basic quantitative thinking skills - bates...

Bohlen and Austin Text PRELIMINARY DRAFT 1/6/2005

Basic QuantitativeThinking Skills

1.1 Quantitative Thinking in Environmental ScienceLike it or not, quantitative thinking forms the basis of most technical discussion ofenvironmental issues. Regulators express emissions standards in quantitative terms – andjustify them on the basis of elaborate models of risks to human health. Fisheries scientistsrely on mathematical models to determine population levels (and sustainable harvest levels)for fish populations. Engineers use equations that describe aspects of water flow in urbanenvironments to design stormwater management structures. Conservation biologists usepopulation dynamic models to guide management strategies for endangered species.

Science and associated quantitative methods form a dominant mode of discourse, both inenvironmental science and in many other fields of modern life. Like it or not, familiarity withthe conventions and principals of quantitative thinking is essential for participating in a largepart of today’s discussions of environmental issues. If you want to be a full participant in theenvironmental policy debates in the 21st century, you will need, at a minimum, to be able tounderstand – and criticize – the quantitative arguments marshaled in defense of one or anotherpolicy proposals.

Most fields of modern science rely on quantitative reasoning in one form or another, andenvironmental science is no exception. But environmental science is an interdisciplinaryfield, in which scientists trained in a variety of disciplines take part. The conventions for useof quantitative thinking and, perhaps more importantly, the conventions for how quantitativeresults are communicated, vary from discipline to discipline. Ecologists, for example, arefrequently well trained in multivariate statistical methods, while hydrologists may have littlestatistical training, but have a thorough grounding in mathematical models of watermovement. Environmental chemists, engineers, atmospheric physicists and so on each bringtheir own particular approach to quantitative thinking to their environmental work.

What then, forms the core of quantitative thinking skills for environmental scientists? Whilewe suspect that no two environmental scientists would completely agree on this question, wethink a fundamental foundation in quantitative reasoning includes the following:

1. Familiarity with common conventions of quantitative presentation in thesciences, such as use of the metric system, understanding of scientificnotation and comfort with the concept of significant figures.

2. Facility with basic skills of numerical manipulation as used in the sciencesincluding facility with unit conversions and dimensional analysis, andfamiliarity and even a degree of comfort with use of exponents and logs.

3. Understanding of the use of models to elucidate the logical implications oftheory, and to express those consequences in way that permit testing oftheoretical ideas.

4. Understanding of statistics to the extent of appreciating the role ofuncertainty in the sciences, grasps the relationship among different sources


of uncertainty (such as environmental heterogeneity and measurementerror), and can think intelligently about how to design a study to collectreliable environmental data.

Many additional quantitative skills are used in environmental science, but beyond a relativelysmall core, the particular skills vary from discipline to discipline. Ecologists might go on toget extensive training in statistical methods, while engineers would be more likely to studystatics and strength of materials. Chemists would work with thermodynamics and kinetics ofchemical reactions, while hydrologists would study models of fluid flow through porousmedia.

This chapter is intended to help increase your quantitative literacy, specifically in the contextof environmental sciences. It is divided into five sections:

1. This introduction,

2. A review of fundamental skills of quantitative thinking in the sciences

3. A discussion of the role of mathematical models in science

4. A discussion of a particular class of mathematical models called stock andflow models that are widely used in science, and

5. An introduction to statistical principals

In keeping with the overall goals of this textbook, the material we present here should setyou up to improve your ability to think critically about environmental issues

2 Tools and Tricks: Fundamental Skills for Quantitative Thinking in the Sciences

The skills we review in section 1.2 of this chapter form the basic building blocks of scientificcomputation. Because of the widely disparate backgrounds of students taking environmentalscience classes, it is likely that for some of you these skills will already be second nature. Forothers they will be vaguely remembered details from a science course taken back in highschool. But for a few of you, they may entirely new.

