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<!DOCTYPE html>
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<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
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<meta name="author" content="Joya Danyluk and Joe Hobart" />
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<title>Foundational Mathematics for Water Engineering</title>
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<body>
<header id="title-block-header">
<h1 class="title">Foundational Mathematics for Water Engineering</h1>
<p class="subtitle">MATH 128</p>
<p class="author">Joya Danyluk and Joe Hobart</p>
<p class="date">Fall 2026</p>
</header>
<nav id="TOC" role="doc-toc">
<ul>
<li><a href="#welcome" id="toc-welcome">Welcome</a>
<ul>
<li><a href="#learning-goals" id="toc-learning-goals">Learning
goals</a></li>
<li><a href="#software" id="toc-software">Software</a></li>
</ul></li>
<li><a href="#the-basics" id="toc-the-basics">The Basics</a>
<ul>
<li><a href="#real-numbers-properties"
id="toc-real-numbers-properties">Real numbers and their properties</a>
<ul>
<li><a href="#translating-signs-carefully"
id="toc-translating-signs-carefully">Translating signs
carefully</a></li>
<li><a href="#practice-problems" id="toc-practice-problems">Practice
Problems</a></li>
</ul></li>
<li><a href="#bedmas-fractions" id="toc-bedmas-fractions">BEDMAS and
fractions</a>
<ul>
<li><a href="#complex-numerical-fractions"
id="toc-complex-numerical-fractions">Complex numerical
fractions</a></li>
<li><a href="#units-conversion-factors-and-precision"
id="toc-units-conversion-factors-and-precision">Units, conversion
factors, and precision</a></li>
<li><a href="#practice-problems-1" id="toc-practice-problems-1">Practice
Problems</a></li>
</ul></li>
<li><a href="#sets-interval-notation"
id="toc-sets-interval-notation">Sets, inequalities, and interval
notation</a>
<ul>
<li><a href="#converting-among-representations"
id="toc-converting-among-representations">Converting among
representations</a></li>
<li><a href="#practice-problems-2" id="toc-practice-problems-2">Practice
Problems</a></li>
</ul></li>
<li><a href="#absolute-value-distance"
id="toc-absolute-value-distance">Absolute value and distance</a>
<ul>
<li><a href="#practice-problems-3" id="toc-practice-problems-3">Practice
Problems</a></li>
</ul></li>
<li><a href="#exponents-roots" id="toc-exponents-roots">Exponents,
roots, and rational exponents</a>
<ul>
<li><a href="#simplifying-roots" id="toc-simplifying-roots">Simplifying
roots</a></li>
<li><a href="#practice-problems-4" id="toc-practice-problems-4">Practice
Problems</a></li>
</ul></li>
<li><a href="#algebraic-expressions-polynomials"
id="toc-algebraic-expressions-polynomials">Algebraic expressions and
polynomials</a>
<ul>
<li><a href="#polynomials-in-two-variables"
id="toc-polynomials-in-two-variables">Polynomials in two
variables</a></li>
<li><a href="#domains-of-algebraic-expressions"
id="toc-domains-of-algebraic-expressions">Domains of algebraic
expressions</a></li>
<li><a href="#practice-problems-5" id="toc-practice-problems-5">Practice
Problems</a></li>
</ul></li>
<li><a href="#expanding-factoring"
id="toc-expanding-factoring">Expanding and factoring</a>
<ul>
<li><a href="#a-factoring-decision-process"
id="toc-a-factoring-decision-process">A factoring decision
process</a></li>
<li><a href="#practice-problems-6" id="toc-practice-problems-6">Practice
Problems</a></li>
</ul></li>
<li><a href="#polynomial-division-zeros"
id="toc-polynomial-division-zeros">Polynomial division and rational
zeros</a>
<ul>
<li><a href="#long-and-synthetic-division"
id="toc-long-and-synthetic-division">Long and synthetic
division</a></li>
<li><a href="#finding-all-rational-zeros"
id="toc-finding-all-rational-zeros">Finding all rational zeros</a></li>
<li><a href="#practice-problems-7" id="toc-practice-problems-7">Practice
Problems</a></li>
</ul></li>
<li><a href="#equations-initial-inequalities"
id="toc-equations-initial-inequalities">Equations and an initial look at
inequalities</a>
<ul>
<li><a href="#checking-and-classifying-results"
id="toc-checking-and-classifying-results">Checking and classifying
results</a></li>
<li><a href="#practice-problems-8" id="toc-practice-problems-8">Practice
Problems</a></li>
</ul></li>
<li><a href="#lines-linear-applications"
id="toc-lines-linear-applications">Lines and linear applications</a>
<ul>
<li><a href="#intercepts-and-related-lines"
id="toc-intercepts-and-related-lines">Intercepts and related
lines</a></li>
<li><a href="#practice-problems-9" id="toc-practice-problems-9">Practice
Problems</a></li>
</ul></li>
<li><a href="#fractional-rational-expressions"
id="toc-fractional-rational-expressions">Fractional and rational
expressions</a>
<ul>
<li><a href="#a-complete-simplification-routine"
id="toc-a-complete-simplification-routine">A complete simplification
