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Chapter 1 — College Algebra and Pre-calculus

Part I — Mathematical Foundations · 3–6 weeks

What you will learn

This chapter covers the algebra and precalculus that every later chapter assumes: functions and their graphs; polynomial, rational, exponential, and logarithmic functions; trigonometric functions; summation notation; and an informal notion of a limit. Exponentials and logarithms are the components of the functions that convert the raw outputs of a large language model (LLM) into probabilities, and of the loss function used to train it. Sines and cosines are used by the Transformer architecture of Chapter 12 to encode the position of each word, and summation notation appears in nearly every formula of the machine learning literature. The informal treatment of limits prepares the definitions of the derivative and the integral in Chapter 2.

Topics

  • Real numbers, exponents, radicals, and factoring
  • Equations and inequalities
  • Functions: notation, domain, range, and graphs
  • Linear and quadratic functions
  • Polynomial and rational functions
  • Function transformations, composition, and inverses
  • Exponential and logarithmic functions and the laws of logarithms
  • Trigonometric functions, the unit circle, and the basic identities
  • Sequences, series, and summation (sigma) notation
  • An informal notion of limits, as preparation for Chapter 2

Resources

Suggested path. Use Khan Academy College Algebra as the main course, with the matching chapters of the OpenStax Precalculus textbook as the written reference and source of extra exercises. Continue with the Khan Academy Precalculus units on composite and inverse functions, trigonometry, series, and limits and continuity; the Lockdown Math lectures can run in parallel with the trigonometry and logarithm units. Finish with the Algebra/Trig Review of Paul's Online Math Notes as a readiness check for Chapter 2, and with lesson 1 of The Essence of Calculus. When time is short, skip the Coursera specialization, the Stewart textbook, and the Khan Academy Precalculus units on conic sections, vectors, and matrices (Chapter 3 covers vectors and matrices).

Online courses (MOOCs)

  • Khan Academy — College Algebra (free; videos + self-grading exercises, unit tests, and a course challenge; start here: linear, quadratic, polynomial, rational, exponential, and logarithmic functions; the Khan Academy math hub lists the earlier algebra courses for review of any weak skill).
  • Khan Academy — Precalculus (free; videos + self-grading exercises and unit tests; the second pass: composite and inverse functions, trigonometry, complex numbers, series, and limits and continuity).
  • Professor Leonard — Precalculus: College Algebra/Trigonometry (free YouTube playlist; more than 70 hours of full-length classroom lectures with worked examples; an alternative to the Khan Academy videos).
  • Johns Hopkins University (Coursera) — Algebra: Elementary to Advanced by Joseph W. Cutrone (free to audit, paid certificate; optional; 3 courses, about 4 weeks at 10 hours/week; equations, functions, polynomials, and roots with graded quizzes).

Books

  • Book: Jay Abramson et al., Precalculus (2nd ed., OpenStax, 2021) — official page (free PDF and web; start here: chapters 1–4 for functions, polynomials, exponentials, and logarithms, 5–7 for trigonometry, 11 for sequences and series, and 12 for an introduction to limits).
  • Book: Jay Abramson et al., College Algebra (2nd ed., OpenStax, 2021) — official page (free PDF and web; chapter 1, "Prerequisites", and chapter 2, "Equations and Inequalities", review exponents, radicals, and factoring).
  • Book: Carl Stitz and Jeff Zeager, Precalculus (3rd corrected ed., 2013) — official page (free PDF, CC BY-NC-SA; more rigorous than the OpenStax text, with a large set of exercises and answers; chapters 1–6 for functions and logarithms, 9 for sequences, 10–11 for trigonometry).
  • Book: James Stewart, Lothar Redlin, and Saleem Watson, Precalculus: Mathematics for Calculus (8th ed., Cengage, 2024) — author page (paid; optional; a widely adopted US precalculus textbook, in the same notation as Stewart's Calculus, which is cited in Chapter 2).
  • Book (reference): Paul Dawkins, Paul's Online Math Notes (Lamar University) — Algebra and the Algebra/Trig Review (free web notes with worked problems; the review is a diagnostic set of the algebra and trigonometry problems that Calculus I assumes).

Lectures, papers and articles

  • Lecture series: Grant Sanderson (3Blue1Brown), Lockdown Math (2020; free; recorded live lectures; start here: quadratics, trigonometry, complex numbers, logarithms, and the number e, presented graphically).
  • Lecture series: Grant Sanderson (3Blue1Brown), The Essence of Calculus (2017; free; lesson 1, watched at the end of this chapter, previews limits and rates of change; Chapter 2 uses the full series).

Milestone

Score at least 90% on the Khan Academy College Algebra course challenge and on the Khan Academy Precalculus unit tests for trigonometry, series, and limits and continuity. Then solve on paper, without notes: expand log(a^2 b / c) with the laws of logarithms and solve 3 · 2^x = 48 for x; write the sum of k · 2^k for k from 1 to 5 in sigma notation and evaluate it; and sketch y = 2 sin(x − π/2) + 1 starting from the graph of sin x. Verify each answer with a graphing calculator or a computer algebra system.

Estimated time

3–6 weeks.