Basic Quantitative
Thinking Skills
1.1 Quantitative Thinking in Environmental Science
Like it or not, quantitative thinking forms the basis of most technical discussion of environmental issues. Regulators express emissions standards in quantitative terms – and justify them on the basis of elaborate models of risks to human health. Fisheries scientists rely on mathematical models to determine population levels (and sustainable harvest levels) for fish populations. Engineers use equations that describe aspects of water flow in urban environments to design stormwater management structures. Conservation biologists use population dynamic models to guide management strategies for endangered species. Science and associated quantitative methods form a dominant mode of discourse, both in environmental science and in many other fields of modern life. Like it or not, familiarity with the conventions and principals of quantitative thinking is essential for participating in a large part of today’s discussions of environmental issues. If you want to be a full participant in the environmental policy debates in the 21st century, you will need, at a minimum, to be able to
understand – and criticize – the quantitative arguments marshaled in defense of one or another policy proposals.
Most fields of modern science rely on quantitative reasoning in one form or another, and environmental science is no exception. But environmental science is an interdisciplinary field, in which scientists trained in a variety of disciplines take part. The conventions for use of quantitative thinking and, perhaps more importantly, the conventions for how quantitative results are communicated, vary from discipline to discipline. Ecologists, for example, are frequently well trained in multivariate statistical methods, while hydrologists may have little statistical training, but have a thorough grounding in mathematical models of water
movement. Environmental chemists, engineers, atmospheric physicists and so on each bring their own particular approach to quantitative thinking to their environmental work.
What then, forms the core of quantitative thinking skills for environmental scientists? While we suspect that no two environmental scientists would completely agree on this question, we think a fundamental foundation in quantitative reasoning includes the following:
1. Familiarity with common conventions of quantitative presentation in the sciences, such as use of the metric system, understanding of scientific notation and comfort with the concept of significant figures.
2. Facility with basic skills of numerical manipulation as used in the sciences including facility with unit conversions and dimensional analysis, and familiarity and even a degree of comfort with use of exponents and logs. 3. Understanding of the use of models to elucidate the logical implications of
theory, and to express those consequences in way that permit testing of theoretical ideas.
4. Understanding of statistics to the extent of appreciating the role of
of uncertainty (such as environmental heterogeneity and measurement error), and can think intelligently about how to design a study to collect reliable environmental data.
Many additional quantitative skills are used in environmental science, but beyond a relatively small core, the particular skills vary from discipline to discipline. Ecologists might go on to get extensive training in statistical methods, while engineers would be more likely to study statics and strength of materials. Chemists would work with thermodynamics and kinetics of chemical reactions, while hydrologists would study models of fluid flow through porous media.
This chapter is intended to help increase your quantitative literacy, specifically in the context of 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 and
flow 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 set you 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 scientific computation. Because of the widely disparate backgrounds of students taking environmental science classes, it is likely that for some of you these skills will already be second nature. For others they will be vaguely remembered details from a science course taken back in high school. 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 the material is worthwhile, if only because it can provide you with perspective on the way that the quantitative communication in the sciences have evolved to be a richly interconnected web of ideas. For example, one cannot really understand significant digits without first
understanding scientific notation, which in turn requires that you understand exponents and logarithms. This presents both a challenge to those of you who are new to quantitative communication – you will find that the pieces of the system only make clear sense in the context of understanding the whole – and also an opportunity. If you persevere, and work to understand the tips and tricks we discuss in this section, you will find that the principals of this type of reasoning will become easier to remember as your understanding becomes more complete.
Examples of the application of this thinking
2.1 Exponents and Logarithms
A clear understanding of logarithms and exponents (antilogarithms) is necessary to achieve any fluency with quantitative reasoning, and real comfort with the concepts that underlie them is 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 be deduced directly or indirectly from the basic definition of exponentiation.
2.1.1 Exponents
We will derive basic relationships for exponentiation first, then explore related properties of logarithms. We include this material here on the principal that if you can derive important properties of exponents from first principals, you will not have to simply memorize them. For any 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. In the following examples, the following definitions apply. (The subscripts here are used merely to count the number of terms being multiplied. The ai are equal to a for all values of i.).
