Notation Study Guide

📖 Core Concepts Notation system – a set of graphics, symbols, or abbreviated expressions that conventionally represent technical facts or quantities. Standard notation – agreed‑upon symbols within a field that ensure everyone “speaks the same language.” Arbitrary meaning – symbols are given meanings by convention, not by intrinsic properties. Secondary notation – visual cues (spacing, color, font) that aid readability but don’t change formal meaning. Domain‑specific highlights Chemistry – chemical formulas (e.g., $H2O$) and SMILES strings (e.g., CCO). Computing – Backus‑Naur Form (BNF) & Extended BNF (EBNF) for grammar definition; regular‑expression syntax for pattern matching. Logic – quantifiers (∀, ∃), connectives (∧, ∨, →), truth‑value operators (⊤, ⊥). Mathematics – Cartesian coordinates $(x, y)$, differentiation notations $f'(x)$, $\frac{df}{dx}$, Big‑O $O(\cdot)$, set‑builder $\{x \mid P(x)\}$. Numeral systems – scientific ($a\times10^{n}$), engineering (exponents multiple of 3), binary (base‑2), hexadecimal (base‑16). Physics – Dirac bra–ket $|\psi\rangle$, tensor index notation with subscripts/superscripts. Typographical conventions – Prefix (Polish) $+ab$, Postfix (RPN) $ab+$ notation. Graphical – Feynman diagrams, structural formulas, Venn diagrams, UML diagrams. --- 📌 Must Remember Chemical formula: element symbols + subscripts = composition (e.g., $C6H{12}O6$). SMILES: linear text; atoms as letters, bonds implied, = for double, # for triple. BNF rule: <symbol> ::= expression. EBNF adds ?, , +, {} for optional/repeat. Regular expression: . = any char, = 0+ repeats, + = 1+ repeats, ? = optional, [] = character class. Derivative: $f'(x) = \frac{df}{dx}$ – both mean instantaneous rate of change. Big‑O: $f(n)=O(g(n))$ means $|f(n)| \le C\cdot|g(n)|$ for large $n$. Scientific notation: $N = a \times 10^{b}$, with $1 \le a < 10$. Engineering notation: exponent $b$ is a multiple of 3 (e.g., $4.7 \times 10^{6}$ Ω). Binary ↔ Hex conversion: group binary bits in 4s → single hex digit. Prefix vs Postfix: operator before operands (+ a b) vs after (a b +). --- 🔄 Key Processes Convert a decimal to scientific notation Move the decimal to create $1 \le a < 10$. Count moves = exponent $b$ (positive if left, negative if right). Write $a \times 10^{b}$. Write a SMILES string Write atoms in order of connectivity. Insert = for double bonds, # for triple bonds. Use parentheses for branches, numbers for ring closures. Define a grammar in BNF Identify non‑terminals (<expr>), terminals (+, , (, )). Write production: <expr> ::= <term> | <expr> + <term>. Switch from BNF to EBNF Replace recursion with {} (repetition) and [] (optional). Example: <expr> ::= <term> { (“+” | “-”) <term> }. Create a set‑builder description Start with {x | condition}. Clearly state variable, domain, and property. --- 🔍 Key Comparisons Scientific vs Engineering Notation Scientific: exponent any integer; coefficient $1\le a<10$. Engineering: exponent must be multiple of 3; aligns with SI prefixes. Binary vs Hexadecimal Binary: base‑2, digits 0/1, useful for low‑level hardware. Hex: base‑16, digits 0‑9 & A‑F, compact representation of binary groups. Prefix (Polish) vs Postfix (RPN) Prefix: operator first, no parentheses needed. Postfix: operator last, evaluation uses a stack. BNF vs EBNF BNF: only |, ::=, recursion; can be verbose. EBNF: adds , +, ?, {} for optional/repeat, clearer. Big‑O vs Exact Runtime Big‑O: describes upper‑bound growth, ignores constants. Exact runtime: includes coefficients, lower‑order terms. --- ⚠️ Common Misunderstandings “$O(n^2)$ means the algorithm always takes $n^2$ steps.” Wrong: $O(n^2)$ is an upper bound; actual runtime may be $0.5n^2$ or $n\log n$. “SMILES is case‑sensitive only for aromatic atoms.” Actually, case distinguishes element symbols (e.g., C vs c for aromatic carbon). “Prefix notation eliminates the need for parentheses altogether.” True only when the arity of each operator is fixed; ambiguous if operators vary. “Scientific notation always uses a single digit before the decimal.” Correct for standard form; engineering notation relaxes this rule. --- 🧠 Mental Models / Intuition Notation as a “language”: think of each domain’s symbols as words; learning the “grammar” (rules) lets you construct meaningful “sentences” (expressions). Big‑O as a “horizon”: imagine standing on a hill—Big‑O tells you the shape of the landscape far away, not the exact steps you’ll take today. Binary‑to‑Hex grouping: picture a 4‑bit “block” as a single “digit” in a larger base—makes conversion a simple lookup. --- 🚩 Exceptions & Edge Cases Zero in scientific notation: written as $0 \times 10^{0}$ (or simply 0). Negative exponents: $3.5 \times 10^{-4}$ = 0.00035. SMILES aromatic notation: lower‑case atoms (c, n) indicate aromaticity; uppercase (C, N) are aliphatic. EBNF optional repetition: {A} allows zero repetitions; A+ requires at least one. Dirac notation inner product: $\langle \phi | \psi \rangle$ is a scalar; order matters (complex conjugate). --- 📍 When to Use Which Choose scientific vs engineering notation – use scientific for general math; use engineering when aligning with SI prefixes (µF, kΩ). Select prefix vs postfix – prefix for functional programming or Lisp‑style expressions; postfix for stack‑based calculators (HP, RPN). Use BNF vs EBNF – BNF for formal language specifications; EBNF for readability in documentation or teaching. Apply SMILES – quick text entry of molecules in cheminformatics; use structural formulas for classroom illustration. Big‑O analysis – apply when comparing algorithm scalability, not for precise timing predictions. --- 👀 Patterns to Recognize Exponent multiples of 3 → likely engineering notation. Lower‑case letters in SMILES → aromatic ring. Curly braces {} in EBNF → “zero or more” repetitions. Vertical bar | inside set‑builder → “such that” separator. Tensor indices up/down – one up (contravariant), one down (covariant). --- 🗂️ Exam Traps Mistaking $O(n)$ for exact runtime – exam may give $O(n)$ and ask for worst‑case; answer with “upper bound, not exact steps.” Confusing engineering exponent with scientific – a number like $4.7 \times 10^{6}$ could be either; check if exponent is multiple of 3. Reading +ab as infix – in prefix notation it means “add a and b,” not “a plus b.” Assuming all SMILES are case‑insensitive – lower‑case matters for aromaticity; wrong case leads to a different structure. Over‑applying Big‑O – picking $O(n^2)$ when the algorithm is actually $Θ(n\log n)$ loses points; watch for tight bounds in the question. ---