routine</a></li>
<li><a href="#practice-problems-10"
id="toc-practice-problems-10">Practice Problems</a></li>
</ul></li>
<li><a href="#solving-linear-quadratic"
id="toc-solving-linear-quadratic">Solving linear and quadratic
equations</a>
<ul>
<li><a href="#solving-by-factoring"
id="toc-solving-by-factoring">Solving by factoring</a></li>
<li><a href="#using-the-quadratic-formula"
id="toc-using-the-quadratic-formula">Using the quadratic
formula</a></li>
<li><a href="#interpreting-the-discriminant"
id="toc-interpreting-the-discriminant">Interpreting the
discriminant</a></li>
<li><a href="#practice-problems-11"
id="toc-practice-problems-11">Practice Problems</a></li>
</ul></li>
<li><a href="#rearranging-radical-equations"
id="toc-rearranging-radical-equations">Rearranging formulas and radical
equations</a>
<ul>
<li><a href="#formula-and-radical-checks"
id="toc-formula-and-radical-checks">Formula and radical checks</a></li>
<li><a href="#practice-problems-12"
id="toc-practice-problems-12">Practice Problems</a></li>
</ul></li>
<li><a href="#solving-inequalities"
id="toc-solving-inequalities">Solving inequalities</a>
<ul>
<li><a href="#sign-charts-and-tolerances"
id="toc-sign-charts-and-tolerances">Sign charts and tolerances</a></li>
<li><a href="#practice-problems-13"
id="toc-practice-problems-13">Practice Problems</a></li>
</ul></li>
<li><a href="#the-basics-chapter-review"
id="toc-the-basics-chapter-review">Chapter Review</a>
<ul>
<li><a href="#chapter-summary" id="toc-chapter-summary">Chapter
summary</a></li>
<li><a href="#common-mistakes" id="toc-common-mistakes">Common
mistakes</a></li>
<li><a href="#exercises" id="toc-exercises">Exercises</a></li>
<li><a href="#questions-for-discussion"
id="toc-questions-for-discussion">Questions for discussion</a></li>
</ul></li>
</ul></li>
<li><a href="#functions" id="toc-functions">Functions</a>
<ul>
<li><a href="#function-language-domain-range"
id="toc-function-language-domain-range">Function language, evaluation,
domain, and range</a>
<ul>
<li><a href="#evaluating-functions-and-recognizing-graphs"
id="toc-evaluating-functions-and-recognizing-graphs">Evaluating
functions and recognizing graphs</a></li>
<li><a href="#domain-codomain-and-range"
id="toc-domain-codomain-and-range">Domain, codomain, and range</a></li>
<li><a href="#practice-problems-14"
id="toc-practice-problems-14">Practice Problems</a></li>
</ul></li>
<li><a href="#parent-piecewise-rates"
id="toc-parent-piecewise-rates">Parent functions, piecewise models, and
rates of change</a>
<ul>
<li><a href="#parent-functions-and-piecewise-definitions"
id="toc-parent-functions-and-piecewise-definitions">Parent functions and
piecewise definitions</a></li>
<li><a
href="#net-change-average-rate-of-change-and-difference-quotients"
id="toc-net-change-average-rate-of-change-and-difference-quotients">Net
change, average rate of change, and difference quotients</a></li>
<li><a href="#practice-problems-15"
id="toc-practice-problems-15">Practice Problems</a></li>
</ul></li>
<li><a href="#graphical-behaviour-transformations"
id="toc-graphical-behaviour-transformations">Graphical behaviour and
transformations</a>
<ul>
<li><a href="#graphical-equations-inequalities-and-extrema"
id="toc-graphical-equations-inequalities-and-extrema">Graphical
equations, inequalities, and extrema</a></li>
<li><a href="#transforming-parent-graphs"
id="toc-transforming-parent-graphs">Transforming parent graphs</a></li>
<li><a href="#practice-problems-16"
id="toc-practice-problems-16">Practice Problems</a></li>
</ul></li>
<li><a href="#function-operations-composition-inverses"
id="toc-function-operations-composition-inverses">Function operations,
composition, and inverses</a>
<ul>
<li><a href="#arithmetic-with-functions"
id="toc-arithmetic-with-functions">Arithmetic with functions</a></li>
<li><a href="#composition-of-functions"
id="toc-composition-of-functions">Composition of functions</a></li>
<li><a href="#one-to-one-and-onto-functions"
id="toc-one-to-one-and-onto-functions">One-to-one and onto
functions</a></li>
<li><a href="#inverse-functions" id="toc-inverse-functions">Inverse
functions</a></li>
<li><a href="#practice-problems-17"
id="toc-practice-problems-17">Practice Problems</a></li>
</ul></li>
<li><a href="#functions-chapter-review"
id="toc-functions-chapter-review">Chapter Review</a>
<ul>
<li><a href="#chapter-summary-1" id="toc-chapter-summary-1">Chapter
summary</a></li>
<li><a href="#common-mistakes-1" id="toc-common-mistakes-1">Common
mistakes</a></li>
<li><a href="#exercises-1" id="toc-exercises-1">Exercises</a></li>
<li><a href="#questions-for-discussion-1"
id="toc-questions-for-discussion-1">Questions for discussion</a></li>
</ul></li>
</ul></li>
<li><a href="#polynomial-rational-functions"
id="toc-polynomial-rational-functions">Polynomial and Rational
Functions</a>
<ul>
<li><a href="#quadratic-functions-completing-square"