Even if you are familiar with the main concepts covered in this section, a review of thematerial is worthwhile, if only because it can provide you with perspective on the way that thequantitative communication in the sciences have evolved to be a richly interconnected web ofideas. For example, one cannot really understand significant digits without firstunderstanding scientific notation, which in turn requires that you understand exponents andlogarithms. This presents both a challenge to those of you who are new to quantitativecommunication – you will find that the pieces of the system only make clear sense in thecontext of understanding the whole – and also an opportunity. If you persevere, and work tounderstand the tips and tricks we discuss in this section, you will find that the principals ofthis type of reasoning will become easier to remember as your understanding becomes morecomplete.

Examples of the application of this thinking

2.1 Exponents and LogarithmsA clear understanding of logarithms and exponents (antilogarithms) is necessary to achieveany fluency with quantitative reasoning, and real comfort with the concepts that underlie themis of great help in developing your “back of the envelope” thinking skills.


There are relatively few rules for working with logs and anti-logs, and all of them can bededuced directly or indirectly from the basic definition of exponentiation.

2.1.1 ExponentsWe will derive basic relationships for exponentiation first, then explore related properties oflogarithms. We include this material here on the principal that if you can derive importantproperties of exponents from first principals, you will not have to simply memorize them. Forany number a, a raised to the nth power is simply equal to a multiplied times itself n times.

Eqn 1.1:

One can readily deduce the most important properties of exponents from this definition. Inthe following examples, the following definitions apply. (The subscripts here are used merelyto count the number of terms being multiplied. The ai are equal to a for all values of i.).

Eqn 1.2( )( )mm

m

nnn

aaaaa

aaaaa

××××=

××××=

−

−

121

121

L

L

Example 1: Product of two exponents

Eqn 1.3( ) ( )

mnmn

mmnnmn

aaa

aaaaaaaaaa+

−−

=⋅

××××⋅××××=⋅ 121121 LL

Example 2: Ratio of two exponents

Eqn 1.4

Example 3: Meaning of a negative exponentUsing the result of equation Eqn 1.4, and realizing that the logic we applied works equally form>n as for n>m, one can readily determine the meaning of a negative exponent.


Eqn 1.5 ==== −−nn

nn

aa

aaa

100

Other Useful Properties of ExponentsDerivations of the following properties use similar logic, and will be left to the student. Thislist of properties expressed as equations may look overwhelming, but most are simpleconsequences of the definition of exponentiation, and should be self evident if you understandthe principals involved. We offer this list primarily for reference if you have not usedexponentiation recently

Eqn 1.6 ( ) mnnm aa =

Eqn 1.7 ( ) nnn baab =

Eqn 1.8 )0 (provided ≠==

− bbab

a

b

a nnn

nn

Eqn 1.9 )0 (provided1

1

≠=

=

=

=

−−

−−

a,bbaa

b

b

ab

a

b

a nnn

n

n

n

n

nn

Eqn 1.10

2.1.2 LogarithmsStudents frequently have a bit more trouble working with logarithms than they do workingwith exponents. However the two are directly related, and one can derive the major propertiesof logarithms if one keeps the relationship clear. Logarithms are simply the mathematicalinverse of exponentiation. Lets put that concept into a formal definition. In the followingrelationship, a is known as the base of the logarithm.

Eqn 1.11

( )

( ) ( )ya

y

ya

ay ay

axxy

a log and

that implies This

ifonly and if log

log ==

==

Common Notational ConventionsThe notation “log (x)”, written without any subscript denoting the base of the logarithm,

almost always denotes log base 10. (i.e. ( )xx 10log)log( = ). The so-called “natural

logarithm” is the logarithm with a base equal to the irrational number 71828.2≅e .Natural logs turn up in a variety of mathematical contexts and they are conventionally

denoted as ( )xx elog)ln( = . We will follow these conventions in this book.


Example 1: The Logarithm of the product of two numbersWhat is the log of the product of two numbers? The answer is surprisingly simple, and iscentral to the importance of logarithms in many areas of science.

Eqn 1.12

Other Useful Properties of Logarithms(Proofs left to the reader).

