Precalculus

Overview

• Primary reference: OpenStax, Precalculus 2e (free, CC BY)
• Supplementary: OpenStax, College Algebra 2e (free) for the algebra review; Paul’s Online Notes, Algebra and Trig Review (free); 3Blue1Brown, Euler’s formula with introductory group theory (free)
• Prerequisites: high-school algebra (solving linear equations, manipulating fractions and powers)
• Estimated time: 2 to 3 weeks at 10 to 12 h/week; in the course, the first stages of Pass 2

Key Takeaways

• Logarithms turn products into sums and change the unit of information. A model’s loss in nats becomes bits by dividing by $\ln 2$; bits per byte is the comparison every tokenizer and model in the course is judged by.
• A rotation is a multiplication by a unit complex number. Rotating a 2D pair by $\theta$ is multiplying $x + iy$ by $e^{i\theta}$; RoPE applies that to every pair of a query and a key.
• Geometric sequences are frequency ladders. RoPE’s inverse frequencies $b^{-2i/d}$ and ALiBi’s head slopes are geometric sequences; their sums and ratios decide how far a position signal reaches.
• How you evaluate a formula matters as much as the formula. Horner’s rule is the cheapest and most stable way to evaluate a polynomial, and the textbook quadratic formula can lose most of its digits for the small root when $b^2 \gg 4ac$.

How to Study

Read each OpenStax chapter with a notebook open and check every identity numerically before you trust it (math.log, cmath.exp, numpy). In the course, take the solve set S-M00 first: it is the pen-and-paper check of the four build modules, then build M00.1 to M00.4 in order with ol start <ID> and ol check <ID>.

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Concepts & Techniques

Core Insight

Precalculus is the toolbox the rest of the math is written in: exponents and logarithms, trigonometry and complex numbers, sequences and series, and polynomials. Each one has a call site in an LLM system: logarithms measure information, rotations encode position, geometric ladders set frequencies, and polynomials approximate the functions a C kernel cannot call.

1. Exponents, logarithms, and units of information

Key ideas:

• Laws: $a^{m}a^{n} = a^{m+n}$, $\log_b(xy) = \log_b x + \log_b y$, change of base $\log_b x = \ln x / \ln b$.
• Units: a negative log-likelihood summed in nats over a text of $n$ bytes is $\text{bpb} = \text{nll} / (n \ln 2)$ bits per byte, a number comparable across tokenizers.
• Call sites: M11.2 perplexity and bits per byte, L1.6 tokenizer metrics, L6.7 the model-zoo table, C1.

2. Trigonometry, rotations, and complex numbers

Key ideas:

• Unit circle: $(\cos\theta, \sin\theta)$; a 2D rotation is the matrix $\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}$, which preserves length and composes by adding angles: $R(a)R(b) = R(a+b)$.
• Euler’s formula: $e^{i\theta} = \cos\theta + i\sin\theta$, so the same rotation is one complex multiplication.
• Call sites: L5.4 sinusoidal positional encoding, L7.3 RoPE.

3. Sequences, series, and frequency ladders

Key ideas:

• Geometric sequence: $a, ar, ar^2, \dots$ with sum $a(1 - r^n)/(1 - r)$ for $r \ne 1$.
• Ladders: RoPE inv_freq is $b^{-2i/d}$ for $i = 0, \dots, d/2 - 1$; ALiBi slopes are a geometric sequence per head, with a rule for head counts that are not powers of two.
• Call sites: L7.3 RoPE, L7.4 ALiBi and YaRN, M02.2, M10.4 schedules.

4. Polynomials, Horner’s rule, and stable roots

Key ideas:

• Horner: $a_0 + x(a_1 + x(a_2 + \dots))$ uses $n$ multiplications and $n$ additions and is the form every polynomial kernel uses.
• Cancellation: when $b^2 \gg 4ac$ one root of $-b \pm \sqrt{b^2 - 4ac}$ subtracts two nearly equal numbers; compute the large-magnitude root first and get the other from $x_1 x_2 = c/a$.
• Call sites: M02.1 Taylor polynomials, M09.6 the polynomial core of tl_expf.

Course modules

Module	Topic	Kind	Pass
S-M00	Precalculus problem set (checked by SymPy)	solve	2
M00.1	Exponents, logs, change of base, units of information	build	2
M00.2	Trig, unit circle, 2D rotations, complex numbers, Euler’s formula	build	2
M00.3	Sequences, geometric series, frequency ladders	build	2
M00.4	Polynomials, Horner, stable quadratic roots	build	2
M00.5	Functions, inverses, monotonicity, inequalities	solve	solve set

M00.5 (functions, inverses, monotonicity, inequalities) is solve-only: it appears as items of S-M00 and is used by M01.1 and M05.2.

Chapters

#	Module	Chapter	Kind	Pass
1	S-M00	Solve set: functions, exp/log, trig and Euler, series, polynomials, inequalities	solve	2
2	M00.1	Exponents, logs, change of base, units of information	build	2
3	M00.2	Trig, unit circle, 2D rotations, complex numbers, Euler’s formula	build	2
4	M00.3	Sequences, geometric series, frequency ladders	build	2
5	M00.4	Polynomials, Horner, stable quadratic roots	build	2

Connections to Other Tracks

Track	Connection
Calculus 1	limits and derivatives build on functions, exponentials, and logarithms
Numerical Methods and Floating Point	range reduction and polynomial approximation of exp (M09.6) start from Horner’s rule
Information Theory	entropy and perplexity are logarithms with a unit
tinyllm	RoPE, ALiBi, and bits per byte are the first places this math runs