id="toc-quadratic-functions-completing-square">Quadratic functions and
completing the square</a>
<ul>
<li><a href="#vertex-form-and-intercepts"
id="toc-vertex-form-and-intercepts">Vertex form and intercepts</a></li>
<li><a href="#completing-the-square"
id="toc-completing-the-square">Completing the square</a></li>
<li><a href="#practice-problems-18"
id="toc-practice-problems-18">Practice Problems</a></li>
</ul></li>
<li><a href="#polynomial-behaviour-graphing"
id="toc-polynomial-behaviour-graphing">Polynomial behaviour and
graphing</a>
<ul>
<li><a href="#end-behaviour-and-local-extrema"
id="toc-end-behaviour-and-local-extrema">End behaviour and local
extrema</a></li>
<li><a href="#zeros-multiplicity-and-a-systematic-sketch"
id="toc-zeros-multiplicity-and-a-systematic-sketch">Zeros, multiplicity,
and a systematic sketch</a></li>
<li><a href="#practice-problems-19"
id="toc-practice-problems-19">Practice Problems</a></li>
</ul></li>
<li><a href="#rational-functions-asymptotes"
id="toc-rational-functions-asymptotes">Rational functions, holes, and
asymptotes</a>
<ul>
<li><a href="#domains-holes-and-vertical-asymptotes"
id="toc-domains-holes-and-vertical-asymptotes">Domains, holes, and
vertical asymptotes</a></li>
<li><a href="#horizontal-asymptotes"
id="toc-horizontal-asymptotes">Horizontal asymptotes</a></li>
<li><a href="#practice-problems-20"
id="toc-practice-problems-20">Practice Problems</a></li>
</ul></li>
<li><a href="#sketching-rational-functions"
id="toc-sketching-rational-functions">Sketching and interpreting
rational functions</a>
<ul>
<li><a href="#a-systematic-graphing-process"
id="toc-a-systematic-graphing-process">A systematic graphing
process</a></li>
<li><a href="#domain-and-range-from-a-rational-graph"
id="toc-domain-and-range-from-a-rational-graph">Domain and range from a
rational graph</a></li>
<li><a href="#practice-problems-21"
id="toc-practice-problems-21">Practice Problems</a></li>
</ul></li>
<li><a href="#polynomial-rational-functions-review"
id="toc-polynomial-rational-functions-review">Chapter Review</a>
<ul>
<li><a href="#chapter-summary-2" id="toc-chapter-summary-2">Chapter
summary</a></li>
<li><a href="#common-mistakes-2" id="toc-common-mistakes-2">Common
mistakes</a></li>
<li><a href="#exercises-2" id="toc-exercises-2">Exercises</a></li>
<li><a href="#questions-for-discussion-2"
id="toc-questions-for-discussion-2">Questions for discussion</a></li>
</ul></li>
</ul></li>
<li><a href="#exponential-logarithmic-functions"
id="toc-exponential-logarithmic-functions">Exponential and Logarithmic
Functions</a>
<ul>
<li><a href="#exponential-functions-transformations"
id="toc-exponential-functions-transformations">Exponential functions and
transformations</a>
<ul>
<li><a href="#basic-graphs-and-properties"
id="toc-basic-graphs-and-properties">Basic graphs and
properties</a></li>
<li><a href="#transformations-of-exponential-graphs"
id="toc-transformations-of-exponential-graphs">Transformations of
exponential graphs</a></li>
<li><a href="#practice-problems-22"
id="toc-practice-problems-22">Practice Problems</a></li>
</ul></li>
<li><a href="#exponential-models-applications"
id="toc-exponential-models-applications">Exponential models in
applications</a>
<ul>
<li><a href="#discrete-and-continuous-growth-and-decay"
id="toc-discrete-and-continuous-growth-and-decay">Discrete and
continuous growth and decay</a></li>
<li><a href="#logistic-growth" id="toc-logistic-growth">Logistic
growth</a></li>
<li><a href="#practice-problems-23"
id="toc-practice-problems-23">Practice Problems</a></li>
</ul></li>
<li><a href="#logarithmic-functions-laws"
id="toc-logarithmic-functions-laws">Logarithmic functions and laws</a>
<ul>
<li><a href="#definition-graph-and-domain"
id="toc-definition-graph-and-domain">Definition, graph, and
domain</a></li>
<li><a href="#product-quotient-and-power-laws"
id="toc-product-quotient-and-power-laws">Product, quotient, and power
laws</a></li>
<li><a href="#practice-problems-24"
id="toc-practice-problems-24">Practice Problems</a></li>
</ul></li>
<li><a href="#solving-exponential-logarithmic-equations"
id="toc-solving-exponential-logarithmic-equations">Solving exponential
and logarithmic equations</a>
<ul>
<li><a href="#exponential-equations"
id="toc-exponential-equations">Exponential equations</a></li>
<li><a href="#logarithmic-equations-and-domain-checks"
id="toc-logarithmic-equations-and-domain-checks">Logarithmic equations
and domain checks</a></li>
<li><a href="#practice-problems-25"
id="toc-practice-problems-25">Practice Problems</a></li>
</ul></li>
<li><a href="#log-scales-flow-duration"
id="toc-log-scales-flow-duration">Change of base, logarithmic scales,
and flow duration</a>
<ul>
<li><a href="#change-of-base-and-orders-of-magnitude"
id="toc-change-of-base-and-orders-of-magnitude">Change of base and
orders of magnitude</a></li>