Eqn 1.13 ( ) 0a allfor 01log >=a

Eqn 1.14 ( ) 0a allfor 1log >=aa

Eqn 1.15 ( ) ( )yxy

xaaa logloglog −=

Eqn 1.16 ( ) ( )xyx ay

a loglog =

Eqn 1.17 ( ) ( )( )

.base, convenientany for ,log

loglog c

b

aa

c

cb =

(This last relationship is often called the base change formula)

General CommentMuch of the value of logs is a direct result of properties Eqn 1.15 and Eqn 1.16. These twoproperties permit many calculations that involve multiplication and division to be replacedwith the simpler operations of addition and subtraction. These simple properties of logsunderlie the simplicity of the slide rule, a device for rapid calculation of multiplication anddivision that was made obsolete by invention of inexpensive pocket calculators. Logs alsopermit compact display or comparison of data that covers a very large range of values. Avariety of measurable quantities, from acidity of aqueous solutions, to noise levels, to theseverity of earthquakes are conventionally reported in values derived from a logarithmicscale.

2.2 Scientific NotationPerhaps the single most common area in which you will be faced with working withexponents and logs is in dealing with very large or very small numbers in a compact fashion.Scientists and engineers use various forms of “scientific notation” to deal with this situation.Scientific notation logically separates a numerical value into two parts, a coefficient and apower of ten. The coefficient is generally a number with a single digit to the left of thedecimal place (that is it has a value x such that ). The power of ten (or exponent)is always an integer. The product of the coefficient and 10 raised to the exponent produces theoriginal numerical value. Many computer programs and calculators express very large and


very small numbers in a form of scientific notation. Often such numbers are written as thecoefficient, followed by a capital letter “E”, followed by the exponent. Some examples mayhelp clarify the concept.

Eqn 1.18

)6E 43.5 aswritten (sometimes 1043.500000543.0

)6E 697.2 as written (sometimes 10697.2000,697,2

)2E 0345.1 as written (sometimes 100345.145.103

6

6

2

−×=

×=

×=

−

In effect, scientific notation acts to “shift the decimal”, allowing us to do much of our mathwith values between 1 and 10, determining the order of magnitude of the result at the end ofour calculations. While this is of considerable value even today (especially in the context of“back of the envelope” calculation), it was of critical importance in the days before readyavailability of computers and calculators. Slide rules made it relatively quick and easy tomultiply and divide numbers between 1 and 10. But that technology would have been of littleuse without a simple way to handle multiplication and division of larger and smaller numbers.Scientific notation provided a way to transform many calculations into calculations involvingnumbers with only a single digit to the left of the decimal. Today, facility with scientificnotation is part of scientific literacy. Understanding of this notational convention – whileperhaps less critical today than a generation ago – helps to make the metric system of unitsmore comprehensible, aids with rapid calculation, and helps to structure thinking about the“order of magnitude” of quantities of scientific or engineering interest.

2.2.1 Mathematical Operations in Scientific NotationScientific notation makes certain calculations more convenient, but if you are not used toworking in scientific notation, it can take a little getting used to. The rules for calculating inscientific notation are a consequence of the way scientific notation is defined. You should beable to figure these rules out for yourself with a little thought. To describe the basicmathematical operations, we need to define the variables a and b as follows:

Eqn 1.19q

p

yb

xa

10

10

×=

×=

Addition (or Subtraction)We cannot directly add the values of the coefficients, x and y, because they are eachmultiplied by different powers of ten. The coefficient x might be multiplied by ten to producea, while y might be multiplied by millions to produce b. We must write a and b so that theyare both expressed as a value multiplied by the same power of ten. Luckily, this is generallynot too difficult to do.

Lets express the value of a in terms that match the power of ten in which b has beenexpressed, namely the exponent q.

Eqn 1.20( )

( ) qqp

qqp

xa

xa

1010

1010

××=

××=−

−

Now we are in a position to add a and b, expressing both in terms of the same exponent.


Eqn 1.21

Again, an example may help.

