Correct answers alone are rarely enough in academic mathematics. Clear, logical presentation of working — with proper notation, justified steps, and structured proofs — is what separates a distinction from a pass. This guide covers notation standards, step-by-step solution structure, proof writing, and LaTeX basics.
In mathematics, the answer is the smallest part of the mark. Markers assess the logical chain from the given information to the conclusion. A correct answer with no working earns zero in most STEM mathematics modules. An incorrect final answer with correct method and clear reasoning can earn partial or full method marks.
Clear mathematical writing also demonstrates that you understand what you are doing — not just that you ran a calculation. Examiners distinguish between students who follow a memorised procedure and those who can articulate why each step is valid.
| Concept | Correct notation | Common errors to avoid |
|---|---|---|
| Sets | ℝ, ℤ, ℕ, ℂ (or \mathbb in LaTeX) | R, Z, N without the double-stroke |
| Implication | ⟹ (implies); ⟺ (if and only if) | Using "=>" informally without quantifiers |
| For all / there exists | ∀x ∈ S, ∃y such that… | Writing "for all x" in plain text without formal quantifiers in a proof |
| Vectors | v (bold) or v⃗ (arrow); components in column notation | Mixing bold and arrow notation within one document |
| Matrices | A (bold capital) or [aᵢⱼ] | Using A and a interchangeably for a matrix and its entries |
| Functions | f: ℝ → ℝ, f(x) = … | Writing f(x) = y = x² + 1 without defining the domain |
| Limits | lim_{x→a} f(x) = L | Writing lim f(x) = L without the subscript |
| Derivatives | f′(x) or df/dx or ∂f/∂x (partial) | Mixing Leibniz and prime notation without definition |
A well-presented solution reads like a logical argument — each line follows from the previous one by a stated or clearly implied rule. Use the following structure:
x² − 5x + 6 = 0 → x = 2,3 → so x = 2 or x = 3
x² − 5x + 6 = 0
(x − 2)(x − 3) = 0 [factorising]
∴ x = 2 or x = 3
Our maths specialists solve problems with full working, correct notation, and clear step-by-step reasoning — any level from calculus to abstract algebra.
Proofs are the core of pure mathematics and appear increasingly in applied STEM modules (algorithm correctness, statistical theory, signal processing). A proof is a logical argument that a statement is true for all cases covered by its conditions. Every proof must have:
| Technique | When to use | Structure |
|---|---|---|
| Direct proof | When the hypothesis leads naturally to the conclusion | Assume P; reason step-by-step; conclude Q |
| Proof by contradiction | When assuming the negation leads to an absurdity | Assume ¬Q; show this leads to ¬P or a logical contradiction |
| Proof by contrapositive | When ¬Q → ¬P is easier to prove than P → Q | Prove the contrapositive; conclude P → Q holds |
| Mathematical induction | Proving a statement holds for all n ∈ ℕ | Base case (n=1); inductive step (assume n=k, prove n=k+1) |
| Proof by construction | Existence proofs — show the thing exists by building it | Construct an explicit example; verify it satisfies all conditions |
Many STEM programmes require mathematical assignments to be typeset in LaTeX. Even where not required, submitting a LaTeX-typeset problem set signals professionalism. Key commands:
| What | LaTeX | Renders as |
|---|---|---|
| Fraction | \frac{a}{b} | a/b (proper fraction) |
| Square root | \sqrt{x} | √x |
| Summation | \sum_{i=1}^{n} i | Σ from i=1 to n |
| Integral | \int_{a}^{b} f(x)\,dx | ∫ with limits |
| Greek letters | \alpha, \beta, \lambda, \Sigma | α, β, λ, Σ |
| Bold vector | \mathbf{v} | v |
| Real numbers | \mathbb{R} | ℝ |
| Aligned equations | \begin{align}…\end{align} | Multi-line aligned at = |
Use Overleaf for LaTeX. Overleaf (overleaf.com) is a free, browser-based LaTeX editor with real-time preview, templates, and collaboration. You do not need to install anything locally. Most universities offer free Overleaf premium accounts — check with your IT department.
For routine algebraic manipulation, brief justifications or none at all are fine — markers do not need you to explain that 2×3 = 6. Justify the non-trivial steps: applying a specific theorem, switching the order of integration, using L'Hôpital's Rule. The rule of thumb: justify anything a reasonably competent peer could not immediately verify without thinking.
Leave it in exact form unless the question asks for a decimal approximation. √2, π/4, e³, and (3 + √5)/2 are perfectly acceptable final answers. Converting to a messy decimal (1.41421…, 0.78539…) introduces rounding error and is not required. If the question specifies "give your answer to 3 significant figures," then round appropriately and state the precision: x ≈ 2.34 (3 s.f.).
Many STEM programmes accept handwritten problem sets, especially for in-class assignments or worksheets. If submitting handwritten work: write clearly and legibly, use a dark pen (not pencil if scanning), number all pages, align equations at the equals sign, and cross out (do not erase) discarded working so markers can see your process. Scan at 300 dpi minimum; do not photograph at an angle.