Mathematics: Analysis and Approaches HL

Number and algebra

Proof, complex numbers, polynomials, partial fractions, inequalities, and advanced sequences for Mathematics AA HL.

Use proof by mathematical induction Prove divisibility, sums, and recursive statements using a clear induction structure. Operate with complex numbers Add, multiply, divide, and interpret complex numbers in Cartesian form. Use modulus and argument of complex numbers Convert between Cartesian and polar information and interpret points on the Argand diagram. Apply de Moivre theorem Use de Moivre theorem for powers, roots, and polar complex-number reasoning. Use polynomial factor and remainder theorems Apply factor and remainder theorems to polynomial divisibility and unknown coefficients. Decompose rational expressions into partial fractions Set up and interpret partial fraction decompositions for rational expressions. Solve rational inequalities Use sign charts and critical values to solve rational inequalities. Prove sequence and series results Use induction and algebraic reasoning to prove sequence and series formulae. Use counting principles Use multiplication principle, permutations, combinations, and restrictions to count outcomes. Use roots of complex polynomials Connect complex roots, conjugate pairs, factors, and polynomial equations. Use sum and product of roots Use coefficients to reason about sums, products, and unknown parameters in polynomial roots. Use conjugate root theorem Use conjugate roots to complete real-coefficient polynomial factorisations. Use binomial theorem with rational exponents Expand expressions with rational powers and identify coefficients or approximation terms. Solve modulus equations and inequalities Solve absolute value equations and inequalities using intervals and distance interpretation. Analyse recurrence relations Use recurrence definitions to generate terms and interpret long-run behaviour. Use systems of linear equations and matrices Solve and interpret systems using elimination, matrices, and row operations. Use matrix inverses and determinants Use determinants and inverses to solve systems and test singularity. Use eigenvalues and eigenvectors in context Interpret eigenvalues and eigenvectors for transformations and simple dynamic systems. Use proof by contradiction Structure contradiction arguments for algebraic and number properties. Use proof by contraposition Prove implications by proving the contrapositive statement. Analyse infinite geometric series Use convergence conditions and sums of infinite geometric series. Solve advanced logarithmic inequalities Solve logarithmic inequalities with domain restrictions and monotonic reasoning. Use modular arithmetic Work with congruences, remainders, and modular conditions in proof and computation. Solve Diophantine equations Find integer solutions and interpret constraints in linear Diophantine equations. Use Euclidean algorithm and gcd Apply the Euclidean algorithm and Bezout relationships to integer problems. Use advanced counting with restrictions Count arrangements with repeated objects, exclusions, and multiple restrictions. Use inclusion-exclusion principle Apply inclusion-exclusion to overlapping sets and counting contexts. Analyse divisibility proofs Use divisibility rules, factor structure, and proof logic to justify integer results. Use congruence classes Classify integer cases using congruence classes and modular partitions. Solve recurrence relations with closed forms Find and verify closed forms for recurrence relationships. Use generating functions for counting Set up and interpret generating functions for restricted counting. Analyse matrix transformations in algebra Use matrices to represent transformations, systems, and algebraic structure. Use the pigeonhole principle Apply pigeonhole reasoning to prove existence results and counting constraints. Analyse integer partitions Count and interpret integer partitions under restrictions. Use modular inverses Find and use modular inverses in congruence equations. Apply Fermat style modular reasoning Use modular exponent reasoning to simplify powers and justify remainders. Use characteristic equations for recurrences Solve recurrence relations using characteristic equation structure. Use Chinese remainder theorem contexts Solve compatible congruence systems and interpret modular constraints. Analyse algebraic number forms Work with exact algebraic number forms and conjugate relationships. Use advanced proof by induction Extend induction to inequalities, divisibility, and recurrence statements. Solve systems using matrix powers Use matrix powers to model systems and repeated transformations. Analyse eigenvector geometry Interpret eigenvectors and eigenvalues as geometric transformations.