Eqn 1.22( )

( ) 86471047.861052.895.77

1052.81095.77

1052.81010795.7

1052.810795.7

22

22

221

23

=×=×+

×+×

×+××

×+×

Multiplication

Eqn 1.23 qpyxba +××=× 10)(

Division

Eqn 1.24

2.2.2 Orders of MagnitudeScientists often talk about the “order of magnitude” of a value. The order of magnituderefers to the value of the exponent of a numerical value expressed in scientific notation. But itis even more common for scientists to speak of two numbers as differing by a certain numberof orders of magnitude. Two values that differ by an order of magnitude differ by a factor often (101). Values that differ by three orders of magnitude differ by a factor of a thousand(103). This terminology is used in settings in which it does not make sense to pay too muchattention to the exact value of two numbers. For example, the mass of a marble (which issomewhere under 100 g) and the mass of a large automobile (which is about 1 metric ton, or1000 kg, or 1,000,000 g) differ by approximately six orders of magnitude. Given these hugediscrepancies in mass, for most practical problems, it really would not be too important toknow the exact weight of the marble. The terminology is used as a sort of shorthand thatgives scientists a quick reference for the approximate relative magnitude of different values.

2.3 Significant DigitsNumbers can express quantities with arbitrary precision (you can always just keep addingdigits to the right of the decimal place). Unfortunately, we can measure any real-valuedquantity, and many integer-valued quantities with only limited precision, so it is easy toexpress values with more precision than they deserve. Somehow, scientists mustcommunicate to one another the actual precision with which numbers have been measured orestimated. The most complete way of doing that is to report each value with a quantitativeestimate of the uncertainty of that number (such as its standard error), but in many contextsthat is both labor intensive and unnecessary. The convention of reporting numbers with anappropriate number of significant digits evolved as a second, less laborious (albeit lessprecise) way of keeping track of and communicating the precision of numbers.

The idea behind the convention of significant digits is simple. If there is substantialuncertainty in the value you report at a certain order of magnitude (e.g. tens, hundreds, orthousands), don’t report any digits at a lesser order of magnitude. The digits that count assignificant are those that would appear in the coefficient if the number were written in


scientific notation, with the proviso that zeros can be added to the right of the decimal toindicate increased precision. Similar rules apply to numbers larger than one, but becausezeros are often used to indicate place (and thus powers of ten) the usage can be ambiguous.

Eqn 1.25

2.3.1 Significant Digits in Measured QuantitiesThe rules for determining how many significant digits to use in your own work are easiest tounderstand in the context of taking measurements. As a general rule, one should recordmeasurements to the first digit that you must estimate, given the measuring technology beingemployed. In other words, you record all the digits about which you are reasonably certain,and just the first digit about which you have some uncertainty. Applying this standardfrequently calls for a little judgment.

Most commercially available meter sticks (the metric equivalent of a yardstick) have divisionsthat one can interpret to the nearest millimeter. One may be able to estimate measurements ata slightly finer resolution by interpolating between the divisions. But manufacturingtolerances for meter sticks are often poor, and division marks on poorly made ones areinaccurately printed. Moreover, meter sticks are often used in field science to make quickmeasurements of variable or imprecisely defined properties such as water depth or height ofvegetation. Thus depending on the accuracy with which the meter stick was produced and thetype of measurement being taken it might be appropriate to record measurements to thenearest centimeter (as for measuring the depth of water in a rapidly flowing stream) or nearestmillimeter (length of a plant stem or leaf). Finer resolution (while superficially possible byinterpolating between the divisions on the scale) will seldom be appropriate.

2.3.2 Significant Digits in Calculated QuantitiesMany quantities in science are not measured directly, but calculated from other measuredvalues, and here the rules for determining the number of significant digits to report in yourwork are more complex. For example, specific gravity is calculated by combining ameasurement of volume (which has a certain accuracy) with a measurement of mass (also oflimited precision). The rules for determining the number of significant digits in calculatedvalues differ for addition or subtraction and multiplication and division.

