Number and Algebra

Introduction to Number and Algebra

The Number and Algebra section is not just a collection of techniques — it is the intellectual engine of the entire AA syllabus. Every equation you will later differentiate, every probability you will compute, every model you will fit rests on the algebraic fluency you build here. But this section also pushes the boundary of what you think a "number" is: you begin with familiar sequences and series, move through the logic of proof that establishes truth with certainty, and arrive — at HL — at complex numbers, where $i^2=-1$ unlocks a number system in which every polynomial equation has a root. The common thread is structure: learning to see numbers not as isolated values but as objects with properties, relationships, and symmetries that can be manipulated, generalised, and proved.

Section Map — What Is Covered

• Sequences and Series (1.1): arithmetic and geometric sequences, finite sums $S_n$, and the convergence of infinite geometric series to a finite limit when $|r|<1$
• Exponents and Logarithms (1.2a): the laws that govern powers, logarithms defined as inverse operations, and the strategy of taking logs to solve exponential equations
• The Binomial Theorem (1.2b): Pascal's triangle, binomial coefficients $\\binom{n}{r}$, and the systematic expansion of $(a+b)^n$ for positive integer n
• Mathematical Proof (1.3): deductive reasoning from definitions, building watertight algebraic arguments, and the asymmetry of proof versus disproof (a single counterexample suffices)
• Counting Principles (1.4): permutations where order matters, combinations where it does not, and the connection between binomial coefficients and counting
• Extended Binomial Expansion (1.5, HL): when n is negative or fractional, the expansion becomes an infinite series that converges only for $|x|<1$
• Proof by Mathematical Induction (1.6, HL): the domino-logic technique — base case plus inductive step — that proves statements for infinitely many integers in two movements
• Complex Numbers (1.7–1.9, HL): Cartesian form, the conjugate and modulus, polar and Euler forms, De Moivre's theorem, roots of unity, and the relationship between polynomial roots and coefficients