I. The Language of Algebra, with Quantity, Units, and Accuracy
001 What Quantity Is
002 Magnitude and Multitude
003 Quantity Is Measured Only by Quantity of Its Own Kind
004 The Unit of Measure
005 Fixing a Unit Where There Is None in Nature
006 Number as a Collection of Units
007 Number as a Proportion to the Unit
008 The Numerical Value of a Quantity
009 The Same Quantity Under Different Units
010 Abstract Number and Concrete Number
011 Mathematics, the Science of Quantity
012 Naming the Quantity Before Measuring It
013 Choosing a Unit Fit for the Purpose
014 Compound Units
015 Keeping Units Consistent Through a Computation
016 The Unit as a Check on the Reasoning
017 The Limit of Exactness in Every Measurement
018 Choosing a Level of Accuracy
019 A Result Is No More Exact Than Its Measurements
020 What a Theorem Requires
021 What a Problem Requires
022 What Algebra Is
023 Arithmetic and Algebra Distinguished
024 Why Letters Are Used
025 Known and Unknown Quantities
026 A Sign Stands for Words
027 The Sign of Equality
028 The Sign of Addition
029 The Sign of Subtraction
030 The Order of the Terms Is Indifferent
031 The Sign of Multiplication
032 Factors and the Continued Product
033 The Sign of Division
034 The Signs of Inequality and of Infinity
035 The Signs as Marks of Quality
036 Positive and Negative Quantities
037 Negative Quantity Is Less Than Nothing
038 The Two Series Reckoned from Zero
039 Integers, and What Lies Between Them
040 A Quantity and Its Opposite Destroy Each Other
041 The Absolute Value of a Quantity
042 Taking a Greater from a Less
043 The Numeral Coefficient
044 The Literal Coefficient
045 What a Power Is
046 The Exponent
047 What a Root Is
048 What an Algebraic Expression Is
049 The Term, or Monomial
050 The Polynomial
051 Binomial, Trinomial, and Residual Quantity
052 The Numerical Value of an Algebraic Expression
053 Like and Unlike Quantities
054 The Parenthesis
055 The Reciprocal
II. Equations and What They Assert
056 The Equation, and Its Place in Algebra
057 The Degree of an Equation
058 The Identical Equation
059 An Equation as the Statement of a Question
060 The Root of an Equation
061 The Axioms on Which Solution Rests
062 Transposition
063 Clearing an Equation of Fractions
064 The Unknown Connected by Addition or Subtraction
065 The Unknown Connected by Multiplication or Division
066 The Rule for the Solution of Simple Equations
067 Equations Containing the Absolute Value of the Unknown
068 What It Is to State a Problem in Algebraic Language
069 Naming the Unknown
070 Finding the Statement of Equality in the Situation
071 Expressing Each Quantity of the Problem by the Letter
072 Writing the Equation
073 Reading the Equation Back Against the Situation
074 Proving the Root Against the Problem, Not Against the Equation
075 Problems Producing Simple Equations of One Unknown
III. Inequalities
076 Why Unequal Quantities Must Be Compared
077 The Same Sense and the Contrary Sense
078 Adding or Subtracting the Same Quantity to Both Members
079 Adding Two Inequalities of the Same Sense
080 Why Two Inequalities May Not Be Subtracted
081 Multiplying or Dividing by a Positive Number
082 Multiplying or Dividing by a Negative Number
083 Changing the Signs of All the Terms
084 Raising Both Members to a Power, or Extracting a Root
085 Finding the Limit of the Unknown
086 Inequalities Containing an Absolute Value
087 When the Absolute Value Is Less, and When Greater, Than a Given Quantity
088 The Sum of Two Squares Exceeds Twice Their Product
IV. The Reasoning of Problems
089 Generalization
090 The Literal Equation
091 A Rule Obtained by Solving in Letters
092 When the Sum and the Difference Are Given
093 The Problem of Two Workmen
094 Deducing a Rule for Dividing
095 Negative Solutions
096 The Discussion of a Problem
097 The Problem of the Couriers
098 The Independent Equation
