Symbols and Notation

Learning Objectives

By the end of this section, you will be able to:

  • Perform unit conversions both in the SI and English units.
  • Explain the most common prefixes in the SI units and be able to write them in scientific notation.

The range of objects and phenomena studied in physics is immense. From the incredibly short lifetime of a nucleus to the age of the Earth, from the tiny sizes of sub-nuclear particles to the vast distance to the edges of the known universe, from the force exerted by a jumping flea to the force between Earth and the Sun, there are enough factors of 10 to challenge the imagination of even the most experienced scientist. Giving numerical values for physical quantities and equations for physical principles allows us to understand nature much more deeply than does qualitative description alone. To comprehend these vast ranges, we must also have accepted units in which to express them. And we shall find that (even in the potentially mundane discussion of meters, kilograms, and seconds) a profound simplicity of nature appears—most physical quantities can be expressed as combinations of only four fundamental physical quantities: length, mass, time, and electric current.

We define a physical quantity either by specifying how it is measured or by stating how it is calculated from other measurements. For example, we define distance and time by specifying methods for measuring them, whereas we define average speed by stating that it is calculated as distance traveled divided by time of travel.

Measurements of physical quantities are expressed in terms of units , which are standardized values. For example, the length of a race, which is a physical quantity, can be expressed in units of meters (for sprinters) or kilometers (for distance runners). Without standardized units, it would be extremely difficult for scientists to express and compare measured values in a meaningful way. (See Figure 1.16 .)

There are two major systems of units used in the world: SI units (also known as the metric system) and English units (also known as the customary or imperial system). English units were historically used in nations once ruled by the British Empire and are still widely used in the United States. Virtually every other country in the world now uses SI units as the standard; the metric system is also the standard system agreed upon by scientists and mathematicians. The acronym “SI” is derived from the French Système International .

SI Units: Fundamental and Derived Units

Table 1.1 gives the fundamental SI units that are used throughout this textbook. This text uses non-SI units in a few applications where they are in very common use, such as the measurement of blood pressure in millimeters of mercury (mm Hg). Whenever non-SI units are discussed, they will be tied to SI units through conversions.

It is an intriguing fact that some physical quantities are more fundamental than others and that the most fundamental physical quantities can be defined only in terms of the procedure used to measure them. The units in which they are measured are thus called fundamental units . In this textbook, the fundamental physical quantities are taken to be length, mass, time, and electric current. (Note that electric current will not be introduced until much later in this text.) All other physical quantities, such as force and electric charge, can be expressed as algebraic combinations of length, mass, time, and current (for example, speed is length divided by time); these units are called derived units .

Units of Time, Length, and Mass: The Second, Meter, and Kilogram

The SI unit for time, the second (abbreviated s), has a long history. For many years it was defined as 1/86,400 of a mean solar day. More recently, a new standard was adopted to gain greater accuracy and to define the second in terms of a non-varying, or constant, physical phenomenon (because the solar day is getting longer due to very gradual slowing of the Earth’s rotation). Cesium atoms can be made to vibrate in a very steady way, and these vibrations can be readily observed and counted. In 1967 the second was redefined as the time required for 9,192,631,770 of these vibrations. (See Figure 1.17 .) Accuracy in the fundamental units is essential, because all measurements are ultimately expressed in terms of fundamental units and can be no more accurate than are the fundamental units themselves.

The SI unit for length is the meter (abbreviated m); its definition has also changed over time to become more accurate and precise. The meter was first defined in 1791 as 1/10,000,000 of the distance from the equator to the North Pole. This measurement was improved in 1889 by redefining the meter to be the distance between two engraved lines on a platinum-iridium bar now kept near Paris. By 1960, it had become possible to define the meter even more accurately in terms of the wavelength of light, so it was again redefined as 1,650,763.73 wavelengths of orange light emitted by krypton atoms. In 1983, the meter was given its present definition (partly for greater accuracy) as the distance light travels in a vacuum in 1/299,792,458 of a second. (See Figure 1.18 .) This change defines the speed of light to be exactly 299,792,458 meters per second. The length of the meter will change if the speed of light is someday measured with greater accuracy.

The Kilogram

The SI unit for mass is the kilogram (abbreviated kg); it was previously defined to be the mass of a platinum-iridium cylinder kept with the old meter standard at the International Bureau of Weights and Measures near Paris. Exact replicas of the previously defined kilogram are also kept at the United States’ National Institute of Standards and Technology, or NIST, located in Gaithersburg, Maryland outside of Washington D.C., and at other locations around the world. The determination of all other masses could be ultimately traced to a comparison with the standard mass. Even though the platinum-iridium cylinder was resistant to corrosion, airborne contaminants were able to adhere to its surface, slightly changing its mass over time. In May 2019, the scientific community adopted a more stable definition of the kilogram. The kilogram is now defined in terms of the second, the meter, and Planck's constant, h (a quantum mechanical value that relates a photon's energy to its frequency).

Electric current and its accompanying unit, the ampere, will be introduced in Electric Current, Resistance, and Ohm's Law when electricity and magnetism are covered. The initial modules in this textbook are concerned with mechanics, fluids, heat, and waves. In these subjects all pertinent physical quantities can be expressed in terms of the fundamental units of length, mass, and time.

Metric Prefixes

SI units are part of the metric system . The metric system is convenient for scientific and engineering calculations because the units are categorized by factors of 10. Table 1.2 gives metric prefixes and symbols used to denote various factors of 10.

Metric systems have the advantage that conversions of units involve only powers of 10. There are 100 centimeters in a meter, 1000 meters in a kilometer, and so on. In nonmetric systems, such as the system of U.S. customary units, the relationships are not as simple—there are 12 inches in a foot, 5280 feet in a mile, and so on. Another advantage of the metric system is that the same unit can be used over extremely large ranges of values simply by using an appropriate metric prefix. For example, distances in meters are suitable in construction, while distances in kilometers are appropriate for air travel, and the tiny measure of nanometers are convenient in optical design. With the metric system there is no need to invent new units for particular applications.

The term order of magnitude refers to the scale of a value expressed in the metric system. Each power of 10 10 , and so forth are all different orders of magnitude. All quantities that can be expressed as a product of a specific power of 10 10 are said to be of the same order of magnitude. For example, the number 800 800 can be written as 8 × 10 2 8 × 10 2 , and the number 450 450 can be written as 4. 5 × 10 2 . 4. 5 × 10 2 . Thus, the numbers 800 800 and 450 450 are of the same order of magnitude: 10 2 . 10 2 . Order of magnitude can be thought of as a ballpark estimate for the scale of a value. The diameter of an atom is on the order of 10 − 10  m, 10 − 10  m, while the diameter of the Sun is on the order of 10 9  m. 10 9  m.

The Quest for Microscopic Standards for Basic Units

The fundamental units described in this chapter are those that produce the greatest accuracy and precision in measurement. There is a sense among physicists that, because there is an underlying microscopic substructure to matter, it would be most satisfying to base our standards of measurement on microscopic objects and fundamental physical phenomena such as the speed of light. A microscopic standard has been accomplished for the standard of time, which is based on the oscillations of the cesium atom.

The standard for length was once based on the wavelength of light (a small-scale length) emitted by a certain type of atom, but it has been supplanted by the more precise measurement of the speed of light. If it becomes possible to measure the mass of atoms or a particular arrangement of atoms such as a silicon sphere to greater precision than the kilogram standard, it may become possible to base mass measurements on the small scale. There are also possibilities that electrical phenomena on the small scale may someday allow us to base a unit of charge on the charge of electrons and protons, but at present current and charge are related to large-scale currents and forces between wires.

Known Ranges of Length, Mass, and Time

The vastness of the universe and the breadth over which physics applies are illustrated by the wide range of examples of known lengths, masses, and times in Table 1.3 . Examination of this table will give you some feeling for the range of possible topics and numerical values. (See Figure 1.19 and Figure 1.20 .)

Unit Conversion and Dimensional Analysis

It is often necessary to convert from one type of unit to another. For example, if you are reading a European cookbook, some quantities may be expressed in units of liters and you need to convert them to cups. Or, perhaps you are reading walking directions from one location to another and you are interested in how many miles you will be walking. In this case, you will need to convert units of feet to miles.

Let us consider a simple example of how to convert units. Let us say that we want to convert 80 meters (m) to kilometers (km).