<li><a href="#semi-log-log-log-and-flow-duration-graphs"
id="toc-semi-log-log-log-and-flow-duration-graphs">Semi-log, log-log,
and flow-duration graphs</a></li>
<li><a href="#practice-problems-26"
id="toc-practice-problems-26">Practice Problems</a></li>
</ul></li>
<li><a href="#exponential-logarithmic-functions-review"
id="toc-exponential-logarithmic-functions-review">Chapter Review</a>
<ul>
<li><a href="#chapter-summary-3" id="toc-chapter-summary-3">Chapter
summary</a></li>
<li><a href="#common-mistakes-3" id="toc-common-mistakes-3">Common
mistakes</a></li>
<li><a href="#exercises-3" id="toc-exercises-3">Exercises</a></li>
<li><a href="#questions-for-discussion-3"
id="toc-questions-for-discussion-3">Questions for discussion</a></li>
</ul></li>
</ul></li>
<li><a href="#trigonometric-functions"
id="toc-trigonometric-functions">Trigonometric Functions</a>
<ul>
<li><a href="#angles-radians-right-triangles"
id="toc-angles-radians-right-triangles">Angles, radians, and right
triangles</a>
<ul>
<li><a href="#degree-and-radian-measure"
id="toc-degree-and-radian-measure">Degree and radian measure</a></li>
<li><a href="#right-triangle-ratios"
id="toc-right-triangle-ratios">Right-triangle ratios</a></li>
<li><a href="#practice-problems-27"
id="toc-practice-problems-27">Practice Problems</a></li>
</ul></li>
<li><a href="#special-reference-unit-circle"
id="toc-special-reference-unit-circle">Special angles, reference angles,
and the unit circle</a>
<ul>
<li><a href="#special-triangles-and-reference-angles"
id="toc-special-triangles-and-reference-angles">Special triangles and
reference angles</a></li>
<li><a href="#trigonometric-functions-of-any-angle"
id="toc-trigonometric-functions-of-any-angle">Trigonometric functions of
any angle</a></li>
<li><a href="#practice-problems-28"
id="toc-practice-problems-28">Practice Problems</a></li>
</ul></li>
<li><a href="#trigonometric-graphs-transformations"
id="toc-trigonometric-graphs-transformations">Trigonometric graphs and
transformations</a>
<ul>
<li><a href="#parent-graphs-and-domains"
id="toc-parent-graphs-and-domains">Parent graphs and domains</a></li>
<li><a href="#amplitude-period-and-phase-shift"
id="toc-amplitude-period-and-phase-shift">Amplitude, period, and phase
shift</a></li>
<li><a href="#practice-problems-29"
id="toc-practice-problems-29">Practice Problems</a></li>
</ul></li>
<li><a href="#inverse-trigonometric-functions"
id="toc-inverse-trigonometric-functions">Inverse trigonometric
functions</a>
<ul>
<li><a href="#principal-values-and-solving-angles"
id="toc-principal-values-and-solving-angles">Principal values and
solving angles</a></li>
<li><a href="#practice-problems-30"
id="toc-practice-problems-30">Practice Problems</a></li>
</ul></li>
<li><a href="#trigonometric-identities-formulas"
id="toc-trigonometric-identities-formulas">Trigonometric identities and
angle formulas</a>
<ul>
<li><a href="#fundamental-identities"
id="toc-fundamental-identities">Fundamental identities</a></li>
<li><a href="#even-odd-addition-and-subtraction-formulas"
id="toc-even-odd-addition-and-subtraction-formulas">Even, odd, addition,
and subtraction formulas</a></li>
<li><a href="#practice-problems-31"
id="toc-practice-problems-31">Practice Problems</a></li>
</ul></li>
<li><a href="#trigonometric-functions-review"
id="toc-trigonometric-functions-review">Chapter Review</a>
<ul>
<li><a href="#chapter-summary-4" id="toc-chapter-summary-4">Chapter
summary</a></li>
<li><a href="#common-mistakes-4" id="toc-common-mistakes-4">Common
mistakes</a></li>
<li><a href="#exercises-4" id="toc-exercises-4">Exercises</a></li>
<li><a href="#questions-for-discussion-4"
id="toc-questions-for-discussion-4">Questions for discussion</a></li>
</ul></li>
</ul></li>
<li><a href="#statistics" id="toc-statistics">Statistics</a>
<ul>
<li><a href="#data-variables-populations-samples"
id="toc-data-variables-populations-samples">Data, variables,
populations, and samples</a>
<ul>
<li><a href="#data-matrices-and-variable-types"
id="toc-data-matrices-and-variable-types">Data matrices and variable
types</a></li>
<li><a href="#populations-samples-parameters-and-statistics"
id="toc-populations-samples-parameters-and-statistics">Populations,
samples, parameters, and statistics</a></li>
<li><a href="#practice-problems-32"
id="toc-practice-problems-32">Practice Problems</a></li>
</ul></li>
<li><a href="#sampling-studies-experiments"
id="toc-sampling-studies-experiments">Sampling, observational studies,
and experiments</a>
<ul>
<li><a href="#bias-and-sampling-methods"
id="toc-bias-and-sampling-methods">Bias and sampling methods</a></li>
<li><a href="#observational-studies-and-experiments"
id="toc-observational-studies-and-experiments">Observational studies and
experiments</a></li>
<li><a href="#practice-problems-33"
id="toc-practice-problems-33">Practice Problems</a></li>
</ul></li>
<li><a href="#centre-variability" id="toc-centre-variability">Measures
