My book will no doubt be flawed in many ways of which I am not yet aware, but there is one “sin” that I have intentionally committed, and for which I shall not repent: many of the arguments are not rigorous, at least as they stand. This is a serious crime if one believes that our mathematical theories are merely elaborate metal constructs, precariously hoisted aloft. Then rigor becomes the nerve-racking balancing act that prevents the entire structure from crashing down around us. But suppose one believes, as I do, that our mathematical theories are attempting to capture aspects of a robust Platonic world that is not of our making. I would the contend that an initial lack of rigor is a small price to pay if it allows the reader to see into this world more directly and pleasurably than would otherwise be possible.

—Tristan Needham, Visual Complex Analysis xi

combinatorics

• it’s time to party (intro motivational problems)
• counting: orderly versus disorderly
• combinatorics to polynomials segue
• some miscellaneous example problems with binomial expansions, etc.

algebraic synæsthesia

• parent functions and their linear transformations
• Polynomials (and sketching them!)
  • some fun factoring problems, as review
  • Persuading Parabolas into Vertex Form (completing the square as a path to enlightenment)
  • Perfecting our Perception of Parabolas (writing assignment to derive some cool parabola theorems) and related: Andrew’s favorite mathematical writing tip
  • calculus-free cubic optimization!
  • Taylor’s Theorem (Math 3 version) (what if we had infinitely-long polynomials?!?)
  • Finding the roots of all polynomials (the linear, quadratic, and cubic formulas, and a soupçon of what’s beyond!)
  • the QUARTIC formula!!!! (formulas calculated by Sage and typeset by \(\LaTeX\) in a tiniest-font 72-inch wide PDF)
• Rational functions (what if we could divide polynomials?)
  • Blowing up fractions! (a fun algebraic technique, more blandly known as partial fraction decomposition)
  • some fun rational function sketching problems

logarithms

• The World Made From Modular Machines (notes on inverse functions, as a primer)
• exponent rules reference sheet (svg)
• fun intro exploratory logs problems
• Logarithms! (intro lengthy exposition and proofs of some basic properties)
• Andrew’s online visual logarithms calculator/pun
• LOGPOCALYPSE
  • logpocalype redux (solution notes and further problems)
• tricky quadratic logarithms problem (and solution)
• log base play (and solution notes)

trig

• Radians
  • printable fun-fun radians reference poster: svg, pdf
  • some example radians to-and-from degrees conversions
• printable fun-fun right-triangle trig functions reference poster: svg, pdf
• Trig Functions, Turbocharged (the unit circle definition of trig functions)
  • printable fun-fun unit circle trig functions reference poster: svg, pdf
  • some example unit circle-based calculations
• the pythagorean theorem (noah schweber’s crayon-on-wall proof)
• trig identities
• trig equations
  • examples of quadratic trig equations
  • examples of trig equations with stuff inside the argument
• trig word problems
  • trig word problems example solutions
• the super-pythagorean theorem: inducement, notes
• Sum and difference identities: Part I, Part II
• Write Your Own Word Problem

complex nummies

• Numbers, Complicated (introductory notes on complex numbers)
• Let’s multiply things by \(i\)! (exploratory problems)
• Multiplication by \(i\) as rotation
• Complex roots, in a Cartesian sense (project to figure out the cubic/quartic roots of \(i\) in rectangular)
• When It’s \(64\) (short fun problem to compute all of the six sixth roots of \(64\) in rectangular) (solution notes)
• Let’s draw powers of \(i\) and stuff! (exploratory problems to motivate polar coordinates)
• Circumpolar Navigation (notes on polar coordinates)
• Polar Tetris (Andrew’s circular version of Tetris)
• Hilariously Hideous Complex Fractions (fun small exploratory problem involving iterated complex fractions)
• Exponential Improvements (notes on the exponential form and Euler’s Identity)
  • notes on the complex nummies derivation of the sine/cosine sum-and-difference formulas
• \(\sqrt{\text{Complex!}}\) (exploratory problems to figure out complex roots)
• Numbers, Spinning and Unspinning (notes on complex roots)
• Complex^{ComplexComplex…} (notes on complex exponentiation) (also available in zine form)
• Cole’s Conjecture and Maya’s Musing (solution notes)
• Prime numbers aren’t prime (exploratory problems on Gaussian integers)

sequences and serieseses

• sequences and series cold open
• sequences and series basic vocabulary and some fun problems
• A Whole Party of Infinite Friends (some very fun very diverse exploratory intro series problems):
  • problems
  • detailed solution notes
• does \(1/n^2\) converge?
• when does \(n^p\) converge?

calculus!!!

• In the Beginning Was The Derivative
• the derivative of \(x^n\)
  • poster version
• Dffrntn Shrtcts (differentiation laws)
  • some fun problems with the product rule
  • fun theorem about polynomials, their derivatives, their zeroes, and their repeated and non-repeated roots
• trig derivatives (statements, not proofs)
• Oh shoot, we need a more rigorous understanding of limits!
  • limits, intuitively
  • limits, more rigorously
• okay, now we can actually prove that the derivative of sine is cosine!
• differentiating logs and exponentials
  • some alternate proofs of basic differentiation laws
• what’s the derivative of inverse tangent?
• (Implicit) Differentiation
• Related rates
• Optimization
• Integration
  • Some totally unremarkable problems with integrals (exploratory problems)
  • Integrals as net area
• Antiderivatives (just… several hundred antiderivative problems!)
• Putting Humpty-Dumpty Back Together (undoing the chain rule)
• Undoing the product rule (integration by parts )
  • Integration by parts: some fun examples
  • Integrating By Parts By Parts By Parts By Parts… (iterated integration by parts)
• infinite integrals (people call these “improper integrals,” which is silly)
• taylor series intro!
  • taylor series derivation!
  • taylor series, more algorithmically
  • but sometimes you can only polynomialize functions over finite ranges
  • can we think about the error term?
• how long are curvy lines?
• higher integrals

linalg

• linear maps