Addition (and Subtraction)The result can be no more accurate (in value of the least significant digit, not number ofdigits) than the least accurate of the values added together. Thus the result of the addition oftwo values is only as accurate as the less accurate of the two numbers being added together.To give you an example, it only makes sense to add a weight of a few grams a weightexpressed in metric tonnes (1000 kg, or 1 x 106 g) if the weight in tonnes was measured to thenearest gram. If it was measured only to the nearest kilogram, it would makes little sense toadd just a few grams to that amount, since the measurement of the larger object’s mass couldeasily be off by several hundred grams.


Eqn 1.26 4.1051.10427.1 =+

Addition is only as accurate as the less accurate number being added together. The moreaccurate number – 1.27 in this case – is rounded to match the less accurate one beforecarrying out the addition.

Multiplication and DivisionThe results of both multiplication and division have the same number of significant digits asthe LESS ACCURATE of the numbers you started with.

Eqn 1.27

digit)t significan (one 4.0.5

95.1

digits)t significna (two 39.00.5

95.1

1032.1132103.10427.1 2

=

=

×==×

Often, these rules will appear counter intuitive, especially when it means reporting a resultwith fewer significant figures.

Eqn 1.28

2.3.3 Complex CalculationsIn complex calculations, including many statistical calculations, if you are using a computeror pocket calculator, you should carry out your computations with all available digits, andround the result to the appropriate number of significant digits only at the end of yourcalculations. Rounding of intermediate results can introduce significant numerical errors.

2.4 The Metric System and SI UnitsThe system of measurements commonly called the “metric system” is the standard set of unitsfor scientific activity worldwide. The modern incarnation of this internally consistent systemof scientific units is the International System of Units (or System Internationaledes Unités,leading to the abbreviation SI). SI units now dominate international commerce and industry,and are the dominant units for engineering practice, at least outside of the United States.Becoming comfortable with SI units will take you a long way towards being able to be a fullparticipant in technical discussion on environmental issues.

For Americans who have grown up using Imperial units (feet, gallons, pounds), learning towork with metric units is very much like learning to use a foreign language. As you are firstworking with these units, you may find yourself mentally translating from metric to imperialunits and back again. As you gain fluency in use of metric units, you will find it easier tothink in terms of the metric units directly. If you go on to use metric measurements on aregular basis, you may even find yourself wanting to buy about _ kg (instead of about 1pound) of eggplant for dinner.


2.4.1 System International UnitsSI units are defined by international agreements going back to 1875. Changes in the SIsystem needed to reflect modern technologies and changes in scientific understnanding aremade under the auspices of an international convention. All SI units are all based on sevenbasic units. These units are assumed to be independent of one another in the sense that onecould (at least in principal) redefine the unit of any one quantity without affecting the othersix. The seven basic quantities and the SI units used to measure them are given in thefollowing table:

SI Base Units

Base quantity Name Symbol

length meter m

mass kilogram kg

time second s

electric current ampere A

thermodynamic temperature Kelvin K

amount of substance mole mol

luminous intensity candela cd

Derived UnitsAll other SI units are defined as combinations of the seven basic units. Volume hasdimensions of length cubed, so the most natural unit of measurement of volume is the metercubed. Velocity has dimensions of distance per unit time, and thus is most directly measuredin meters per second. Some combinations of the seven basic units are used so frequently that,as a matter of convenience, they have been given names. Examples include the coulomb (aunit of electrical charge, equal to A·s), The joule (a unit of energy, equal to kg·m2s-2), and theVolt ( a measure of electrical potential, equal to m2·kg·s-3·A-1).

Using Prefixes to Rescale Basic and Derived UnitsMany basic and derived units turn out to be unwieldy in certain situations, even in everydaypractice. For example, it becomes tedious to write travel distances, such as the distance fromPortland, Maine to Washington, D.C. in terms of meters. We would tend to write the distancebetween the two cities as 890 kilometers, not as 890,000 meters. The situation gets even moreunwieldy for astronomers, measuring interstellar distances, or atomic physicists, estimatingdistances within atoms. The SI system of units include an internationally agreed upon list ofprefixes that permit people to scale both basic and derived units for greater convenience forparticular purposes.