of centre and variability</a>
<ul>
<li><a href="#mean-median-and-mode" id="toc-mean-median-and-mode">Mean,
median, and mode</a></li>
<li><a href="#variance-and-standard-deviation"
id="toc-variance-and-standard-deviation">Variance and standard
deviation</a></li>
<li><a href="#practice-problems-34"
id="toc-practice-problems-34">Practice Problems</a></li>
</ul></li>
<li><a href="#quartiles-outliers-shape"
id="toc-quartiles-outliers-shape">Quartiles, outliers, and distribution
shape</a>
<ul>
<li><a href="#five-number-summaries-and-boxplots"
id="toc-five-number-summaries-and-boxplots">Five-number summaries and
boxplots</a></li>
<li><a href="#skewness-and-choosing-summaries"
id="toc-skewness-and-choosing-summaries">Skewness and choosing
summaries</a></li>
<li><a href="#practice-problems-35"
id="toc-practice-problems-35">Practice Problems</a></li>
</ul></li>
<li><a href="#statistics-chapter-review"
id="toc-statistics-chapter-review">Chapter Review</a>
<ul>
<li><a href="#chapter-summary-5" id="toc-chapter-summary-5">Chapter
summary</a></li>
<li><a href="#common-mistakes-5" id="toc-common-mistakes-5">Common
mistakes</a></li>
<li><a href="#exercises-5" id="toc-exercises-5">Exercises</a></li>
<li><a href="#questions-for-discussion-5"
id="toc-questions-for-discussion-5">Questions for discussion</a></li>
</ul></li>
</ul></li>
<li><a href="#glossary" id="toc-glossary">Glossary</a></li>
<li><a href="#appendix-appendices"
id="toc-appendix-appendices">(APPENDIX) Appendices</a></li>
<li><a href="#formula-reference" id="toc-formula-reference">Formula
Reference</a>
<ul>
<li><a href="#formula-algebra-geometry"
id="toc-formula-algebra-geometry">Algebra and geometry</a></li>
<li><a href="#formula-functions-rates"
id="toc-formula-functions-rates">Functions and rates</a></li>
<li><a href="#formula-exponential-logarithmic"
id="toc-formula-exponential-logarithmic">Exponential and logarithmic
models</a></li>
<li><a href="#formula-trigonometry"
id="toc-formula-trigonometry">Trigonometry</a></li>
<li><a href="#formula-statistics"
id="toc-formula-statistics">Descriptive statistics</a></li>
</ul></li>
<li><a href="#references" id="toc-references">References</a></li>
</ul>
</nav>
<h1 class="unnumbered" id="welcome">Welcome</h1>
<!-- Copyright statement to be confirmed. -->
<p>Mathematics is a practical language for describing, measuring, and
improving water systems. In Water Engineering Technology, that language
appears in calculations involving flow, storage, treatment, pumping,
chemical dosing, site geometry, and the interpretation of operational
data. <em>Foundational Mathematics for Water Engineering</em> develops
the mathematical habits needed to work carefully in those settings:
stating assumptions, tracking units, representing relationships clearly,
checking whether an answer is reasonable, and explaining what a result
means.</p>
<p>This book is organized in six chapters. It begins with algebraic
foundations and then develops functions, polynomial and rational models,
exponential and logarithmic models, trigonometry, and descriptive
statistics. Each chapter opens with learning outcomes, then introduces
key terms in definition boxes immediately before they are needed.
Concepts are developed through explanations, worked examples, applied
problems, and practice. Chapter reviews include exercises with immediate
feedback and open-ended questions for discussion.</p>
<p>The central perspective is that a calculation is not complete when a
number appears on a calculator. A useful mathematical result includes a
quantity, appropriate units, a sensible magnitude, and an interpretation
in the setting that produced it. The examples use realistic but
fictional water-engineering situations unless a cited source is
necessary.</p>
<h2 class="unnumbered" id="learning-goals">Learning goals</h2>
<p>By the end of the book, readers should be able to:</p>
<ul>
<li>represent and solve quantitative water-engineering problems with
appropriate algebra, functions, trigonometry, and descriptive
statistics;</li>
<li>interpret graphs, tables, formulas, and calculations in
context;</li>
<li>communicate assumptions, units, and conclusions clearly; and</li>
<li>check the reasonableness of a calculation before using it to support
a decision.</li>
</ul>
<h2 class="unnumbered" id="software">Software</h2>
<p>Optional demonstrations use <a
href="https://www.desmos.com/calculator">Desmos</a> for graphing and
Microsoft Excel for tables, calculations, and charts. Both are tools for
checking and communicating mathematics, not substitutes for
understanding the underlying method.</p>
<h1 id="the-basics">The Basics</h1>
<p>Water engineering calculations depend on reliable algebra. Flow,
storage, concentration, pressure, and treatment problems all require
careful work with numbers, expressions, equations, and inequalities.