Eqn 1.2
(
)
(
m m)
m n n na
a
a
a
a
a
a
a
a
a
×
×
×
×
=
×
×
×
×
=
− − 1 2 1 1 2 1L
L
Example 1: Product of two exponents
Eqn 1.3
(
) (
)
m n m n m m n n m na
a
a
a
a
a
a
a
a
a
a
a
a
+ − −=
⋅
×
×
×
×
⋅
×
×
×
×
=
⋅
1 2L
1 1 2L
1Example 2: Ratio of two exponents
Eqn 1.4
Example 3: Meaning of a negative exponent
Using the result of equation Eqn 1.4, and realizing that the logic we applied works equally for
Eqn 1.5 −
=
−=
=
=
n n n na
a
a
a
a
1
0 0Other Useful Properties of Exponents
Derivations of the following properties use similar logic, and will be left to the student. This list of properties expressed as equations may look overwhelming, but most are simple consequences of the definition of exponentiation, and should be self evident if you understand the principals involved. We offer this list primarily for reference if you have not used exponentiation recently Eqn 1.6
( )
a
m n=
a
mn Eqn 1.7( )
ab =
na
nb
n Eqn 1.8
=
=
(provided
≠
0
)
a
b
−b
b
a
b
a
n n n n n Eqn 1.9(provided
0
)
1
1
≠
=
=
=
=
− − − −a,b
b
a
a
b
b
a
b
a
b
a
n n n n n n n n n Eqn 1.102.1.2 Logarithms
Students frequently have a bit more trouble working with logarithms than they do working with exponents. However the two are directly related, and one can derive the major properties of logarithms if one keeps the relationship clear. Logarithms are simply the mathematical inverse of exponentiation. Lets put that concept into a formal definition. In the following relationship, a is known as the base of the logarithm.
Eqn 1.11
( )
( )( )
y a y y aa
y
a
y
a
x
x
y
aand
log
that
implies
This
if
only
and
if
log
log=
=
=
=
Common Notational Conventions
The notation “log (x)”, written without any subscript denoting the base of the logarithm, almost always denotes log base 10. (i.e.
log(
x
)
=
log
10( )
x
). The so-called “natural logarithm” is the logarithm with a base equal to the irrational numbere
≅
2
.
71828
. Natural logs turn up in a variety of mathematical contexts and they are conventionally denoted asln(
x
)
=
log
e( )
x
. We will follow these conventions in this book.Example 1: The Logarithm of the product of two numbers
What is the log of the product of two numbers? The answer is surprisingly simple, and is central 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
log
a( )
1
=
0
for
all
a
>
0
Eqn 1.14log
a( )
a
=
1
for
all
a
>
0
Eqn 1.15
( )
x
( )
y
y
x
a a alog
log
log
=
−
Eqn 1.16
log
a( )
x
y=
y
log
a( )
x
Eqn 1.17
( )
( )
( )
,
for
any
convenient
base,
.
log
log
log
c
b
a
a
c c b=
(This last relationship is often called the base change formula)
General Comment
Much of the value of logs is a direct result of properties Eqn 1.15 and Eqn 1.16. These two properties permit many calculations that involve multiplication and division to be replaced with the simpler operations of addition and subtraction. These simple properties of logs underlie the simplicity of the slide rule, a device for rapid calculation of multiplication and division that was made obsolete by invention of inexpensive pocket calculators. Logs also permit compact display or comparison of data that covers a very large range of values. A variety of measurable quantities, from acidity of aqueous solutions, to noise levels, to the severity of earthquakes are conventionally reported in values derived from a logarithmic scale.
2.2 Scientific Notation
Perhaps the single most common area in which you will be faced with working with
exponents 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 a power of ten. The coefficient is generally a number with a single digit to the left of the decimal 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 the original 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 the coefficient, followed by a capital letter “E”, followed by the exponent. Some examples may help clarify the concept.
Eqn 1.18
)
6
E
43
.
5
as
written
(sometimes
10
43
.
5
00000543
.
0
)
6
E
697
.
2
as
written
(sometimes
10
697
.
2
000
,
697
,
2
)
2
E
0345
.
1
as
written
(sometimes
10
0345
.
1
45
.
103
6 6 2−
×
=
×
=
×
=
−In effect, scientific notation acts to “shift the decimal”, allowing us to do much of our math with values between 1 and 10, determining the order of magnitude of the result at the end of our 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 ready availability of computers and calculators. Slide rules made it relatively quick and easy to multiply and divide numbers between 1 and 10. But that technology would have been of little use 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 involving numbers with only a single digit to the left of the decimal. Today, facility with scientific notation is part of scientific literacy. Understanding of this notational convention – while perhaps less critical today than a generation ago – helps to make the metric system of units more 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 Notation
Scientific notation makes certain calculations more convenient, but if you are not used to working in scientific notation, it can take a little getting used to. The rules for calculating in scientific notation are a consequence of the way scientific notation is defined. You should be able to figure these rules out for yourself with a little thought. To describe the basic
mathematical operations, we need to define the variables a and b as follows:
Eqn 1.19 q p
y
b
x
a
10
10
×
=
×
=
Addition (or Subtraction)
We cannot directly add the values of the coefficients, x and y, because they are each
multiplied by different powers of ten. The coefficient x might be multiplied by ten to produce
a, while y might be multiplied by millions to produce b. We must write a and b so that they
are both expressed as a value multiplied by the same power of ten. Luckily, this is generally not too difficult to do.