The prefixes are given in the following table. Note that the values on the left of the table areused to build new units that are larger than the basic SI units, while the prefixes on the rightare used to build units that are smaller that the basic SI units. The symbols used for theprefixes that correspond to factors greater than a factor of 1000 (103) are all capitalized.That’s easy to remember – big letters indicate big units.

We can see from the table on the next page that the familiar unit of distance, the kilometer,corresponds to 103 meters, or 1000 meters, while the millimeter corresponds to 10-3 meters.Wavelengths of light are often measured in nanometers, which correspond to 10-9 meters, orone billionth of a meter. Thus we could express the distance between Portland, Maine andWashington, D.C. in any of the following ways (note that all values are expressed to only twosignificant digits). Of course, some of these are much more convenient than others – which isthe whole point of using the prefixes.


Eqn 1.28

Prefixes for Large Units Prefixes for Small Units

Factor Name Symbol Factor Name Symbol

1024 yotta Y 10-1 deci d

1021 zetta Z 10-2 centi c

1018 exa E 10-3 milli m

1015 peta P 10-6 micro µ

1012 tera T 10-9 nano n

109 giga G 10-12 pico p

106 mega M 10-15 femto f

103 kilo k 10-18 atto a

102 hecto h 10-21 zepto z

101 deka da 10-24 yocto y

One can use prefixes with any of the named basic and derived units of the SI. The onlyexception (and it is a fairly self-evident one) is for units of mass. One should name units ofmass with reference not to the basic unit of mass, the kilogram, but with reference to thegram, even though the gram is not formally one of the seven basic units of the SI. Thus indiscussions of the global carbon cycle, one would speak in terms petagrams of carbon, notterakilograms.

Additional Units not Formally Part of SIMany units that are used everyday in environmental science are not formally part of the SIsystem, but are currently accepted by international authorities for use with the SI units. Theseinclude such units as the liter (1000 cm3, or 1 dm3, or 10-3 m3), the metric ton (sometimeswritten tonne, equal to 1000 kg, or 1 Mg), the minute (60 seconds), and the hectare (the areaof a square 100m on a side, 10000 m2, or 0.01 km2).

For Further InformationThe U.S. National Institute of Standards and Technology maintains a web site that providesadditional details on SI units at the URL: http://physics.nist.gov/cuu/Units/index.html

2.5 Dimensional AnalysisDimensional analysis is a profoundly useful technique for reasoning from what you know towhat you need to know. It can also be used to generate insights unavailable any other way. Itplays an especially important role in certain fields. For example, in fluid dynamics,dimensionless coefficients (Reynolds number, Froude number, etc.) capture important aspectsof fluid flows.

The basic principal of dimensional analysis is simple. If you have an equation relating twoquantities, the dimensions (length, mass, time, etc.) of the quantities on the two sides of theequation must be similar.


2.5.1 Don’t Memorize Formulas – Learn PrincipalsStudents beginning to learn quantitative subjects often find themselves memorizing formulas.But memorization has a severe disadvantage – you tend to forget what you memorize ratherquickly. You may get through the final exam, but you are unlikely to have effective workingknowledge of the material a year or two after you finish the course.

If you learn underling principals and a few simple approaches for reasoning from what youDO know to what you need to know, you will retain working knowledge quite a bit longer.Furthermore, using these strategies for learning will help you better understand the materialand provide a quick way for you to check or supplement your memory during exams.

Because of the interdisciplinary nature of environmental science, it is likely that many of youwill be called upon at some time in your career – whether as scientists, policymakers, orinformed citizens – to reason about something you have not studied in years. Thus it isimportant for you to retain – or be able to regenerate – your working knowledge. In thiscontext, dimensional analysis is an extremely valuable tool for being able to generate orregenerate working understanding.

2.6 Unit Conversions

basic quantitative thinking skills - bates...

Documents