STEM & LaTeX (Math Solving)

Prerequisites: Single-variable calculus (derivatives and integrals) and comfort with algebraic manipulation. No LaTeX experience needed; that is taught here. If the worked integrals below are unfamiliar, this track is not the one to start on.

A right answer reached the wrong way is still a failure

Mathematical annotation is demanding because it requires a kind of attention most people don’t default to: verifying reasoning steps rather than outcomes.

An AI can reach a correct numerical answer through an invalid process. It can apply a real technique to the wrong problem. It can skip steps that conceal an error. It can state a correct conclusion from premises that don’t support it. In all of these cases, the final answer might look fine, but the training signal is wrong.

This module covers two skills: LaTeX (the formatting language for math) and mathematical reasoning evaluation (verifying that AI solutions are not just numerically correct but logically sound). Both are required for STEM annotation at the Specialist and Subject Matter Expert tiers.

LaTeX has a small syntax you use constantly

LaTeX is the standard for rendering mathematical notation in AI training contexts. Inline math uses single dollar signs; display (block) math uses double dollar signs or \[ \].

Inline vs. display mode

Inline: The formula is $E = mc^2$ and it appears within the sentence.

Display mode (centered, larger):
$$
E = mc^2
$$

Common math constructs

Fractions:

$$\frac{numerator}{denominator}$$
$$\frac{d}{dx}\left(\frac{x^2+1}{x-1}\right)$$

Exponents and subscripts:

$$x^{n+1}$$           % exponent with multiple characters needs braces
$$a_i$$               % single-character subscript
$$\sum_{i=1}^{n} i^2$$   % summation with bounds: braces required for multi-char bounds

Square roots and nth roots:

$$\sqrt{x^2 + y^2}$$
$$\sqrt[3]{8} = 2$$

Greek letters: \alpha, \beta, \gamma, \delta, \epsilon, \theta, \lambda, \mu, \sigma, \pi, \omega, \Sigma (uppercase), \Delta (uppercase)

Operators: \cdot (multiplication dot, also used for vector dot products), \times (multiplication cross, also used for vector cross products), \leq, \geq, \neq, \approx, \equiv, \in, \subset, \cup, \cap, \infty

Matrices:

$$
A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}
$$

Common LaTeX errors to catch in AI output

Missing braces on multi-character exponents or subscripts: x^n+1 renders as $x^n + 1$, not $x^{n+1}$. The exponent only applies to the single character immediately following ^ unless braces wrap the full expression. The correct form is x^{n+1}.

Wrong operator: x * y in LaTeX renders literally as x * y. The correct forms are x \cdot y or x \times y. For vectors, \cdot and \times specifically mean the dot product and cross product; for scalars they’re just multiplication.

Mismatched delimiters: \left( must be paired with \right). An AI that opens a \left[ and closes with \right) produces a rendering error.

Forgetting math mode: Writing theta in prose instead of $\theta$ means the Greek letter is not rendered. It appears as the literal word “theta.”

Incorrect spacing: In LaTeX, spaces inside math mode are ignored. Use \, for thin space, \quad for medium space, \qquad for large space where readability requires it.

The total is given by $\sum_i=1^n x_i^2 + 1$.

$\sum_{i=1}^{n} x_i^2 + 1$

Evaluating mathematical reasoning step by step

The verification protocol

For each step in an AI’s mathematical solution:

1. Identify the operation: What mathematical rule or technique is being applied? (chain rule, integration by parts, completing the square, polynomial long division, etc.)
2. Verify the algebra: Does the arithmetic check out? Substituting numbers into symbolic steps is often the fastest verification method.
3. Check for skipped steps: An AI that jumps from step 2 to step 5 may be concealing an error in the omitted steps. If you cannot reconstruct the missing steps yourself, flag the gap.
4. Verify the conclusion: Does the final answer follow from the final step? Does it have the right units (for physics problems)? Is it in the required form?

Example: correct reasoning vs. flawed reasoning

Return to $\int x e^x \, dx$ from the opening exercise, and put the two solutions side by side.

Correct: integration by parts with $u = x$, $dv = e^x dx$, giving $du = dx$, $v = e^x$, and therefore $x e^x - e^x + C$. Each step follows from the technique, and the conclusion holds.

Flawed: $e^x \cdot \frac{x^2}{2} + C$, produced by treating $e^x$ as a constant and integrating $x$ alone, which is a technique that does not exist. The error isn’t numerical; it’s a misapplication of a rule. Your rationale should name the specific mistake: “The AI treated $e^x$ as a constant factor, which is invalid since $e^x$ is a function of $x$. The correct technique is integration by parts.”

Olympiad-level and competition math

High-end STEM annotation tasks often involve competition mathematics (AMC, AIME, Putnam, IMO level). These require proof-based reasoning, not just computation.

Proof structure: A valid proof must establish a claim for all cases, not just example cases. An AI that “proves” a universal statement by checking three examples has produced an invalid proof.

Common proof techniques:

• Mathematical induction: Base case + inductive step. Both must be established. A common AI error is proving only the base case, or stating the inductive hypothesis without proving the inductive step follows from it.
• Proof by contradiction: Assume the negation, derive a genuine contradiction. The contradiction must be explicit. Saying “this is absurd” without showing why is not a proof.
• Constructive proof: Exhibit the object whose existence is claimed. The AI must verify the object satisfies all stated conditions, not just assert it does.
• Pigeonhole principle: If $n+1$ items are placed into $n$ containers, at least one container holds more than one item. Recognizing when this applies is the skill; applying it correctly requires identifying the “items” and “containers” precisely.

Combinatorics: Watch for AI errors in overcounting (not dividing by symmetries when objects are indistinguishable) or undercounting (treating distinguishable objects as identical).

Screening assessments test four kinds of quantitative reasoning

Estimation: Fermi problems (“How many gas stations are in the US?”). The skill is structured decomposition and reasonable assumptions, not knowing the answer. Evaluate whether the AI breaks the problem into tractable sub-estimates and combines them coherently.

Probability: Basic Bayes’ theorem, conditional probability, expected value. AI errors here often involve confusing $P(A|B)$ with $P(B|A)$: the base rate fallacy. Watch for it explicitly in medical and legal probability problems.

Logic: Syllogisms, valid vs. sound arguments, common fallacies. Relevant for evaluating AI reasoning quality across domains.

Dimensional analysis: In physics and engineering tasks, checking that units are consistent is a first-pass sanity check on any formula. An AI that produces a velocity in kg·m is wrong regardless of the numerical value.

Physics and engineering notation

For physics and engineering annotation tasks:

• Vectors are written in bold (\mathbf{F}) or with an arrow (\vec{F}), not as plain scalars
• Units belong outside math mode or inside \text{}: write $5 \text{ m/s}$, not $5 m/s$ (where m and s are interpreted as variables, not unit symbols)
• Standard derivatives: $\frac{dy}{dx}$; partial derivatives: $\frac{\partial f}{\partial x}$
• Scientific notation: $6.022 \times 10^{23}$, not 6.022e23, which is code syntax, not math notation

LaTeX math flashcards

Technical

Right answer, wrong reasoning still fails

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Technical

The LaTeX brace rule

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Technical

Dimensional analysis as a first-pass check

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Technical

Contradiction vs. contrapositive

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Technical

Display math vs. inline math

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Technical

Greek letters in LaTeX

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• Where to go next: Markdown and JSON Formatting Mastery covers the other formatting language you will be scored on.