Lets express the value of a in terms that match the power of ten in which b has been expressed, namely the exponent q.
Eqn 1.20
(
)
(
p q)
q q q px
a
x
a
10
10
10
10
×
×
=
×
×
=
− −Eqn 1.21
Again, an example may help.
Eqn 1.22
(
)
(
77
.
95
8
.
52
)
10
86
.
47
10
8647
10
52
.
8
10
95
.
77
10
52
.
8
10
10
795
.
7
10
52
.
8
10
795
.
7
2 2 2 2 2 2 1 2 3=
×
=
×
+
×
+
×
×
+
×
×
×
+
×
Multiplication Eqn 1.23a
b
x
y
p+q×
×
=
×
(
)
10
Division Eqn 1.242.2.2 Orders of Magnitude
Scientists often talk about the “order of magnitude” of a value. The order of magnitude refers to the value of the exponent of a numerical value expressed in scientific notation. But it is even more common for scientists to speak of two numbers as differing by a certain number of orders of magnitude. Two values that differ by an order of magnitude differ by a factor of ten (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 much
attention to the exact value of two numbers. For example, the mass of a marble (which is somewhere under 100 g) and the mass of a large automobile (which is about 1 metric ton, or 1000 kg, or 1,000,000 g) differ by approximately six orders of magnitude. Given these huge discrepancies in mass, for most practical problems, it really would not be too important to know the exact weight of the marble. The terminology is used as a sort of shorthand that gives scientists a quick reference for the approximate relative magnitude of different values.
2.3 Significant Digits
Numbers can express quantities with arbitrary precision (you can always just keep adding digits to the right of the decimal place). Unfortunately, we can measure any real-valued quantity, and many integer-valued quantities with only limited precision, so it is easy to express values with more precision than they deserve. Somehow, scientists must
communicate to one another the actual precision with which numbers have been measured or estimated. The most complete way of doing that is to report each value with a quantitative estimate of the uncertainty of that number (such as its standard error), but in many contexts that is both labor intensive and unnecessary. The convention of reporting numbers with an appropriate number of significant digits evolved as a second, less laborious (albeit less precise) way of keeping track of and communicating the precision of numbers. The idea behind the convention of significant digits is simple. If there is substantial uncertainty in the value you report at a certain order of magnitude (e.g. tens, hundreds, or thousands), don’t report any digits at a lesser order of magnitude. The digits that count as significant 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 to indicate increased precision. Similar rules apply to numbers larger than one, but because zeros 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 Quantities
The rules for determining how many significant digits to use in your own work are easiest to understand in the context of taking measurements. As a general rule, one should record measurements to the first digit that you must estimate, given the measuring technology being employed. 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 standard frequently calls for a little judgment.
Most commercially available meter sticks (the metric equivalent of a yardstick) have divisions that one can interpret to the nearest millimeter. One may be able to estimate measurements at a slightly finer resolution by interpolating between the divisions. But manufacturing tolerances for meter sticks are often poor, and division marks on poorly made ones are inaccurately printed. Moreover, meter sticks are often used in field science to make quick measurements of variable or imprecisely defined properties such as water depth or height of vegetation. Thus depending on the accuracy with which the meter stick was produced and the type of measurement being taken it might be appropriate to record measurements to the nearest centimeter (as for measuring the depth of water in a rapidly flowing stream) or nearest millimeter (length of a plant stem or leaf). Finer resolution (while superficially possible by interpolating between the divisions on the scale) will seldom be appropriate.
2.3.2 Significant Digits in Calculated Quantities
Many quantities in science are not measured directly, but calculated from other measured values, and here the rules for determining the number of significant digits to report in your work are more complex. For example, specific gravity is calculated by combining a measurement of volume (which has a certain accuracy) with a measurement of mass (also of limited precision). The rules for determining the number of significant digits in calculated values 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 of digits) than the least accurate of the values added together. Thus the result of the addition of two 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 weight expressed in metric tonnes (1000 kg, or 1 x 106 g) if the weight in tonnes was measured to the nearest gram. If it was measured only to the nearest kilogram, it would makes little sense to add just a few grams to that amount, since the measurement of the larger object’s mass could easily be off by several hundred grams.
Eqn 1.26
1
.
27
+
104
.
1
=
105
.
4
Addition is only as accurate as the less accurate number being added together. The more accurate number – 1.27 in this case – is rounded to match the less accurate one before carrying out the addition.
Multiplication and Division
The results of both multiplication and division have the same number of significant digits as the 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
.
0
0
.
5
95
.
1
10
32
.
1
132
103
.
104
27
.
1
2=
=
×
=
=
×
Often, these rules will appear counter intuitive, especially when it means reporting a result with fewer significant figures.