The first thing to do is to list the units that you have and the units that you want to convert to. In this case, we have units in meters and we want to convert to kilometers .

Next, we need to determine a conversion factor relating meters to kilometers. A conversion factor is a ratio expressing how many of one unit are equal to another unit. For example, there are 12 inches in 1 foot, 100 centimeters in 1 meter, 60 seconds in 1 minute, and so on. In this case, we know that there are 1,000 meters in 1 kilometer.

Now we can set up our unit conversion. We will write the units that we have and then multiply them by the conversion factor so that the units cancel out, as shown:

Note that the unwanted m unit cancels, leaving only the desired km unit. You can use this method to convert between any types of unit.

Click Appendix C for a more complete list of conversion factors.

Example 1.1

Unit conversions: a short drive home.

Suppose that you drive the 10.0 km from your school to home in 20.0 min. Calculate your average speed (a) in kilometers per hour (km/h) and (b) in meters per second (m/s). (Note: Average speed is distance traveled divided by time of travel.)

First we calculate the average speed using the given units. Then we can get the average speed into the desired units by picking the correct conversion factor and multiplying by it. The correct conversion factor is the one that cancels the unwanted unit and leaves the desired unit in its place.

Solution for (a)

(1) Calculate average speed. Average speed is distance traveled divided by time of travel. (Take this definition as a given for now—average speed and other motion concepts will be covered in a later module.) In equation form,

(2) Substitute the given values for distance and time.

(3) Convert km/min to km/h: multiply by the conversion factor that will cancel minutes and leave hours. That conversion factor is 60 min/hr 60 min/hr

Discussion for (a)

To check your answer, consider the following:

(1) Be sure that you have properly cancelled the units in the unit conversion. If you have written the unit conversion factor upside down, the units will not cancel properly in the equation. If you accidentally get the ratio upside down, then the units will not cancel; rather, they will give you the wrong units as follows:

which are obviously not the desired units of km/h.

(2) Check that the units of the final answer are the desired units. The problem asked us to solve for average speed in units of km/h and we have indeed obtained these units.

(3) Check the significant figures. Because each of the values given in the problem has three significant figures, the answer should also have three significant figures. The answer 30.0 km/hr does indeed have three significant figures, so this is appropriate. Note that the significant figures in the conversion factor are not relevant because an hour is defined to be 60 minutes, so the precision of the conversion factor is perfect.

(4) Next, check whether the answer is reasonable. Let us consider some information from the problem—if you travel 10 km in a third of an hour (20 min), you would travel three times that far in an hour. The answer does seem reasonable.

Solution for (b)

There are several ways to convert the average speed into meters per second.

(1) Start with the answer to (a) and convert km/h to m/s. Two conversion factors are needed—one to convert hours to seconds, and another to convert kilometers to meters.

(2) Multiplying by these yields

Discussion for (b)

If we had started with 0.500 km/min, we would have needed different conversion factors, but the answer would have been the same: 8.33 m/s.

You may have noted that the answers in the worked example just covered were given to three digits. Why? When do you need to be concerned about the number of digits in something you calculate? Why not write down all the digits your calculator produces? The module Accuracy, Precision, and Significant Figures will help you answer these questions.

Nonstandard Units

While there are numerous types of units that we are all familiar with, there are others that are much more obscure. For example, a firkin is a unit of volume that was once used to measure beer. One firkin equals about 34 liters. To learn more about nonstandard units, use a dictionary or encyclopedia to research different “weights and measures.” Take note of any unusual units, such as a barleycorn, that are not listed in the text. Think about how the unit is defined and state its relationship to SI units.

Check Your Understanding

Some hummingbirds beat their wings more than 50 times per second. A scientist is measuring the time it takes for a hummingbird to beat its wings once. Which fundamental unit should the scientist use to describe the measurement? Which factor of 10 is the scientist likely to use to describe the motion precisely? Identify the metric prefix that corresponds to this factor of 10.

The scientist will measure the time between each movement using the fundamental unit of seconds. Because the wings beat so fast, the scientist will probably need to measure in milliseconds, or 10 − 3 10 − 3 seconds. (50 beats per second corresponds to 20 milliseconds per beat.)

One cubic centimeter is equal to one milliliter. What does this tell you about the different units in the SI metric system?

The fundamental unit of length (meter) is probably used to create the derived unit of volume (liter). The measure of a milliliter is dependent on the measure of a centimeter.

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Chapter 1 – measurements in chemistry.

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Section 1: Chemistry and Matter

What is chemistry, physical and chemical properties, elements and compounds, states of matter, section 2: how scientists study chemistry  , the scientific method      , section 3: scientific notation  , video tutorial, practice problems  , section 4: units of measurement , international system of units and the metric system                    , derived si units      , section 5: making measurements in the lab  , precision vs. accuracy                                                                            , significant figures                                                                                     , exact numbers, rules of rounding                                                                                     , video tutorial   , calculations with significant figures                                                , conversions and the importance of units                                       , conversion factors, chapter summary.

Everything around us is made up of chemicals.  From the color that makes a rose so red to the gasoline that fills our cars and the silicon chips that power our computers and cell phones…Chemistry is everywhere! Understanding how chemical molecules form and interact to create complex structures enables us to harness the power of chemistry and use it, just like a toolbox, to create many of the modern advances that we see today.  This includes advances in medicine, communication, transportation, building infrastructure, food science and agriculture, and nearly every other technical field that you can imagine.

Chemistry is one branch of science. Science is the process by which we learn about the natural universe by observing, testing, and then generating models that explain our observations. is the process by which we learn about the natural universe by observing, testing, and then generating models that explain our observations. Because the physical universe is so vast, there are many different branches of science (Figure 1.1). Thus, chemistry is the study of matter, biology is the study of living things, and geology is the study of rocks and the earth. Mathematics is the language of science, and we will use it to communicate some of the ideas of chemistry.

Although we divide science into different fields, there is much overlap among them. For example, some biologists and chemists work in both fields so much that their work is called biochemistry. Similarly, geology and chemistry overlap in the field called geochemistry. Figure 1.1  shows how many of the individual fields of science are related.

Figure 1-1

Figure 1.1:  The Relationships Between Some of the Major Branches of Science. Chemistry lies more or less in the middle, which emphasizes its importance to many branches of science.

Physical vs. Chemical Properties

Part of understanding matter is being able to describe it. One way chemists describe matter is to assign different kinds of properties to different categories. The properties that chemists use to describe matter fall into two general categories. Physical properties are characteristics that describes matter, such as boiling point, melting point and color.  Physical Changes, such as melting a solid into a liquid, do not alter the chemical structure of that matter.  Chemical properties are characteristics that describe how the chemical structure of matter changes during a chemical reaction. An example of a chemical property is flammability—a materials ability to burn—because burning (also known as combustion) changes the chemical composition of a material.

Any sample of matter that has the same physical and chemical properties throughout the sample is called a substance. There are two types of substances. A substance that cannot be broken down into chemically simpler components is an element.  Aluminum, which is used in soda cans, is an element. A substance that can be broken down into chemically simpler components (because it has more than one element) is a compound. Water is a compound composed of the elements hydrogen and oxygen. Today, there are about 118 elements in the known universe which are organized on a fundamental chart called the Periodic Table of Elements (Fig. 1.2). In contrast, scientists have identified tens of millions of different compounds to date.

The smallest part of an element that maintains the identity of that element is called an atom. Atoms are extremely tiny; to make a line 1 inch long, you would need 217 million iron atoms! Similarly, the smallest part of a compound that maintains the identity of that compound is called a molecule. Molecules are composed of atoms that are attached together and behave as a unit (Fig. 1.2). Scientists usually work with millions of atoms and molecules at a time.  When a scientist is working

Figre 1-2a

Figure 1.2:  ( Upper Panel) The Periodic Table of the Elements is an organized chart that contains all of the known chemical elements. ( Lower Panel ) To the left of the arrow is shown one atom of oxygen and two atoms of hydrogen. Each of these represent single elements. When they are combined on the righthand side, they form a single molecule of water (H 2 O). Note that water is defined as a compound, because each single molecule is made up of more than one type of element, in this case, one atom of oxygen with two atoms of hydrogen.

with large numbers of atoms or molecules at a time, the scientist is studying the macroscopic view of the universe. However, scientists can also describe chemical events on the level of individual atoms or molecules, which is referred to as the microscopic viewpoint. We will see examples of both macroscopic and microscopic viewpoints throughout this book (Figure 1.3).