Following the sequence of the MATH 128 Fall 2026 notes, this chapter
begins with real numbers and arithmetic, then develops sets, exponents,
roots, polynomials, equations, lines, rational expressions, and
inequalities. Units and physical restrictions remain part of every
calculation. A solution is complete only when its algebra is correct,
its units are appropriate, and its meaning is reasonable in context.</p>
<div class="learning-outcomes">
<p><strong>Learning outcomes.</strong> By the end of this chapter, you
should be able to:</p>
<ul>
<li>apply real-number, fraction, exponent, and root properties;</li>
<li>represent sets and ranges using inequalities and interval
notation;</li>
<li>simplify, expand, factor, and divide algebraic expressions;</li>
<li>apply the Remainder, Factor, and Rational Zeros Theorems;</li>
<li>solve and check linear, quadratic, radical, and formula-based
equations;</li>
<li>determine equations of lines and interpret slope; and</li>
<li>solve linear, polynomial, rational, and absolute-value
inequalities.</li>
</ul>
</div>
<h2 id="real-numbers-properties">Real numbers and their properties</h2>
<p>Measurements, counts, calculated values, and model outputs all belong
to a common number system.</p>
<div class="definition-box">
<p><strong>Real number.</strong> A number represented by a point on the
real number line. The set of all real numbers is denoted by <span
class="math inline">\(\mathbb{R}\)</span>.</p>
</div>
<div class="definition-box">
<p><strong>Rational number.</strong> A number that can be written as
<span class="math inline">\(a/b\)</span>, where <span
class="math inline">\(a\)</span> and <span
class="math inline">\(b\)</span> are integers and <span
class="math inline">\(b\ne0\)</span>.</p>
</div>
<p>The numbers used in measurement and calculation are real numbers,
denoted by <span class="math inline">\(\mathbb{R}\)</span>. They include
natural numbers, whole numbers, integers, rational numbers, and
irrational numbers. A rational number can be written as <span
class="math inline">\(a/b\)</span>, where <span
class="math inline">\(a\)</span> and <span
class="math inline">\(b\)</span> are integers and <span
class="math inline">\(b\ne0\)</span>. Irrational numbers, including
<span class="math inline">\(\sqrt2\)</span> and <span
class="math inline">\(\pi\)</span>, cannot be written in that form.</p>
<p>For real numbers <span class="math inline">\(a\)</span>, <span
class="math inline">\(b\)</span>, and <span
class="math inline">\(c\)</span>, addition and multiplication are
commutative and associative. The distributive property is</p>
<p><span class="math display">\[a(b+c)=ab+ac.\]</span></p>
<p>The commutative property changes order, while the associative
property changes grouping. Neither property permits arbitrary
rearrangement of subtraction or division. For instance, <span
class="math inline">\(8-3\)</span> is not equal to <span
class="math inline">\(3-8\)</span>. The properties of negatives follow
from multiplication by <span class="math inline">\(-1\)</span>:</p>
<p><span class="math display">\[(-1)a=-a,\qquad -(-a)=a,\qquad
(-a)(-b)=ab,\]</span></p>
<p><span class="math display">\[-(a+b)=-a-b,\qquad
-(a-b)=-a+b.\]</span></p>
<p>These identities are especially important when a negative sign
appears outside a long formula. A reliable practice is to regard the
sign as a factor of <span class="math inline">\(-1\)</span> and
distribute it explicitly.</p>
<h3 id="translating-signs-carefully">Translating signs carefully</h3>
<p>The negative sign can indicate a negative number, subtraction, or
multiplication by <span class="math inline">\(-1\)</span>. These
meanings are related but should not be blurred. In <span
class="math inline">\(-3^2\)</span>, the exponent applies before the
negative sign, so the value is <span
class="math inline">\(-(3^2)=-9\)</span>. In <span
class="math inline">\((-3)^2\)</span>, the parentheses make <span
class="math inline">\(-3\)</span> the base, so the value is 9.
Parentheses are therefore essential when a negative value is substituted
into a power.</p>
<p>Engineering sign conventions should be stated before calculation. A
negative elevation can mean below a chosen datum. A negative change in
tank depth can mean that depth decreased. Neither represents a negative
physical length. The sign communicates direction relative to a
reference.</p>
<div class="worked-example">
<p><span class="worked-example-title"><strong>Worked Example: Evaluating
a negative squared correction</strong></span></p>
<p>A calibration calculation contains <span
class="math inline">\(-0.4^2\)</span>. Because the exponent is evaluated
first,</p>
<p><span class="math display">\[-0.4^2=-(0.4^2)=-0.16.\]</span></p>
<p>If the intended base were <span class="math inline">\(-0.4\)</span>,
the expression would need to be written <span
class="math inline">\((-0.4)^2=0.16\)</span>.</p>
</div>
<div class="try-it">
<p><strong>Try It.</strong> Evaluate <span
class="math inline">\(-1.5^2\)</span> and <span
class="math inline">\((-1.5)^2\)</span>.</p>
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
<span class="math inline">\(-1.5^2=-2.25\)</span>, while <span
class="math inline">\((-1.5)^2=2.25\)</span>.