Eqn 1.28
2.3.3 Complex Calculations
In complex calculations, including many statistical calculations, if you are using a computer or pocket calculator, you should carry out your computations with all available digits, and round the result to the appropriate number of significant digits only at the end of your calculations. Rounding of intermediate results can introduce significant numerical errors.
2.4 The Metric System and SI Units
The system of measurements commonly called the “metric system” is the standard set of units for scientific activity worldwide. The modern incarnation of this internally consistent system of 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 full participant in technical discussion on environmental issues.
For Americans who have grown up using Imperial units (feet, gallons, pounds), learning to work with metric units is very much like learning to use a foreign language. As you are first working with these units, you may find yourself mentally translating from metric to imperial units and back again. As you gain fluency in use of metric units, you will find it easier to think in terms of the metric units directly. If you go on to use metric measurements on a regular basis, you may even find yourself wanting to buy about _ kg (instead of about 1 pound) of eggplant for dinner.
2.4.1 System International Units
SI units are defined by international agreements going back to 1875. Changes in the SI system needed to reflect modern technologies and changes in scientific understnanding are made under the auspices of an international convention. All SI units are all based on seven basic units. These units are assumed to be independent of one another in the sense that one could (at least in principal) redefine the unit of any one quantity without affecting the other six. The seven basic quantities and the SI units used to measure them are given in the following 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 Units
All other SI units are defined as combinations of the seven basic units. Volume has dimensions of length cubed, so the most natural unit of measurement of volume is the meter cubed. Velocity has dimensions of distance per unit time, and thus is most directly measured in 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 (a unit of electrical charge, equal to A·s), The joule (a unit of energy, equal to kg·m2s-2), and the Volt ( a measure of electrical potential, equal to m2·kg·s-3·A-1).
Using Prefixes to Rescale Basic and Derived Units
Many basic and derived units turn out to be unwieldy in certain situations, even in everyday practice. For example, it becomes tedious to write travel distances, such as the distance from Portland, Maine to Washington, D.C. in terms of meters. We would tend to write the distance between the two cities as 890 kilometers, not as 890,000 meters. The situation gets even more unwieldy for astronomers, measuring interstellar distances, or atomic physicists, estimating distances within atoms. The SI system of units include an internationally agreed upon list of prefixes that permit people to scale both basic and derived units for greater convenience for particular purposes.
The prefixes are given in the following table. Note that the values on the left of the table are used to build new units that are larger than the basic SI units, while the prefixes on the right are used to build units that are smaller that the basic SI units. The symbols used for the prefixes 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, or
one billionth of a meter. Thus we could express the distance between Portland, Maine and Washington, D.C. in any of the following ways (note that all values are expressed to only two significant digits). Of course, some of these are much more convenient than others – which is the 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 only exception (and it is a fairly self-evident one) is for units of mass. One should name units of mass with reference not to the basic unit of mass, the kilogram, but with reference to the gram, even though the gram is not formally one of the seven basic units of the SI. Thus in discussions of the global carbon cycle, one would speak in terms petagrams of carbon, not terakilograms.
Additional Units not Formally Part of SI
Many units that are used everyday in environmental science are not formally part of the SI system, but are currently accepted by international authorities for use with the SI units. These include such units as the liter (1000 cm3, or 1 dm3, or 10-3 m3), the metric ton (sometimes written tonne, equal to 1000 kg, or 1 Mg), the minute (60 seconds), and the hectare (the area of a square 100m on a side, 10000 m2, or 0.01 km2).
For Further Information
The U.S. National Institute of Standards and Technology maintains a web site that provides additional details on SI units at the URL: http://physics.nist.gov/cuu/Units/index.html
2.5 Dimensional Analysis
Dimensional analysis is a profoundly useful technique for reasoning from what you know to what you need to know. It can also be used to generate insights unavailable any other way. It plays an especially important role in certain fields. For example, in fluid dynamics,
dimensionless coefficients (Reynolds number, Froude number, etc.) capture important aspects of fluid flows.
The basic principal of dimensional analysis is simple. If you have an equation relating two quantities, the dimensions (length, mass, time, etc.) of the quantities on the two sides of the equation must be similar.
2.5.1 Don’t Memorize Formulas – Learn Principals
Students beginning to learn quantitative subjects often find themselves memorizing formulas. But memorization has a severe disadvantage – you tend to forget what you memorize rather quickly. You may get through the final exam, but you are unlikely to have effective working knowledge 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 you DO 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 material and 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 you will be called upon at some time in your career – whether as scientists, policymakers, or informed citizens – to reason about something you have not studied in years. Thus it is important for you to retain – or be able to regenerate – your working knowledge. In this context, dimensional analysis is an extremely valuable tool for being able to generate or regenerate working understanding.