Figure 1-3

Figure 1.3:  How many molecules are needed for a period in a sentence? Although we do not notice it from a macroscopic perspective, matter is composed of microscopic particles so tiny that billions of them are needed to make a speck that we can see with the naked eye.  The X25 and X400,000,000 indicate the number of times the image is magnified.

A material composed of two or more substances is a mixture. In a mixture, the individual substances maintain their chemical identities. Many mixtures are obvious combinations of two or more substances, such as a mixture of sand and water. Such mixtures are called heterogeneous mixtures.  In some mixtures, the components are so intimately combined that they act like a single substance even though they are not. Mixtures with a consistent composition throughout are called homogeneous mixtures  Homogeneous mixtures that are mixed so thoroughly that neither component can be observed independently of the other are called solutions. Sugar dissolved in water is an example of a solution. A metal alloy, such as steel, is an example of a solid solution. Air, a mixture of mainly nitrogen and oxygen, is a gaseous solution.

Homogeneous vs heterogeneous.jpg

Figure 1.4:  Heterogeneous vs. Homogeneous Mixtures.  A mixture contains more than one substance.  In the upper panel you see an example of a heterogeneous mixture of oil and water.  The mixture is heterogeneous because you can visibly see two different components in the mixture.  In the lower panel, you see an example of a homogeneous mixture, coffee.  It is homogeneous because you cannot distinguish the many different components that make up a cup of coffee (water; caffeine; coffee alkaloids and tannins).  It looks the same throughout.  If the mixture is homogeneous and is also see through or clear, it is called a solution. In our example, the coffee is a solution; however, a concentrated espresso may be very opaque and would only be homogeneous mixture, not a solution.

Another way to classify matter is to describe it as a solid, a liquid, or a gas, which was done in the examples of solutions, above. These three descriptions, each implying that the matter has certain physical properties, represent the three phases of matter. A solid has a definite shape and a definite volume. Liquids have a definite volume but not a definite shape; they take the shape of their containers. Gases have neither a definite shape nor a definite volume, and they expand to fill their containers. We encounter matter in each phase every day. In fact, we regularly encounter water in all three phases: ice (solid), water (liquid), and steam (gas).

We know from our experience with water that substances can change from one phase to another if the conditions are right. Typically, varying the temperature of a substance (and, less commonly, the pressure exerted on it) can cause a phase change or a physical process in which a substance goes from one phase to another (Figure 1.5). Phase changes have particular names depending on what phases are involved, as summarized in Table 1.1.


Figure 1.5. Analyzing Phase Changes.  ( Upper panel ) A photo of boiling water demonstrates the phase change of water from the liquid to the gaseous phase.  Note that phase changes are a physical property of a molecule.  The water is still chemically the same (H 2 O) in the solid, liquid, or gaseous state. ( Lower panel ) Change in temperature can cause phase changes .  Above is the temperature scale for the phase changes of water.  If you add heat to solid ice, water will melt at 0 o C and boil at 100 o C.  If you remove heat from gaseous water, it will condense into the liquid state at 100 o C and freeze at 0 o C.

In summary, Figure 1.6 “The Classification of Matter” illustrates the relationships between the different ways matter can be classified.

Figure 1-6

Figure 1.6 The Classification of Matter.  Matter can be classified in a variety of ways depending on its properties

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Section 2: How Scientists Study Chemistry

The scientific method.

How do scientists work? Generally, they follow a process called the scientific method. The scientific method is an organized procedure for learning answers to questions. To find the answer to a question (for example, “Why do birds fly toward Earth’s equator during the cold months?”), a scientist goes through the following steps, which are also illustrated in Figure 1.7.

Figure 1.7

Figure 1.7 The General Steps of the Scientific Method.   The steps may not be as clear-cut in real life as described here, but most scientific work follows this general outline.

Propose a hypothesis. A scientist generates a testable idea, or hypothesis, to try to answer a question or explain how the natural universe works. Some people use the word theory in place of hypothesis, but the word hypothesis is the proper word in science. For scientific applications, the word theory is a general statement that describes a large set of observations and data. A theory represents the highest level of scientific understanding, and is built from a wide array of factual knowledge or data.

Test the hypothesis. A scientist evaluates the hypothesis by devising and carrying out experiments to test it. If the hypothesis passes the test, it may be a proper answer to the question. If the hypothesis does not pass the test, it may not be a good answer.

Refine the hypothesis if necessary. Depending on the results of experiments, a scientist may want to modify the hypothesis and then test it again. Sometimes the results show the original hypothesis to be completely wrong, in which case a scientist will have to devise a new hypothesis.

Not all scientific investigations are simple enough to be separated into these three discrete steps. But these steps represent the general method by which scientists learn about our natural universe.

1 2 1 the importance of units worksheet answers

Section 3: Scientific Notation

The study of chemistry can involve numbers that are very large. It can also involve numbers that are very small. Writing out such numbers and using them in their long form is problematic, because we would spend far too much time writing zeroes, and we would probably make a lot of mistakes! There is a solution to this problem. It is called scientific notation. 

Scientific notation allows us to express very large and very small numbers using powers of 10.

Recall that:

10 0 = 1   10 1 = 10   10 2 = 100

10 3 = 1000        10 4 = 10000       10 5 = 100000.

As you can see, the power to which 10 is raised is equal to the number of zeroes that follow the 1. This will be helpful for determining which exponent to use when we express numbers using scientific notation.

Let us take a very large number:

     579, 000, 000, 000

and express it using scientific notation.

First, we find the coefficient, which is a number between 1 and 10 that will be multiplied by 10 raised to some power.

Our coefficient is: 5.79

This number will be multiplied by 10 that is raised to some power. Now let us figure out what power that is.

We can do this by counting the number of positions that stand between the end of the original number and the new position of the decimal point in our coefficient.

        5 . 7 9 0 0 0 0 0 0 0 0 0            

 ↑                          ↑, how many positions are there.

We can see that there are 11 positions between our decimal and the end of the original number. This means that our coefficient, 5.79, will be multiplied by 10 raised to the 11th power.

Our number expressed in scientific notation is:

      5.79 x 10 11

But what about very small numbers?

You may recall that:

   10 -1 = 0.1            10 -2 = 0.01      10 -3 = 0.001

10 -4 = 0.0001         10 -5 = 0.00001.

The number of spaces to the right of the decimal point for our 1 is equal to the number in the exponent that is behind the negative sign. This is useful to keep in mind when we express very small numbers in scientific notation.

Here is a very small number:

Let us express this number using scientific notation.

Our coefficient will be 6.42

This number will be multiplied by 10 raised to some power, which will be negative. Let us figure out the correct power. We can figure this out by counting how many positions stand between the decimal point in our coefficient and the decimal point in our original number.

  0 . 0 0 0 0 6 4 2 ↑             ↑

How many positions.

There are 5 positions between our new decimal point and the decimal point in the original number, so our coefficient will be multiplied by 10 raised to the negative 5th power.

Our number written in scientific notation is:

6.42 x 10 -5

You can use these methods to express any large or small number using scientific notation.


Section 4: units of measurement, international system of units and the metric system.


The International System of Units, abbreviated SI from the French Système International D’unités, is the main system of measurement units used in science. Since the 1960s, the International System of Units has been internationally agreed upon as the standard metric system. The SI base units are based on physical standards. The definitions of the SI base units have been and continue to be modified and new base units added as advancements in science are made. Each SI base unit except the kilogram is described by stable properties of the universe.

There are seven base units, which are listed in Table 1.2.  Chemistry primarily uses five of the base units: the mole for amount, the kilogram for mass, the meter for length, the second for time, and the kelvin for temperature. The degree Celsius ( o C) is also commonly used for temperature. The numerical relationship between kelvins and degrees Celsius is as follows

K = o C + 273

The size of each base unit is defined by international convention. For example, the kilogram is defined as the quantity of mass of a special metal cylinder kept in a vault in France (Figure 1.8). The other base units have similar definitions. The sizes of the base units are not always convenient for all measurements. For example, a meter is a rather large unit for describing the width of something as narrow as human hair. Instead of reporting the diameter of hair as 0.00012 m or even 1.2 × 10 -4 m, SI also provides a series of prefixes that can be attached to the units, creating units that are larger or smaller by powers of 10, known as the metric system.