</details>
</div>
<div class="worked-example">
<p><span class="worked-example-title"><strong>Worked Example: Applying
the distributive property</strong></span></p>
<p>Simplify <span class="math inline">\(3(8-2y)\)</span>:</p>
<p><span class="math display">\[3(8-2y)=24-6y.\]</span></p>
<p>Every term inside the parentheses is multiplied by 3.</p>
</div>
<div class="try-it">
<p><strong>Try It.</strong> Simplify <span
class="math inline">\(-4(2-3x)\)</span>.</p>
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
<span class="math inline">\(-8+12x\)</span>.
</details>
</div>
<div class="worked-example">
<p><span class="worked-example-title"><strong>Worked Example:
Classifying a measured value</strong></span></p>
<p>A recorded turbidity reading of 2.75 can be written as <span
class="math inline">\(275/100=11/4\)</span>. It is therefore rational
and real. It is not an integer.</p>
</div>
<div class="try-it">
<p><strong>Try It.</strong> Classify <span
class="math inline">\(-6\)</span>, <span
class="math inline">\(\sqrt7\)</span>, and <span
class="math inline">\(4/9\)</span> using the smallest applicable number
set.</p>
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
<span class="math inline">\(-6\)</span> is an integer, <span
class="math inline">\(\sqrt7\)</span> is irrational, and <span
class="math inline">\(4/9\)</span> is rational.
</details>
</div>
<div class="worked-example">
<p><span class="worked-example-title"><strong>Worked Example:
Interpreting a signed water-level change</strong></span></p>
<p>A reservoir level changes by <span
class="math inline">\(-0.18\)</span> m in the morning and by <span
class="math inline">\(+0.07\)</span> m in the afternoon. The net change
is</p>
<p><span class="math display">\[-0.18+0.07=-0.11\ \text{m}.\]</span></p>
<p>The negative result means that the final level is 0.11 m below the
initial level.</p>
</div>
<div class="try-it">
<p><strong>Try It.</strong> A wet well level changes by <span
class="math inline">\(-0.24\)</span> m and then by <span
class="math inline">\(+0.31\)</span> m. Find and interpret the net
change.</p>
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
<span class="math inline">\(-0.24+0.31=0.07\)</span> m, so the final
level is 0.07 m above the initial level.
</details>
</div>
<div class="applied-problem">
<p><strong>Applied Problem: Tracking a reservoir level.</strong> A
reservoir begins the day at 6.42 m relative to its operating datum.
Withdrawal lowers the level by 0.38 m, inflow raises it by 0.21 m, and
evaporation lowers it by another 0.03 m. Determine the final level and
the net change.</p>
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
The final level is <span
class="math inline">\(6.42-0.38+0.21-0.03=6.22\)</span> m. The net
change is <span class="math inline">\(6.22-6.42=-0.20\)</span> m, so the
reservoir finishes 0.20 m below its starting level.
</details>
</div>
<h3 id="practice-problems">Practice Problems</h3>
<ol type="1">
<li>Simplify <span class="math inline">\(5(3-2x)\)</span>.
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
<span class="math inline">\(15-10x\)</span>.
</details></li>
<li>Name the smallest applicable set for 0.
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
The whole numbers.
</details></li>
<li>Is <span class="math inline">\(0.125\)</span> rational? Explain.
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
Yes. <span class="math inline">\(0.125=1/8\)</span>.
</details></li>
<li>Simplify <span class="math inline">\(-(7-y)\)</span>.
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
<span class="math inline">\(-7+y\)</span>.
</details></li>
<li>A measured flow is 12.4 L/s. Identify the smallest familiar number
set containing 12.4.
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
It is rational because <span class="math inline">\(12.4=62/5\)</span>.
</details></li>
<li>A tank level falls 0.35 m and then rises 0.12 m. Find the net
change.
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
<span class="math inline">\(-0.35+0.12=-0.23\)</span> m. The level falls
by 0.23 m overall.
</details></li>
<li>A pressure correction is written <span
class="math inline">\(-(1.8-0.6)\)</span>. Simplify it.
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
<span class="math inline">\(-(1.8-0.6)=-1.2\)</span>.
</details></li>
<li>A flow measurement is <span class="math inline">\(\sqrt{50}\)</span>
L/s. Is the exact value rational or irrational?
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
<span class="math inline">\(\sqrt{50}=5\sqrt2\)</span>, which is
irrational.
