long-form-math

What this skill does

# Long-Form Mathematics

Write mathematics the way the best textbooks do: with motivation, intuition, scratch work, and post-proof reflection. Every proof tells a story. Every definition earns its place.

## When to Use

- Writing or explaining proofs
- Introducing mathematical definitions, theorems, or concepts
- Tutoring someone in mathematics
- Creating educational math content (lecture notes, textbook-style explanations)
- Answering "why" questions about mathematical results
- Working through problem sets with a student

## Core Philosophy

| Principle | Meaning |
|-----------|---------|
| Understanding over economy | A longer explanation that builds understanding beats a terse proof that sacrifices clarity |
| Show the thinking | Make the invisible reasoning process visible -- scratch work, false starts, technique selection |
| Words between symbols | Never present bare equation chains; every step gets connective text explaining WHY |
| Math is communication | Writing quality matters as much as correctness; the reader must be convinced AND enlightened |

## Proof Writing Workflow

Every proof follows three phases. Do not skip phases 1 and 3.

### Phase 1: Pre-Proof Strategy

Before writing the proof, show the reader how the proof was discovered.

**Scratch Work.** Work backward from the desired conclusion. Show the exploratory reasoning that reveals what to prove and how. Label this section explicitly.

> Example: "We want to show |a_n - 0| < e. Unraveling, this means 1/n < e, so n > 1/e. This tells us to pick N = ceil(1/e)."

**Technique Selection.** State which proof technique to use and WHY.

| Try this technique... | When... |
|----------------------|---------|
| Direct proof | The hypothesis gives enough structure to reach the conclusion |
| Contrapositive | The hypothesis yields a complicated expression; starting from ~Q is simpler |
| Contradiction | The result sounds "negative" (no, never, impossible, does not exist) |
| Construction | The result asserts existence ("there exists...") |
| Cases | The domain splits naturally (even/odd, positive/negative/zero) |
| Both directions | The result is biconditional ("if and only if") |

**Proof Idea.** For long proofs, give a plain-English summary of the high-level strategy before diving into details.

> Example: "The idea is to show |a - b| < e for all e > 0, forcing |a - b| = 0."

### Phase 2: The Proof

Write the polished proof following these structural rules:

1. **State assumptions first.** Open with "Assume that..." or "Let... be..."
2. **Type every variable.** When introducing k, say "where k is an integer." Never leave a variable untyped.
3. **Justify each step.** Explain WHY each step is taken, not just WHAT happens. Use phrases like "By [theorem]...", "Since [established fact]...", "By definition of..."
4. **Words between equations.** Connect every equation to the next with explanatory text. Vary transitional words: therefore, thus, hence, consequently, so, it follows that.
5. **Conclude explicitly.** End by stating what has been shown: "Therefore n^2 is even, as desired."

**Correct pattern:**

> Since x is even, we can write x = 2k for some integer k. Then x^2 = (2k)^2 = 4k^2 = 2(2k^2). Since 2k^2 is an integer, it follows that x^2 is even.

**Incorrect pattern:**

> x = 2k
> x^2 = 4k^2 = 2(2k^2)
> Therefore x^2 is even.

The incorrect version is a bare equation chain with no connective text and no typing of variables.

### Phase 3: Post-Proof Analysis

After the proof, reflect on it.

- **The Moral.** State what the result teaches us in plain English. ("The moral: two sets can have the same size even though one is a proper subset of the other.")
- **Could it be simpler?** If a more elegant proof exists, show it.
- **Alternative approaches.** If a different technique illuminates a different aspect, present it.
- **Reverse engineering.** Show how the polished proof was constructed from scratch work -- how we went from goal-backward exploration to hypothesis-forward presentation.

## Exposition Patterns

When introducing a new topic or definition, follow this sequence:

### The Introduction Sequence

```
Motivating problem  -->  Intuition  -->  Formal definition  -->  Examples + Non-examples
```

1. **Motivating problem first.** Open with a paradox, puzzle, question, or historical problem that creates NEED for the concept. Never open with a bare definition.

   > "So, what do you think? Does slow and steady win the race? Zeno's paradox tells us..."

2. **Intuition before formalism.** Give the informal idea first. Build understanding of what the definition captures before stating it precisely.

   > "What the definition means is that eventually the points get 'arbitrarily close' to a."

3. **Formal definition.** Now state the precise definition. The reader is ready for it.

4. **Examples and non-examples.** Immediately follow with concrete examples that satisfy the definition AND examples that fail it, explaining why they fail.

   > "The integers Z almost form a field; they only fail the second half of Axiom 5."

### Additional Exposition Techniques

- **Questions that drive investigation.** Pose numbered questions that structure sections around intellectual curiosity. ("Does there exist a function continuous at one point but not at another?")
- **Historical storytelling.** Weave mathematical history into the exposition as narrative, not sidebar decoration. The story of Hippasus and the irrationals, Cantor and infinity, Archimedes and exhaustion -- these ARE the exposition.
- **Progressive disclosure.** Reference future topics: "We will come back to this." Build concepts incrementally. State a "moral" after each result, with each moral slightly more surprising than the last.
- **The almost-correct proof.** Show a proof attempt that fails, diagnose WHY it fails, then fix it. This teaches more than a perfect proof alone.
- **Investigation tables.** Create tables with "???" entries that get filled in as the exposition progresses, giving the reader an ongoing investigation to follow.

## Mathematical Writing Rules

Quick reference for clean mathematical prose. These rules apply to ALL mathematical writing in this style.

### Symbol Rules

| Rule | Bad | Good |
|------|-----|------|
| Never start a sentence with a symbol | "x^2 - 6x + 8 = 0 has two roots." | "The equation x^2 - 6x + 8 = 0 has two roots." |
| Separate adjacent symbols with words | "With the exception of a, b is the only root." | "With the exception of a, the number b is the only root." |
| Use words, not logical shorthand | "forall x, exists y s.t. ..." | "For every x, there exists a y such that..." |
| Explain every new symbol | "Then n = 2k + 1." | "Then n = 2k + 1, where k is an integer." |
| Use frozen symbols conventionally | Using m for a function | Use m, n for integers; f, g for functions; A, B for sets |
| Use consistent symbol families | "x = 2a and y = 2r" | "x = 2a and y = 2b" (or "x = 2r and y = 2s") |

### Language Rules

| Rule | Bad | Good |
|------|-----|------|
| Use "we" as default pronoun | "I will now show..." | "We will now show..." |
| Avoid "clearly/obviously" | "Clearly, n^2 is positive." | "Since n is nonzero, n^2 is positive." |
| Use "each/every" not "any" | "For any integer n..." | "For every integer n..." |
| Never use "Since...then" | "Since n is odd, then n^2 is odd." | "Since n is odd, it follows that n^2 is odd." |
| "Since" = established fact | "If n is odd" (when n is already known odd) | "Since n is odd" (for established facts) |
| Vary connective words | "Therefore... Therefore... Therefore..." | "Therefore... Thus... It follows that... Hence..." |
| Write complete sentences | Bare equation with no punctuation | Every equation is part of a sentence, ending with a period |

### Formatting Rules

- **Inline vs. display:** Short expressions inline; long expressions as centered displays.
- **Line breaks:** When breaking an expression, end the first line with the operator (+, -, =, <).
- *