Figure 1.8 The Kilogram. The standard for the kilogram is a platinum-iridium cylinder kept in a speacial vault in France.  Source: Wikimedea (https://commons.wikimedia.org/wiki/File:National_prototype_kilogram_K20_replica.jpg)

Common prefixes and their multiplicative factors are listed in Table 1.3 “Prefixes Used with SI Units”. (Perhaps you have already noticed that the base unit kilogram is a combination of a prefix, kilo- meaning 1,000 ×, and a unit of mass, the gram.) Some prefixes create a multiple of the original unit: 1 kilogram equals 1,000 grams (or 1 kg = 1,000 g), and 1 megameter equals 1,000,000 meters (or 1 Mm = 1,000,000 m). Other prefixes create a fraction of the original unit. Thus, 1 centimeter equals 1/100 of a meter, 1 millimeter equals 1/1,000 of a meter, 1 microgram equals 1/1,000,000 of a gram, and so forth.


The basic unit of mass in the International System of Units is the kilogram. A kilogram is equal to 1000 grams. A gram is a relatively small amount of mass and so larger masses are often expressed in kilograms. When very tiny amounts of matter are measured, we often use milligrams which are equal to 0.001 gram. There are numerous larger, smaller, and intermediate mass units that may also be appropriate. At the end of the 18th century, a kilogram was the mass of a liter of water. In 1889, a new international prototype of the kilogram was made of a platinum-iridium alloy. The kilogram is equal to the mass of this international prototype, which is held in Paris, France.

Mass and weight are not the same thing. Although we often use the terms mass and weight interchangeably, each one has a specific definition and usage. The mass of an object is a measure of the amount of matter in it. The mass (amount of matter) of an object remains the same regardless of where the object is placed. For example, moving a brick to the moon does not cause any matter in it to disappear or be removed.

The weight of an object is determined by the force that gravitation exerts upon the object.  The weight is equal to the mass of the object times the local acceleration of gravity. Thus, on the Earth, weight is determined by the force of attraction between the object and the Earth. Since the force of gravity is not the same at every point on the Earth’s surface, the weight of an object is not constant. The gravitational pull on the object varies depending on where the object is with respect to the Earth or other gravity-producing object. For example, a man who weighs 180 pounds on Earth would weigh only 45 pounds if he were in a stationary position, 4,000 miles above the Earth’s surface. This same man would weigh only 30 pounds on the moon because the moon’s gravity is only one-sixth that of Earth. The mass of this man, however, would be the same in each situation. For scientific experiments, it is important to measure the mass of a substance rather than the weight to retain consistency in the results regardless of where you are performing the experiment.

The SI unit of length is the meter. In 1889, the definition of the meter was a bar of platinum-iridium alloy stored under conditions specified by the International Bureau of Standards. In 1960, this definition of the standard meter was replaced by a definition based on a wavelength of krypton-86 radiation. In 1983, that definition was replaced by the following: the meter is the length of the path traveled by light in a vacuum during a time interval of a second.


When used in a scientific context, the words heat and temperature do NOT mean the same thing. Temperature represents the average kinetic energy of the particles that make up a material. Increasing the temperature of a material increases its thermal energy. Thermal energy is the sum of the kinetic and potential energy in the particles that make up a material. Objects do not “contain” heat; rather they contain thermal energy. Heat is the movement of thermal energy from a warmer object to a cooler object. When thermal energy moves from one object to another, the temperature of both objects change.

A thermometer is a device that measures temperature. The name is made up of “thermo” which means heat and “meter” which means to measure. The temperature of a substance is directly proportional to the average kinetic energy it contains. In order for the average kinetic energy and temperature of a substance to be directly proportional, it is necessary that when the temperature is zero, the average kinetic energy must also be zero. It was necessary for use in calculations in science for a third temperature scale in which zero degrees corresponds with zero kinetic energy, that is, the point where molecules cease to move. This temperature scale was designed by Lord Kelvin. Lord Kelvin stated that there is no upper limit of how hot things can get, but there is a limit as to how cold things can get. In 1848, William Lord Kelvin developed the idea of absolute zero, which is the temperature at which molecules stop moving and therefore, have zero kinetic energy. This is known as the Kelvin temperature scale.

The Celcius scale is based on the  freezing point and boiling point of water.  Thus, 0 o C is the freezing point of water, whereas 100 o C is the boiling point of water. Most of us are familiar with temperatures that are below the freezing point of water. It should be apparent that even though the air temperature may be -5 o C, the molecules of air are still moving (i.e. 0 o C is not absolute zero). Substances like oxygen gas and nitrogen gas have already melted and boiled to vapor at temperatures below -150 o C.

The Fahrenheit scale is also defined by the freezing point and boiling points of water.  However, the scale is different from that of the Kelvin and Celsius scales.  In the Fahrenheit scale, the freezing point of water is 32 o F and the boiling point of water is 212 o F. To convert between the Fahrenheit scale and the Celsius scale , the following conversions can be used:

[ o C] = ([ o F] -32) ×  5/9        or        [ o F] = [ o C] ×  9/5 + 32

The Kelvin temperature scale has its zero at absolute zero (determined to be -273.15 o C), and uses the same degree scale as the Celsius scale. Therefore, the mathematical relationship between the Celsius scale and the Kelvin scale is

K = o C + 273.15

In the case of the Kelvin scale, the degree sign is not used. Temperatures are expressed simply as 450 K, and are always positive.

The SI unit for time is the second. The second was originally defined as a tiny fraction of the time required for the Earth to orbit the Sun. It has since been redefined several times. The definition of a second (established in 1967 and reaffirmed in 1997) is: the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom.

Chemists use the term mole to represent a large number of atoms or molecules. Just as a dozen implies 12 things, a mole (mol) represents 6.022 × 10 23 things. The number 6.022 × 10 23 , called Avogadro’s number after the 19th-century chemist Amedeo Avogadro, is the number we use in chemistry to represent macroscopic amounts of atoms and molecules. Thus, if we have 6.022 × 10 23 Oxygen atoms, we say we have 1 mol of Oxygen atoms. If we have 2 mol of Na atoms, we have 2 × (6.022 × 10 23 ) Na atoms, or 1.2044 × 10 24 Na atoms. Similarly, if we have 0.5 mol of benzene (C 6 H 6 ) molecules, we have 0.5 × (6.022 × 10 23 ) C 6 H 6 molecules, or 3.011 × 10 23 C 6 H 6 molecules.

Derived SI Units

Derived units are combinations of SI base units. Units can be multiplied and divided, just as numbers can be multiplied and divided. For example, the area of a square having a side of 2 cm is 2 cm × 2 cm, or 4 cm2 (read as “four centimeters squared” or “four square centimeters”). Notice that we have squared a length unit, the centimeter, to get a derived unit for area, the square centimeter.


Volume is an important quantity that uses a derived unit. Volume is the amount of space that a given substance occupies and is defined geometrically as length × width × height. Each distance can be expressed using the meter unit, so volume has the derived unit m × m × m, or m 3 (read as “meters cubed” or “cubic meters”). A cubic meter is a rather large volume, so scientists typically express volumes in terms of 1/1,000 of a cubic meter. This unit has its own name—the liter (L). A liter is a little larger than 1 US quart in volume. (Table 1.4)  gives approximate equivalents for some of the units used in chemistry.) As shown in Figure 1.9 “The Liter”, a liter is also 1,000 cm 3 . By definition, there are 1,000 mL in 1 L, so 1 milliliter and 1 cubic centimeter represent the same volume.

1 mL = 1 cm 3

Volume figure

Figure 1.9:  The Liter.   A liter is defined as a cube that is 10 cm (1/10th of a meter) on a side.  A milliliter, 1/1000th of a liter, is equal to 1 cubic centimeter (1 cm 3 ).

Energy, another important quantity in chemistry, is the ability to perform work. Moving a box of books from one side of a room to the other side, for example, requires energy. It has a derived unit of kg·m 2 /s 2 . (The dot between the kg and m 2 units implies the units are multiplied together and then the whole term is divided by s 2 .) Because this combination is cumbersome, this collection of units is redefined as a joule (J), which is the SI unit of energy. An older unit of energy, the calorie (cal), is also widely used. There are:

4.184 J = 1 cal

Note that this differs from our common use of the big ‘Calorie’or ‘Cal’ listed on food packages in the United States. The big ‘Cal’ is actually a kilocalorie or kcal  (Fig 1.10)  Note that all chemical processes or reactions occur with a simultaneous change in energy and that energy can be stored in chemical bonds.