</details></li>
</ol>
<h2 id="bedmas-fractions">BEDMAS and fractions</h2>
<p>Reliable calculations require an agreed order for interpreting
operations and a consistent way to work with parts of a whole.</p>
<div class="definition-box">
<p><strong>Fraction.</strong> A quotient <span
class="math inline">\(a/b\)</span> in which <span
class="math inline">\(a\)</span> is the numerator, <span
class="math inline">\(b\)</span> is the denominator, and <span
class="math inline">\(b\ne0\)</span>.</p>
</div>
<p>BEDMAS records the order of operations: brackets, exponents, division
and multiplication from left to right, then addition and subtraction
from left to right. Fractions follow the same rules. To multiply
fractions, multiply their numerators and denominators. To divide,
multiply by the reciprocal. To add or subtract, first use a common
denominator.</p>
<p>For nonzero denominators,</p>
<p><span class="math display">\[\frac ab\frac cd=\frac{ac}{bd},\qquad
\frac ab\div\frac cd=\frac ab\frac dc,\]</span></p>
<p>and</p>
<p><span class="math display">\[\frac ab+\frac
cb=\frac{a+c}{b}.\]</span></p>
<p>When denominators differ, use the least common denominator rather
than adding denominators. A fraction bar also acts as a grouping symbol.
Everything in its numerator is evaluated as one group and everything in
its denominator as another. In an engineering calculation, keep several
digits during intermediate work and round only the final result to a
precision justified by the data.</p>
<h3 id="complex-numerical-fractions">Complex numerical fractions</h3>
<p>A complex fraction has one or more fractions in its numerator or
denominator. Simplify the numerator and denominator separately, or
multiply both by the least common denominator of all smaller fractions.
For example,</p>
<p><span class="math display">\[
\frac{\frac16+\frac49}{\frac24-\frac2{10}}
=\frac{\frac{11}{18}}{\frac3{10}}
=\frac{11}{18}\cdot\frac{10}{3}
=\frac{55}{27}.
\]</span></p>
<p>The equality at every stage makes the order visible. Entering an
entire complex fraction into a calculator without grouping parentheses
can produce a different expression.</p>
<div class="worked-example">
<p><span class="worked-example-title"><strong>Worked Example: Comparing
two fractional operating periods</strong></span></p>
<p>A pump operates for <span class="math inline">\(3/8\)</span> of an
hour at one setting and <span class="math inline">\(5/12\)</span> of an
hour at another. The total operating time is</p>
<p><span
class="math display">\[\frac38+\frac5{12}=\frac9{24}+\frac{10}{24}=\frac{19}{24}\
\text{h}.\]</span></p>
</div>
<div class="try-it">
<p><strong>Try It.</strong> A backwash cycle uses <span
class="math inline">\(2/15\)</span> h for draining and <span
class="math inline">\(1/10\)</span> h for rinsing. Find the combined
time.</p>
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
<span class="math inline">\(2/15+1/10=4/30+3/30=7/30\)</span> h, or 14
min.
</details>
</div>
<div class="worked-example">
<p><span class="worked-example-title"><strong>Worked Example: Combining
fractions of capacity</strong></span></p>
<p>If two operating periods process <span
class="math inline">\(1/6\)</span> and <span
class="math inline">\(4/9\)</span> of a tank capacity, then</p>
<p><span
class="math display">\[\frac16+\frac49=\frac3{18}+\frac8{18}=\frac{11}{18}.\]</span></p>
</div>
<div class="try-it">
<p><strong>Try It.</strong> Combine <span
class="math inline">\(1/4\)</span> and <span
class="math inline">\(2/3\)</span> of a tank capacity.</p>
<details class="answer-dropdown">
<summary>
<strong>Check Your Work</strong>
</summary>
<span class="math inline">\(1/4+2/3=3/12+8/12=11/12\)</span>.
</details>
</div>
<h3 id="units-conversion-factors-and-precision">Units, conversion
factors, and precision</h3>
<p>Units may be treated as algebraic factors. A conversion factor is a
ratio equal to 1, such as <span class="math inline">\(1000\ \text{L}/1\
\text{m}^3\)</span>. Arrange it so the unwanted unit cancels. For
example,</p>
<p><span class="math display">\[
2.4\ \text{m}^3\left(\frac{1000\ \text{L}}{1\ \text{m}^3}\right)=2400\
\text{L}.
\]</span></p>
<p>This cancellation shows why multiplying by 1000 is appropriate. If
the units do not cancel as intended, the conversion factor is upside
down or the relationship is incomplete. A rate such as litres per minute
is a fraction, so multiplying by minutes produces litres.</p>
<p>Exact numbers, such as 1000 L in 1 m³, do not limit precision.
Measured values do. Retain guard digits during a calculation and round
the final result once. A reported answer should not imply more precision
than the measurements support. Estimation provides a quick
reasonableness check: a basin roughly 20 m by 8 m by 3 m should have a
volume near 480 m³, not 48 m³ or 4800 m³.</p>
<div class="worked-example">
<p><span class="worked-example-title"><strong>Worked Example:
Calculating a rectangular basin volume</strong></span></p>
<p>A clearwell is 18.0 m long, 7.5 m wide, and filled to a depth of 3.2
m. Using <span class="math inline">\(V=LWD\)</span>,</p>
<p><span class="math display">\[
V = (18.0)(7.5)(3.2) = 432\ \text{m}^3.
\]</span></p>
<p>The numerical result is 432, and its unit is cubic metres because
three lengths were multiplied.</p>
</div>
<div class="try-it">
<p><strong>Try It.</strong> Find the volume of a basin measuring 12 m by