099 The Indeterminate Equation
100 Cases of Indetermination
101 Impossible Problems
V. Functions
102 What a Function Is
103 The Domain and the Range
104 The Independent and the Dependent Quantity
105 Function Notation
106 Reading a Statement in Function Notation in Terms of Its Situation
107 The Graph of a Function
108 A Function Given as a Rule, a Table, a Graph, or in Words
109 Comparing Two Functions Presented Differently
110 The Domain Fixed by the Situation
111 What an Intercept Tells
112 Where a Function Increases and Where It Decreases
113 Where a Function Is Positive and Where Negative
114 The Greatest and the Least Values a Function Takes
115 Symmetry in a Function
116 End Behavior
117 What a Rate of Change Is
118 The Average Rate of Change Over an Interval
119 A Constant Rate of Change
120 Writing a Function from a Description
121 A Function Defined by a Recursive Process
122 Combining Functions by Addition and Multiplication
123 A Sequence as a Function
124 The Arithmetic Sequence, Recursively and Explicitly
125 The Geometric Sequence, Recursively and Explicitly
VI. The Coordinate Plane
126 Why Position Must Be Fixed by Number
127 The Two Axes and the Origin
128 The Co-ordinates of a Point
129 The Signs of the Co-ordinates, and the Four Quarters of the Plane
130 The Distance Between Two Points
131 Any Single Equation Between the Co-ordinates Represents a Locus
132 Tracing a Locus by Means of Points
133 An Equation Wanting the Absolute Term Passes Through the Origin
134 Two Simultaneous Equations Represent Determinate Points
135 The Equation of a Right Line Is Always of the First Degree
136 Every Equation of the First Degree Represents a Right Line
137 The Inclination of a Line, and What Measures It
138 The Equation in Terms of Slope and Intercept
139 The Equation in Terms of the Two Intercepts
140 The Line Through Two Fixed Points
141 The Angle Between Two Right Lines
142 The Condition That Two Lines Be Parallel
143 The Condition That Two Lines Be Perpendicular
144 The Line Through a Given Point Parallel to a Given Line
145 The Line Through a Given Point Perpendicular to a Given Line
146 Solving a System by Graph
147 The Graph of a Linear Inequality as a Half-Plane
148 The Graph of a System of Linear Inequalities
149 Choosing the Scale and the Origin of a Graph
150 Reading a Graph of Data
VII. Systems
151 Equations Containing Two Unknown Quantities
152 What Elimination Is
153 Elimination by Substitution
154 Elimination by Comparison
155 Elimination by Addition and Subtraction
156 Problems Producing Equations of Two Unknown Quantities
157 Equations Containing Three or More Unknown Quantities
158 The Three Methods Applied to Three Unknowns
159 Problems Producing Equations of Three or More Unknowns
VIII. Polynomials
160 What Algebraic Addition Is
161 Adding Similar Quantities with Like Signs
162 Adding Similar Quantities with Unlike Signs
163 Adding Quantities Not Similar
164 What Algebraic Subtraction Is
165 The Difference of Two Similar Quantities
166 The Difference of Quantities Not Similar
167 Addition and Subtraction as One Operation
168 Comparing Two Negative Quantities
169 Algebraic and Arithmetical Subtraction Compared
170 What Algebraic Multiplication Is
171 The Exponents in Multiplication
172 Multiplying by Parts
173 The Rule of the Signs
174 What a Negative Multiplier Signifies
175 Multiplying a Polynomial by a Monomial
176 Multiplying a Polynomial by a Polynomial
177 What Algebraic Division Is
178 Division as the Inverse of Multiplication
179 The Exponents in Division
180 The Signs in Division
181 Dividing a Polynomial by a Monomial
182 Dividing a Polynomial by a Polynomial
183 The Square of the Sum of Two Quantities
184 The Square of the Difference of Two Quantities
185 The Product of the Sum and the Difference
186 The Negative Exponent
187 The Zero Exponent
188 The Difference of Two Squares Divided by Their Difference