Figure 1.10: The Difference between kilocalories in Scientific and Common Use .  Calories represented on food packaging actually refer to kilocalories in scientific terms.

Density is defined as the mass of an object divided by its volume; it describes the amount of matter contained in a given amount of space.


Thus, the units of density are the units of mass divided by the units of volume: g/cm3 or g/mL (for solids and liquids, respectively), g/L (for gases), kg/m3, and so forth. For example, the density of water is about 1.00 g/mL, while the density of mercury is 13.6 g/mL. Mercury is over 13 times as dense as water, meaning that it contains over 13 times the amount of matter in the same amount of space. The density of air at room temperature is about 1.3 g/L.

Section 5: Making Measurements in the Lab

Precision vs. accuracy.

It is important to note the different terminology we use when talking in science. One such set of terminology is precision and accuracy. Although precision and accuracy are often used interchangeably in the non-scientific community, the difference between the terms is extremely important to realize. Precision tells you how close two measurements are to one another, while accuracy tells you how close a measurement is to the known value. A measurement can be precise while not being accurate, or accurate but not precise; the two terms are NOT related. A good analogy can be found in a game of darts (Fig. 1.11).  A player who always hits the same spot just to the left of the dart board would be precise but not very accurate. However, a dart player who is all over the board but hits the center of the board on average would be accurate but not precise. A good darts player, just like a good scientist, wants to be both precise and accurate.


Figure 1.11: Difference Between Accuracy and Precision.  A game of darts can be used to show the difference between accuracy and precision. 

Adapted from: https://upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Reliability_and_validity.svg/717px-Reliability_and_validity.svg.png

Typically within the laboratory, accuracy is a measure of how well your equipment is calibrated.  For example, if your balance is not calibrated correctly, you can make very precise, repeated measurements, but the measurements will not represent the true value. Precision, on the otherhand, is usually determined by how careful the scientist is in making measurements.  If you are careless and spill part of your sample on the way, your measurements in repeated experiments will not be precise even if your balance is accurate.

Significant Figures

It is important to realize that values in scientific measurements are never 100% accurate. Our instruments only measure to a certain level of accuracy.  Thus, we can pick different instruments to make a measurement based upon the level of accuracy we need for the experiment. Due to the inherent inaccuracy in any measured number we must keep track of the different levels of accuracy each number has with significant figures. Significant figures of a measured quantity are defined as all the digits known with certainty and the first uncertain, or estimated, digit. It makes no sense to report any digits after the first uncertain one, so it is the last digit reported in a measurement. Zeros are used when needed to place the significant figures in their correct positions. Thus, zeros may or may not be significant figures. Significant figures apply in the real world, as they allow us to quantify the accuracy of any type of measurement. To identify how many numbers in a measurement have significance, you can follow a discreet set of rules, shown below and to the right.


Figure 1.12: Measuring an Object to the Correct Number of Significant Figures. How many digits should be shown in this measurement?

The correct answer is 3! The two that you know for sure + the estimated position…for this reading it would be close to 1.37

Think about this

Exact numbers are numbers that are not measured by a scientific instrument.  They are either used as definitions to define a concept or terminology, or they are made by counting the total of something present.  An example of an exact number, would be the number of eggs in a carton or a defined unit such as there are 100 cm in 1 m. Exact numbers, such as the number of people in a  room,  DO NOT affect the number of significant figures in calculations made with measured values.


Rules of Rounding

In scientific operations, the rules of rounding may be a little bit different than the ones you are used to using.  Normal rounding rules suggest that if a number is 4 or below, it should be rounded down to the lower number, whereas if it is 5 or higher, it should be rounded up.  However, note that 5 is right in the middle and causes a problem when using these conventional rounding rules.  If you have a large dataset of numbers that you need to round, using this rounding rule will lead to bias in your dataset (i.e. 4/9th of the time you will be rounding down, and 5/9th of the time you will be rounding up).  In a large dataset, this bias is unacceptable.

In Scientific Rounding, we typically use a rule called ‘Rounding to the Even.’ In this rounding system the rules are the same for 4 and below, you round down to the lower number, and for 6 and above you round up to the higher number.  However, if the number you are rounding is 5, then you round to the even number.  This helps to alleviate the sample bias that can occur when rounding large datasets.


Calculations with Significant Figures

The first thing to realize before performing any calculations in science is that all measured numbers are only as good as the instrument used to measure them. Even with the best instrument available the measured number will never be 100% exact. Scientists use the “good enough” rule of precision, meaning that we accept an inherent amount of imprecision from every measurement we take as long as the final result is close enough to where we want it to be. This concept becomes dangerous when we begin to use these “good enough” numbers for any calculations, if we aren’t careful to keep track of our significant figures our numbers can quickly lose their “good enough” status. To protect their “good enough” numbers, the scientific community has set forth certain rules for performing any calculations; in this section we need only concern ourselves with two very important rules: the Addition/Subtraction rule, and the Multiplication/Division rule.

Addition/Subtraction Rule:

  • Find the number with the least number of decimals and keep track of the number of decimal places
  • Perform the addition/subtraction
  • Round the final answer to the least number of decimals found in Step 1

1 2 1 the importance of units worksheet answers

Multiplication/Division Rule:

  • Count the number of significant figures in each number (keep track of the number of significant figures)
  • Perform the multiplication/division
  • Round your final answer to the lowest number of significant figures found in step 1

1 2 1 the importance of units worksheet answers

Calculating Complicated Problems:

  • Using the order of operations, break the problem up into multiple steps
  • Perform any addition/subtraction steps following the Addition/Subtraction rule (Do not round yet, just keep track of the correct number of decimals when finding the number of significant figures)
  • Perform multiplication/division using the Multiplication/Division rule
  • Round the final answer to the correct number of significant figures

Conversions and the Importance of Units

The ability to convert from one unit to another is an important skill. For example, a nurse with 50 mg aspirin tablets who must administer 0.2 g of aspirin to a patient, needs to know that 0.2 g equals 200 mg, so that 4 tablets are needed. Fortunately, there is a simple way to convert from one unit to another.

If you learned the SI units and prefixes described in Section 1.4 Units of Measurement”, then you know that 1 cm is 1/100th of a meter or:

100 cm = 1 m

Suppose we divide both sides of the equation by 1 m (both the number and the unit; Note that it is critically important to always write out your units!  This avoids confusion and mistakes when making conversions.):

As long as we perform the same operation on both sides of the equals sign, the expression remains an equality. Look at the right side of the equation; it now has the same quantity in the numerator (the top) as it has in the denominator (the bottom). Any fraction that has the same quantity in the numerator and the denominator has a value of 1:


We know that 100 cm is 1 m, so we have the same quantity on the top and the bottom of our fraction, although it is expressed in different units. A fraction that has equivalent quantities in the numerator and the denominator but expressed in different units is called a conversion factor


Note that conversion factors can be written with either term in the numerator or denominator, and used as appropriate for the problem that you want to solve. This is because, both terms are equal to 1

1 2 1 the importance of units worksheet answers

Here is a simple example. How many centimeters are there in 3.55 m? Perhaps you can determine the answer in your head. If there are 100 cm in every meter, then 3.55 m equals 355 cm. To solve the problem more formally with a conversion factor, we first write the quantity we are given, 3.55 m. Then we multiply this quantity by a conversion factor, which is the same as multiplying it by 1. We can write 1 as 100cm/1m  and multiply:


Because m, the abbreviation for meters, occurs in both the numerator and the denominator of our expression, they cancel out. The final step is to perform the calculation that remains once the units have been canceled. Note that it is CRITICAL to retain the right units in the final answer or it will not make sense. A generalized description of this process is as follows:  

quantity (old units) × conversion factor = quantity (new units)

You may be wondering why we use a seemingly complicated procedure for a straightforward conversion. In later studies, the conversion problems you will encounter will not always be so simple. If you can master the technique of applying conversion factors, you will be able to solve a large variety of problems. In the previous example, we used the fraction 100 cm/1 m as a conversion factor. Does the conversion factor 1 m/100 cm also equal 1? Yes, it does; it has the same quantity in the numerator as in the denominator (except that they are flip-flopped). Why did we not use that conversion factor? If we had used the second conversion factor, the original unit would not have canceled, and the result would have been meaningless. Here is what we would have gotten:               



You can see that none of the units cancelled out. For the answer to be meaningful, we have to construct the conversion factor in a form that causes the original unit to cancel out. Figure 1.13 “A Concept Map for Conversions” shows a concept map for constructing a proper conversion.

conversion figure

Figure 1.13 A Concept Map for Conversions.  This is how you construct a conversion factor to convert from one unit to another.

