Mathematics — A First Course

Chapters

1. Sets
   • A rule that always answers yes or no
   • Two ways to name a set, and the set with nothing in it
   • Finding out what a description really contains
   • Being inside a set, and being contained by one
   • The real line's own subsets
   • The set of every way to choose
   • Fixing the frame, and drawing what's inside it
   • What two sets share, and what combining them gives
   • Everything a set leaves out
   • Two overlapping groups, and the one fact you can't skip
   • Three overlapping groups, and when the numbers don't add up
2. Relations and Functions
   • A Rule That Cannot Contradict Itself
   • Building Every Possible Pairing
   • Writing Down a Rule Between Two Sets
   • Does Every Target Get Hit?
   • The One Rule That Decides Whether It Is a Function
   • Functions Written as a Single Formula
   • Rules Written in Cases
   • When Two Functions Do Not Already Share Their Ground
3. Trigonometric Functions
   • More than a corner of a triangle
   • A unit the circle hands you for free
   • Sine and cosine, freed from the right angle
   • What the picture already decided
   • Six functions, and what each one refuses to do
   • One proof that pays for the rest of the chapter
   • The angle added to itself
   • A sum in disguise as a product
   • One equation, and every moment it is true
   • Every trick at once
4. The Principle of Mathematical Induction
   • When Enough Cases Are Never Enough
   • A Chain Reaction, Made Precise
   • The Principle, Stated and Defended
   • Sums That Add Up Exactly
   • Sums That Collapse
   • Proving an Inequality Instead of an Equality
   • A Property, Not Just a Sum
   • What the Proof Covers, and What It Doesn't
5. Complex Numbers and Quadratic Equations
   • A Puzzle No Pair of Real Numbers Can Solve
   • Naming a Number That Squares to Negative One
   • Checking That the Old Algebra Still Works
   • The Cycle of Four, and Where a Familiar Rule Breaks
   • Two New Measurements, Size and a Mirror Image
   • The Inverse, Done Properly, and What It Buys You
   • Reading a Product's Size Without Expanding It
   • A Complex Number Is Also a Point
   • Giving Every Point Exactly One Angle
   • Every Quadratic, Finished
6. Linear Inequalities
   • When "Exactly" Becomes "At Most"
   • The Two Rules, and the One That Bites
   • One Inequality, Three Different Answers
   • Turning Words into Inequalities
   • A Line Splits the Plane in Two
   • Picking a Test Point Without Getting Fooled
   • When Several Constraints Must All Hold at Once
   • One Claim, Three Parts, One Set of Moves
   • Real Ranges: Converting Units and Diluting Mixtures
7. Permutations and Combinations
   • When choices multiply, and when they add
   • The shorthand behind every count in this chapter
   • When the order you choose in changes the count
   • The check the algebra will not do for you
   • Overcounting on purpose, then correcting for it
   • Keeping objects together, keeping them apart
   • The same choices, when order stops mattering
   • Shortcuts a combination count already obeys
   • Splitting "at least" into cases you can actually count
8. The Binomial Theorem
   • The pattern inside every power of a binomial
   • The triangle that already contains every coefficient
   • Why the pattern is not a coincidence
   • Does a minus sign change more than one term?
   • Squeezing one exact fact out of an enormous power
   • Finding term number eight without writing the other seven
   • One middle term, or two
   • The one term that has no x in it at all
   • One coefficient, hiding inside two expansions at once
9. Sequences and Series
   • A rule that predicts every future term
   • When a list becomes a single number
   • The fixed amount: naming a constant-step sequence
   • Adding up a constant climb
   • Splitting a gap into equal steps
   • The fixed factor: naming a constant-multiple sequence
   • Adding up a constant multiply
   • Turning an impostor into a genuine geometric series
   • Splitting a gap by a constant factor, and what stays positive
   • Building a new sum from formulae you already trust
   • When two progressions share a hidden condition
10. Straight Lines
    • Turning a tilt into a number
    • When two slopes agree, or exactly disagree
    • The angle between two lines, in one formula
    • From a point and a direction to an equation
    • Fixing a line by where it crosses the axes
    • Fixing a line by its closest point to the origin
    • One equation behind every form
    • How far a point sits from a line, and from another line like it
    • Where a reflected point lands
11. Conic Sections
    • The angle that decides the curve
    • One point, one distance, one equation
    • Equal to a point, equal to a line
    • How big must the sum be?
    • Which way the difference goes
    • Reading a curve's family off its own equation
    • Choosing where the origin goes
    • A locus with no coordinates to start from
12. Introduction to Three-Dimensional Geometry
    • The third number a flat plan cannot record
    • Building a point's three numbers, and reading them back
    • How far apart two points in space really are
    • What repeated distances reveal about a shape
    • Landing on a point a chosen fraction of the way
    • One ratio that also tests three points for a line
    • The balance point of a triangle's three vertices
    • When a distance condition describes no point at all
13. Limits and Derivatives
    • How fast, exactly, at this instant?
    • Guessing the destination without arriving
    • Which way you arrive changes what you can promise
    • A toolkit, and its first payoff
    • When cancelling will not work, trapping the answer between two you already know
    • Turning two limits into a technique for a hundred more
    • Coming back to the cliff, now with the right tool
    • The derivative is a function too
    • Building new derivatives out of old ones
    • Every trigonometric derivative, for the price of two
    • One proof idea, reaching further than it looks
14. Mathematical Reasoning
    • A Sentence You Can Actually Judge
    • Saying No to a Claim, Precisely
    • What And and Or Actually Promise
    • Every Case, Some Case, and What Denying Them Means
    • The Shapes One Conditional Can Take
    • The Proof That Matches the Claim's Shape
    • Contradiction, and the One Case That Breaks a Claim
15. Statistics
    • The Simplest Measure, and What It Throws Away
    • Why a Deviation Has to Become a Distance
    • When Values Repeat: Weighting Deviations by Frequency
    • Data You Only Know by the Bracket It Falls In
    • Why Squaring, and Why Divide by n
    • The Same Variance, Now Weighted by Frequency
    • The Shortcut Returns, This Time for Spread
    • Comparing Variability When the Units Don't Match
    • Two Ways to Change Every Value at Once
    • Two Sums Remember Everything You Need
16. Probability
    • A Wheel That Is Not What It Looks Like
    • When One Stage Depends on Another
    • Every Question About the Experiment Is a Subset
    • Or, And, and Not: Building New Events from Old Ones
    • When Two Events Cannot Both Happen, and When One Of Them Must
    • Probability, Redefined From Three Rules
    • When Every Outcome Really Is Equally Likely
    • The Complement: The Fast Way to Handle At Least One
    • The Addition Theorem, When Two Events Overlap