Syllabus Map

AA HL Number and Algebra Syllabus Points

The official IB syllabus content covering AHL points 1.10 through 1.16:

  1. 1.10Extension of the binomial theorem to fractional and negative indices $(1+x)^n$ where $n \in \mathbb{Q}$ and $|x| < 1$
  2. 1.11Partial fractions: linear factors, repeated linear factors, and irreducible quadratic factors
  3. 1.12Complex numbers: Cartesian form $z = a+bi$, complex conjugates, modulus-argument form $r(\cos\theta+i\sin\theta)$, Euler form $z = r e^{i\theta}$
  4. 1.13Powers and roots of complex numbers: De Moivre's theorem for integer and rational powers; $n$th roots of complex numbers and roots of unity
  5. 1.14Proof by mathematical induction (sums of series, divisibility, matrix powers, inequalities); proof by contradiction and counterexample
  6. 1.15Solutions of systems of linear equations in up to 3 variables; row operations, Gaussian elimination, unique/infinite/inconsistent planes
  7. 1.16Polynomials: Factor and remainder theorems, Fundamental Theorem of Algebra, sum and product of roots (Vieta's formulas)

Study by topic

AHL Topic 1 Interactive Study Guides

All 6 lesson guides are equipped with our standard 4-tab interface: Study Notes (theory, Argand diagrams & worked proofs), Practice Exercises (10 tiered problems with collapsible solutions), Formula Sheet (formula booklet & pitfall alerts), and Mini Test (interactive diagnostic MCQs & exam FRQs).

1.14 Proof by Mathematical Induction Use the base case, inductive hypothesis, and induction step to prove results about sums, divisibility, matrix powers, and inequalities. Open Interactive Guide 1.12 Complex Numbers: Cartesian, Polar & Euler Argand diagrams, modulus $|z|$ and argument $\arg(z)$, conversions between Cartesian $a+bi$, polar $r\text{cis}\theta$, and Euler's exponential form $r e^{i\theta}$. Open Interactive Guide 1.13 De Moivre’s Theorem & Roots of Unity De Moivre's theorem for powers, finding all $n$ complex roots $z^n = w$, roots of unity, and geometric representation as vertices of regular polygons. Open Interactive Guide 1.15 Systems of Linear Equations & Matrices Solving systems in 3 variables using row reduction / Gaussian elimination, geometric interpretation of 3 planes (unique point, line of intersection, inconsistent). Open Interactive Guide 1.16 Polynomials & Vieta’s Formulas The Fundamental Theorem of Algebra, remainder and factor theorems, conjugate root theorem, and Vieta's relations between roots and coefficients. Open Interactive Guide 1.10, 1.11 Partial Fractions & Extended Binomial Series Decomposing rational expressions into linear and quadratic partial fractions, and expanding $(1+x)^n$ for rational exponents with validity interval $|x| < 1$. Open Interactive Guide

Prerequisite Knowledge

Core SL Foundation Prerequisites

Before studying these AHL extensions, ensure complete familiarity with the 6 standard level Topic 1 lessons:

1.1 Standard Form & Accuracy 1.2 Arithmetic Sequences & Series 1.3, 1.8 Geometric Sequences & Sums 1.4 Financial Applications 1.5, 1.7 Exponents and Logarithms 1.6, 1.9 Simple Proof & Binomial ($n \in \mathbb{N}$)