Chapter Summary


Chapter 1 materials have been adapted and modified from the following creative commons resources unless otherwise noted:  1. Anonymous. (2012) Introduction to Chemistry: General, Organic, and Biological (V1.0). Published under Creative Commons by-nc-sa 3.0. Available at: http://2012books.lardbucket.org/books/introduction-to-chemistry-general-organic-and-biological/index.html 2. Poulsen, T. (2010) Introduction to Chemistry. Published under Creative Commons by-nc-sa 3.0. Available at: http://openedgroup.org/books/Chemistry.pdf 3. OpenStax (2015) Atoms, Isotopes, Ions, and Molecules: The Building Blocks. OpenStax CNX.Available at:  http://cnx.org/contents/be8818d0-2dba-4bf3-859a-737c25fb2c99@12 .

  • Chapter 1: Measurements in Chemistry
  • Chapter 6 – Quantities in Chemical Reactions

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Homeschool Science Corner: The Importance of Units of Measurement

May 14, 2013 by Paige Hudson

Units of Measurement | Homeschool Science Corner at Elemental Blogging

That’s why from the very first time I introduce measurements to my students I always emphasize the importance of not forgetting to write the units. I know from experience how key it is to lay a foundation for remembering these units of measurement early on.

Why are the units of measurement so important?

Knowing the units of measurement that correspond with a number can give you so much more information than a digit sitting there by itself. Units can:

  • Help to show another person the exact amount you have;
  • Assist in solving a mathematical problem, especially in chemistry, where you can follow the units to get to the answer;
  • Show which measurement system the person is using (i.e. metric or standard).

In a nutshell, the unit of measurement in science serves as the supporting pillar upon which a number rests.

What are the two main systems of measurement?

There are two main systems of measurement used in today’s nations:

  • The Metric System –  This system is used in most of the world and it employs units like meters, grams and liters. The system is base 10 and the names are formed with prefixes. It was derived from one of the early French measuring systems.
  • The Standard or Standard American Engineering (SAE) System –  This system is mainly utilized in the United States and it contains units like inches, pounds and gallons. It was derived from an early English measuring system that has its roots in the Roman system of measurements.

In the US, the SAE system of units is more widely used on consumer products and in industrial manufacturing, while the metric system is more widely used in science, medicine and government. So, it’s especially important for American students to be familiar with both systems .

What about converting units between systems?

Every student should know how to convert measurements inside of their most commonly used system of measurement.  By this I mean that they should be familiar with knowing how to convert grams to kilograms or ounces to pounds.  Normally, these conversion factors are taught as a part of their math program.  I also recommend that have your students memorize several basic conversion factors between the two systems . Here’s what every student should know:

  • Pounds to Kilograms: 1 kg = 2.2 lb
  • Gallons to Liters: 1 gal = 3.785 L
  • Feet to Meters : 1 foot = 0.305 m

Want a free set of worksheets to practice these conversions along with a conversion factor cheat sheet? Head to the Printables page !

How do you emphasize units of measurement outside of math class?

One of the key ways to stress the need for units to accompany numbers apart from math class is in the science lab. In other words, when the students record their results in an experiment, make certain that they include the units. In the beginning, you should also take the time to explain why those few letters are so necessary when they record their results .   Remember that  results  are the specific and measurable things that occur in your experiment, so they cannot be truly understood unless the numeral is paired with a unit of measurement.

Here are two more tips for highlighting units of measurement outside of math and science.

  • Call out the amount and the units when you measuring ingredients for a recipe. For example, say we need 1 cup (or 240 milliliters) of milk for this recipe, instead of we need 1 of that big one over there.
  • Talk about the units of distance when going for a ride in the car or when taking a walk. You can say it’s 5 miles (or 8 kilometers) to the store and that will take us about 10 minutes to get there.

There are many situations in our everyday life into which we can weave the concept of units of measurements. Doing so, will instill in our students an importance in the units of measurements that will spill over into their scientific studies.

by Paige Hudson

May 14, 2013 at 8:07 am

We are sticklers for including units of measurement here too. They can convert within each system, but I haven’t had them convert between systems. Thank you for mentioning and including those conversion factors. We will add them to our list of memory work. 🙂 I agree that there are many opportunities in every day life to discuss units of measure and we take the time to do so. One of our favorites is to discuss unit prices at the grocery store. We will look at the unit price, weigh our produce and then calculate the final cost based on the weight. Sometimes we try to estimate the weight as well before putting it on the scale. We also use the unit price to compare sale offers to see which is really a better deal.

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May 14, 2013 at 8:20 am

What a great idea to weave in math and units in the grocery store!

Here’s the last few that I didn’t include the factors for in the post in case you want to add them to your list as well: Miles to Kilometers: 1 mi = 1.61 km Cups to Milliliters: 1 c = 240 mL Inches to Centimeters: 1 in = 2.54 cm Ounces to Grams: 1 oz = 28.3 g

May 14, 2013 at 8:40 am

Thank you! I will add them to the list as well. 🙂

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Key to Measurements workbook series

Key to Measurements Workbooks

Key to Measurement workbooks include a variety of hands-on experiences related to the customary units of measurement. Group projects are included in addition to numerous individual activities. In Book 1, students learn how a linear measurement system is developed and then do activities related to measuring length. Book 2 focuses on length, perimeter, and area measures. In Book 3, the concept of area is further developed, and students are introduced to volume. Book 4 covers a variety of topics. Students experiment with weighing objects and measuring capacity, and they also learn about temperature and time.

Practice makes perfect. Practice math at IXL.com

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1.A: Units and Measurement (Answers)

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Check your Understanding

1.1. \(4.79×10^2\) Mg

1.2. \(3×10^8m/s\)

1.3. \(10^8km^2\)

1.4. The numbers were too small, by a factor of 4.45.

1.5. \(4πr^3/3\)

1.7. \(3×10^4m\) or 30 km. It is probably an underestimate because the density of the atmosphere decreases with altitude. (In fact, 30 km does not even get us out of the stratosphere.)

1.8. No, the coach’s new stopwatch will not be helpful. The uncertainty in the stopwatch is too great to differentiate between the sprint times effectively.

Conceptual Questions

1. Physics is the science concerned with describing the interactions of energy, matter, space, and time to uncover the fundamental mechanisms that underlie every phenomenon.

3. No, neither of these two theories is more valid than the other. Experimentation is the ultimate decider. If experimental evidence does not suggest one theory over the other, then both are equally valid. A given physicist might prefer one theory over another on the grounds that one seems more simple, more natural, or more beautiful than the other, but that physicist would quickly acknowledge that he or she cannot say the other theory is invalid. Rather, he or she would be honest about the fact that more experimental evidence is needed to determine which theory is a better description of nature.

5. Probably not. As the saying goes, “Extraordinary claims require extraordinary evidence.”

7. Conversions between units require factors of 10 only, which simplifies calculations. Also, the same basic units can be scaled up or down using metric prefixes to sizes appropriate for the problem at hand.

9. a. Base units are defined by a particular process of measuring a base quantity whereas derived units are defined as algebraic combinations of base units.

b. A base quantity is chosen by convention and practical considerations. Derived quantities are expressed as algebraic combinations of base quantities.

c. A base unit is a standard for expressing the measurement of a base quantity within a particular system of units. So, a measurement of a base quantity could be expressed in terms of a base unit in any system of units using the same base quantities. For example, length is a base quantity in both SI and the English system, but the meter is a base unit in the SI system only.

11. a. Uncertainty is a quantitative measure of precision. b. Discrepancy is a quantitative measure of accuracy.