189 Two Further Theorems
190 Powers of Ten
191 Expressing a Great Quantity as a Digit Times a Power of Ten
192 Expressing a Small Quantity in the Same Form
193 Multiplying and Dividing Quantities So Expressed
IX. Factoring
194 What a Divisor, or Measure, Is
195 The Prime Number
196 Quantities Prime to Each Other
197 The Composite Quantity
198 Separating a Monomial into Its Prime Factors
199 Separating a Polynomial When One Factor Is a Monomial
200 Factoring by Reversing the Theorems
201 The Perfect-Square Trinomial
202 The Difference of Two Squares
203 The Difference of the Same Powers
204 The Sum and the Difference of Two Cubes
205 The Difference of Even Powers Above the Second
206 What a Quadratic Trinomial Is
207 Factoring a Quadratic Trinomial
208 The Use of Factoring
209 What the Greatest Common Divisor Is
210 The G.C.D. of Monomials
211 The G.C.D. of Two Polynomials
212 Factors Not Common May Be Rejected
213 Multiplying by a Factor Not Common
214 What a Multiple Is
215 The Least Common Multiple
216 Finding the Least Common Multiple
217 The L.C.M. Found by Means of the G.C.D.
X. Algebraic Fractions
218 What an Algebraic Fraction Is
219 Entire, Mixed, and Improper Quantities
220 Simple, Compound, and Complex Fractions
221 Multiplying the Numerator
222 Dividing the Numerator
223 Multiplying the Denominator
224 Dividing the Denominator
225 Multiplying Both Terms
226 Dividing Both Terms
227 Reducing a Fraction to Its Lowest Terms
228 Reducing a Fraction to an Entire or Mixed Quantity
229 Reducing a Mixed Quantity to a Fraction
230 The Signs of a Fraction
231 Reducing an Entire Quantity to Fractional Form
232 Converting a Fraction to One with a Given Denominator
233 Reducing Fractions to a Common Denominator
234 Reducing Fractions to the Least Common Denominator
235 The Addition of Fractions
236 The Subtraction of Fractions
237 Multiplying a Fraction by an Entire Quantity
238 Multiplying a Fraction by a Fraction
239 Dividing a Fraction by an Entire Quantity
240 Dividing by a Fraction
241 Reducing a Complex Fraction to a Simple One
242 The Fractional Equation
243 Clearing a Fractional Equation, and What It Risks
244 Extraneous Roots, and Why They Arise
XI. Roots and Radicals
245 Involution, or the Formation of Powers
246 The Signs of the Different Powers
247 Raising a Monomial to a Given Power
248 Raising a Polynomial to a Given Power
249 Raising a Fraction to a Power
250 Evolution
251 How a Square Is Composed
252 Extracting the Square Root of a Number
253 The Square Root of a Fraction
254 The Perfect Square
255 The Approximate Square Root
256 The Square Root of a Monomial
257 The Square Root of a Polynomial
258 When a Quantity Is Not a Perfect Square
259 Radicals of the Second Degree
260 The Reduction of Radicals
261 The Addition and Subtraction of Radicals
262 The Multiplication and Division of Radicals
263 Rationalizing a Denominator
264 What a Fractional Exponent Means
265 Why the Properties of Exponents Require This Meaning
266 Radicals and Fractional Exponents as Two Notations for One Thing
267 Simple Equations Containing Radicals
268 Why Squaring Both Members May Introduce a Root
XII. Quadratics
269 What a Quadratic Equation Is
270 Pure and Affected Quadratics
271 Every Quadratic Reduced to a Standard Form
272 Solving the Pure Quadratic
273 The Two Roots of a Pure Quadratic
274 The Form of the Affected Quadratic
275 Completing the Square
276 Solving the Affected Quadratic
277 The Quadratic Formula
278 Every Quadratic Has Two Roots
279 The Sum and the Product of the Roots
280 The Discriminant, and What It Foretells of the Roots
281 Equations in Quadratic Form
282 Quadratic Equations Containing Two Unknown Quantities
283 The Graph of a Quadratic Function
284 The Vertex, and the Greatest or Least Value
285 The Axis of Symmetry
286 The Intercepts, and the Roots as Intercepts