13. Check to make sure it makes sense and assess its significance.

15. a. \(10^3\);

b. \(10^5\);

c. \(10^2\);

d. \(10^{15}\);

e. \(10^2\);

f. \(10^{57}\)

17. \(10^2\) generations

19. \(10^{11}\) atoms

21. \(10^3\) nerve impulses/s

23. \(10^{26}\) floating-point operations per human lifetime

25. a. 957 ks;

b. 4.5 cs or 45 ms;

27. a. 75.9 Mm;

29. a. 3.8 cg or 38 mg;

31. a. 27.8 m/s;

33. a. 3.6 km/h;

b. 2.2 mi/h

35. \(1.05×10^5ft^2\)

37. 8.847 km

39. a. \(1.3×10^{−9}m\);

b. 40 km/My

41. \(10^6Mg/μL\)

43. 62.4 \(lbm/ft^3\)

45. 0.017 rad

47. 1 light-nanosecond

49. \(3.6×10^{−4}m^3\)

51. a. Yes, both terms have dimension \(L^2T^{-2}\)

c. Yes, both terms have dimension \(LT^{-1}\)

d. Yes, both terms have dimension \(LT^{-2}\)

53. a. \([v] = LT^{–1}\);

b. \([a] = LT^{–2}\);

c. \([∫vdt]=L\);

d. \([∫adt]=LT^{–1}\);

e. \([\frac{da}{dt}]=LT^{–3}\)

c. \(L^0 = 1\) (that is, it is dimensionless)

57. \(10^{28}\) atoms

59. \(10^{51}\) molecules

61. \(10^{16}\) solar systems

63. a. Volume = \(10^{27}m^3\), diameter is \(10^9\) m.;

b. \(10^{11}\) m

65. a. A reasonable estimate might be one operation per second for a total of \(10^\) in a lifetime.;

b. about \((10^9)(10^{–17}s) = 10^{–8} s\), or about 10 ns

73. a. The number 99 has 2 significant figures; 100. has 3 significant figures.

c. percent uncertainties

77. 7.557 \(cm^2\)

79. a. 37.2 lb; because the number of bags is an exact value, it is not considered in the significant figures;

b. 1.4 N; because the value 55 kg has only two significant figures, the final value must also contain two significant figures

Additional Problems

81. a. \([s_0]=L\) and units are meters (m);

b. \([v_0]=LT^{−1}\) and units are meters per second (m/s);

c. \([a_0]=LT^{−2}\) and units are meters per second squared (\(m/s^2\));

d. \([j_0]=LT^{−3}\) and units are meters per second cubed (\(m/s^3\));

e. \([S_0]=LT^{−4}\) and units are \(m/s^4\);

f. \([c]=LT^{−5}\) and units are \(m/s^5\).

83. a. 0.059%;

c. 4.681 m/s;

85. a. 0.02%;

b. \(1×10^4\) lbm

87. a. 143.6 \(cm^3\);

b. 0.2 \(cm^3\) or 0.14%

Challenge Problems

89. Since each term in the power series involves the argument raised to a different power, the only way that every term in the power series can have the same dimension is if the argument is dimensionless. To see this explicitly, suppose \([x] = L^aM^bT^c\). Then, \([x^n] = [x]^n= L^{an}M^{bn}T^{cn}\). If we want \([x] = [x^n]\), then an = a, bn = b, and cn = c for all n. The only way this can happen is if a = b = c = 0.

Contributors and Attributions

Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a  Creative Commons Attribution License (by 4.0) .

Units of Measure: Worksheets with Answers

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1A: Units, Measurement Uncertainty, and Significant Figures (Worksheet)

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Name: ______________________________

Section: _____________________________

Student ID#:__________________________

Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.

All scientists the world over use metric units. Since 1960, the metric system in use has been the Système International d'Unités , commonly called the SI units. These units facilitate international communication by discouraging use of units peculiar to one culture or another (e.g., pounds, inches, degrees Fahrenheit). But regardless of the units used, we want to have some confidence that our measured and calculated results bear a close relationship to the “true” values. Therefore, we need to understand the limits on our measured and calculated values. One way we convey this is by writing numerical answers with no more and no fewer than the number of digits that are justified by the limits of our ability to measure and know the quantity.

Learning Objective

  • Know the units used to describe various physical quantities
  • Become familiar with the prefixes used for larger and smaller quantities
  • Master the use of unit conversion (dimensional analysis) in solving problems
  • Appreciate the difference between precision and accuracy
  • Understand the relationship between precision and the number of significant figures in a number

Success Criteria

  • Associate units with physical quantities
  • Replace prefixes by multiplying by appropriate numerical factors
  • Be able to use dimensional analysis for unit conversions
  • Report computed values to the correct number of significant figures.

Système International d'Unités (SI Units)

The SI units consist of seven base units and two supplementary units. For now, we will only use the four base units listed below. Later we will talk about two others. We will never use the seventh unit (candela), a unit for luminous intensity.

Any other units can be constructed as a combination of fundamental units. For example, velocity could be measured in meters per second (written m/s or m × s –1 ), and area could be measured in units of meters squared (m 2 ). When a named unit is defined as a combination of base units, it is called a derived unit . For example, the SI unit of energy is the joule (J), which is defined as a kg × m 2 × s –2 . Note that when a unit is named for some scientist (e.g., Joule, Herz, Kelvin) the written name of the unit is not capitalized, but the abbreviation is capitalized.

All metric units can be related to larger or smaller units for the same quantity by use of prefixes that imply multiplication of the stem unit by certain powers of 10. The following prefixes are important to know.

Give the names and their abbreviations for the SI units of length, mass, time, and temperature.

The unit of volume is the liter (L). Why is this not a base SI unit? What kind of SI unit is it?

A student is asked to calculate the mass of calcium oxide produced by heating a certain amount of calcium carbonate. The student ’ s answer of 90.32 is numerically correct, but the instructor marks it wrong. Why?

Write the number of seconds in a day (86,400 s) in exponential notation, using a coefficient that is greater than 1 and less than 10. (This form is called scientific notation and is generally the preferred form of exponential notation, as explained below).

The diameter of a helium atom is about 30 pm. Write this length in meters, using standard scientific notation.

A cubic container is 2.00 cm on each edge. What is its volume in liters? What is its volume in milliliters (mL)? Are your answers reasonable?

Dimensional Analysis

Units can actually help in setting up and solving many problems by using a method called dimensional analysis (also called the factor-label method). In dimensional analysis, a problem is typically viewed as a conversion of a given value in given units into a new value in certain desired units. Mathematically, such problems take on the general form

\[ (\text{given quantity in } \cancel{\text{given units}}) \underbrace{\left(\dfrac{\text{wanted units}}{\cancel{\text{given units}}}\right)}_{\text{conversion factor} \equiv 1} = \text{wanted quantity in wanted units} \nonumber \]

The factor "wanted units/given units" is a conversion factor, which is always a fractional expression of an equivalence relationship between two different units. In carrying out the multiplication and division, the given units cancel out, leaving the wanted units. To apply dimensional analysis, follow this general problem-solving strategy:

  • Identify and record what is know, with its given units;
  • identify what is to be calculated with its units;
  • identify the concepts and/or relationships that connect the given information with what needs to be calculated;
  • set up the solution using unit relationships as one or more conversion factors, such that all units except those desired for the answer cancel;
  • do the mathematics;
  • check or validate your answer by asking yourself if it is a reasonable result.

How many inches is 2.00 cm, given that the inch is defined as exactly 2.54 cm?

  • We know the length in centimeters.
  • We want the length in inches.
  • 1 inch (in.) is exactly 2.54 cm (no uncertainty)
  • Possible conversion factor s are 1 in/2.54 cm and 2.54 cm/1 in) We are starting with cm and want to end up with in, so the first conversion factor will do the job. \[(2.00 \,cm) \underbrace{\left(\dfrac{1\,in}{2.54 \,cm}\right)}_{\text{conversion factor} \equiv 1} = \,? \nonumber \]
  • Do the mathematics. \[(2.0 \cancel{\,cm}) \underbrace{\left(\dfrac{1\,in.}{2.54 \cancel{\,cm}}\right)}_{\text{conversion factor} \equiv 1} = 0.787\, in. \nonumber \] Note that the centimeter units cancel, leaving the desired units of inches.
  • If 2.54 cm is an inch, then 2.00 cm should be a fraction of an inch. So, 0.787 in looks like a reasonable answer.

In general, how can you identify whether or not you have written the correct conversion factor for the problem?

One liter is 1.06 quarts (qt). Write two possible conversion factors from this relationship.

The posted speed limit is 60 mi/hr. You are doing 120 km/hr in your Porsche convertible that you just bought in Germany. Are you speeding? Explain. [1.0 mi = 1.6 km]

In the gym, you slip on two 45-lb barbell plates to a bar that weighs 45 lb. What is the mass of the set-up in kilograms? [1.00 kg = 2.20 lbs]

A table top is 36 in long and 24 in wide. What is the area of the table top in square meters? [1 in = 2.54 cm, exactly]

Accuracy and Precision

Measured quantities always have some experimental error. Therefore, measured quantities are regarded as inexact . The accuracy of a measured quantity is its agreement with a standard or true value. In reality, we generally cannot know the true value of something we wish to measure. We gain confidence that our measured value is close to the truth by repeating the measurement many times. If our repeated measurements yield a set of data that differ very little from each other, we have some confidence that the average of these measured values is close to the true value. The repeatability of the measurements is called its precision . In general, we assume that greater precision in a set of numbers makes it more likely that the average value will be accurate. However, it is possible for a very precise set of values to be inaccurate. For example, a scientist could make the same error in each of a set of measurements, which could happen if a key measuring device were miscalibrated. Conversely, it is possible that a set of widely scattered values (poor precision) could have an average value that is very close to the true value, therefore resulting in high accuracy.

We express the precision of a number by writing all the repeatable digits and the first uncertain digit from a measurement or calculation. The retained digits are called the significant figures (sig. figs.) of the number. The following rules should be used to determine the number of significant figures of a number and to establish the correct number of significant figures in the answer to a calculation.

  • \(2.620\) has 4 sig. figs.
  • \(50.003\) has 5 sig. figs.
  • \(103,000\) has 3 sig. figs.
  • \(103,000.\) has 6 sig. figs.
  • \(0.0012\) has 2 sig. figs.
  • \(0.00070\) has 2 sig. figs.
  • \(2.0070\) has 5 sig. figs.
  • \(1.2 \times 10^{-3}\) has 2 sig. figs.
  • \(7.0 \times 10^{-4}\) has 2 sig. figs
  • In multiplication and division, the answer may have no more significant figures than the number in the chain with the fewest significant figures.

\[\dfrac{(9.97)(6.5)}{4.321} = 15 \,\,\, \text{has 2 sig. figs.} \nonumber \]

  • When adding or subtracting, the answer has the same number of decimal places as the number with the fewest decimal places. The number of significant figures for the result, then, is determined by the usual rules after establishing the appropriate number of decimal places. \[3.0081 +7.41 = 10.4181 \approx 10.42 \,\,\, \text{has 2 decimal places and 2 sig. figs.} \nonumber \] The rules for addition and subtraction may radically alter the number of significant figures for the answer in a chain of mathematical calculations, as the following shows.
  • All integer fractions: \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{7}{9}\)
  • Counted numbers: "15 people"
  • Conversions within a unit system: 12 inches = 1 foot (exactly!)

Relationships between units in different unit systems are usually not exact:

  • 2.2 lb. = 1.0 kg 2 sig. figs.
  • 2.2046223 lb. = 1.0000000 kg 8 sig. figs.

But, the following inter-system conversion factors are now set by definition and are exact:

  • 2.54 cm / 1 inch (exactly)
  • 1 calorie / 4.184 Joules (exactly)

A way of getting around the ambiguity in significant figures for numbers like 103,000 is to use standard scientific exponential notation , consisting of a coefficient whose magnitude is greater than 1 and less than 10 multiplied by the appropriate power of ten. All digits in the coefficient are significant.

  • \(1.03 \times 10^5\) has 3 sig. figs.
  • \(1.030 \times 10^5\) has 4 sig. figs.
  • \(1.0300 \times 10^5\) has 5 sig. figs.
  • \(1.03000 \times 10^5\) has 6 sig. figs.

A one-gram standard reference weight is repeatedly weighed on an analytical balance. The readings from the balance are as follows:

  • \(1.003 \,g \)
  • \(0.9998 \,g\)
  • \(1.005 \,g \)
  • \(0.9995 \,g\)

Comment on the precision and accuracy of these data.

The same one-gram reference weight is weighed on another analytical balance. The readings from this balance are as follows:

  • \(0.9845 \,g\)
  • \(1.0114 \,g\)
  • \(0.9879 \,g\)
  • \(1.0208 \,g\)

The same one-gram reference weight is weighed on a third analytical balance. The readings from this balance are as follows:

  • \(1.237\, g\)
  • \(1.243 \,g\)
  • \(1.238\, g\)
  • \(1.245\, g\)

How is precision represented in reporting a measured value?

How many significant figures are there in each of the following numbers?

  • \(3.000 x 10^2 \)

Use your calculator to carry out the following calculations and report the answers to the correct number of significant figures:

  • \(x = (2)(39.0983) + (2)(51.996) + (7)(15.9994)\) (The first number in each multiplication is an integer)
  • \( x= \dfrac{1.44 \times 10^4}{2.40 \times 10^8}\)
  • \( x= \dfrac{(3.5 \times 10^{-5})(6.2 \times 10^{12}}{3.3 \times 10^{-15}}\)
  • \(x =\sqrt{(7.56 \times 10^{-5})(0.125)}\)
  • \(x = \left [ \dfrac{(0.5622)(3.20 + 8.111)}{621.25} \right]^{1/3}\)

A supermarket in London is selling cod for 12.98 £/kg. If the rate of exchange is $1.6220 = 1.0000 £, what is the price in dollars per pound? 1.000 kg = 2.205 lb

A hollow metal sphere has an outer diameter (o.d.) of 4.366 cm and an inner diameter (i.d.) of 4.338 cm. What is the volume of the metal in the sphere? Express your answer to the proper number of significant figures. [\(V_{sphere} = (4/3)\pi r^3\) ]

  • Measuring Units Worksheets

Measuring Units Worksheet

Children are keen on anything that is new and grabs their attention. They learn best when they are focused and engaged. This is why it is so important to give them the right resources for learning. Measuring units worksheet is an exciting topic because kids can get hands-on with measuring cups, teaspoons, tablespoons, scales, etc., One of the most important aspects of education is mathematics. This includes learning the units of measurement by practising the converting units of measurement word problems worksheets. The ability to measure helps the little ones become more successful in life by empowering them to make more knowledgeable decisions. Without measurement, it would be difficult to know how much an object weighs, how tall someone is, or what time it is.

Measuring Units for kids

Detailed list of free measuring units worksheet pdf for kids.

Kids need to learn about measuring units by regularly practising the measuring units worksheet. You have to teach children the difference between different measuring units by comparing units of measurement worksheets. You can help them get acquainted with measurements of volume, weight, and temperature. Parents should encourage children to measure various things in their environment. Converting units of measurement word problems worksheets help kids understand the difference between each measuring unit. Kids thoroughly enjoy practising these measuring units worksheet answers. Here is a comprehensive list of measuring units worksheets.

Download Converting Units of Measurement Word Problems Worksheets Download PDF

Download Comparing Units of Measurement Worksheets Download PDF

Suggested Article:  Measuring Length Worksheets

Importance of Practising Measuring Units Worksheets

It is essential to practise the measuring units worksheet. There are different types of measurements used in everyday life for measuring length, mass, volume and time. These measuring units worksheets with answers help the little ones identify the right unit for the given problem. Here are a few points that explain the importance of practising the measuring units worksheet.

  • Practising measuring units worksheets is essential for many reasons. It helps you know how one type of unit relates to another, or to work out the size of a number needed to make up a certain unit.
  • The more kids practise, the faster their brain automatically calculates this information when they need it.
  • Converting units of measurement word problems worksheets teach students about such things as how to read a tape measure, convert measurements from one system to another, and how to work with fractions.
  • It also teaches them about common mistakes people make when measuring and what they need to do if they forget which way is north on a compass.

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Frequently Asked Questions on Measuring Units Worksheet

Why are the measuring units worksheets important for kids.

The measuring units worksheet is essential for kids because it helps you know how one type of unit relates to another, or to work out the size of a number needed to make up a certain unit. The more kids practise, the faster their brain automatically calculates this information when they need it.

Is the measuring units worksheet useful for a Class 3 child?

Yes. The measuring units worksheet is helpful for a Class 3 child. Converting units of measurement word problems worksheets teaches students about such things as how to read a tape measure, convert measurements from one system to another, and how to work with fractions.

What are the types of measuring units worksheets?

The types of measuring units worksheet are, converting units of measurement word problems worksheets and comparing units of measurement worksheets.

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  1. Measuring Units Worksheet Answer Key

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