Unit Overview

Number & Algebra

IB Mathematics: Analysis and Approaches HL· 5 min read 📊 18-20% of total IB AA HL exam score

1. Unit at a Glance

This unit progresses from foundational number concepts to increasingly complex algebraic structures, building skills you will use across every other topic in IB AA HL. We start with core number systems, move through sequences, exponents, and combinatorics, then cover HL-specific extensions like complex numbers, matrices, and formal proof.

Many IB exam questions combine concepts from this unit with functions, calculus, or geometry, so mastering these fundamentals is critical for earning high marks on both paper 1 and paper 2.

This unit includes the following sub-topics:

Number representation and number systems

Learn to represent numbers in different bases, classify number sets, and work with approximations and error.

★⏱ 8 min

Sequences and series

Master arithmetic, geometric, and infinite series, including applications to compound interest and recurrence relations.

★★⏱ 10 min

Exponents and logarithms

Explore properties of exponents and logarithms, and solve exponential and logarithmic equations.

★★⏱ 8 min

Counting principles

Apply addition, multiplication principles, permutations, and combinations to solve combinatorics problems.

★★★⏱ 7 min

Binomial theorem

Expand binomial expressions and find specific terms in expansions using the binomial theorem.

★★⏱ 6 min

Proof techniques (HL only)

Learn induction, proof by contradiction, and counterexample for constructing rigorous algebraic proofs.

★★★★⏱ 9 min

Complex numbers fundamentals

Introduce complex numbers, perform operations in Cartesian form, and solve polynomials with complex roots.

★★★⏱ 8 min

Polar and exponential form of complex numbers

Convert between forms of complex numbers and perform multiplication/division in polar form.

★★★⏱ 9 min

De Moivre's theorem

Apply De Moivre's theorem to find powers and roots of complex numbers for algebraic and geometric problems.

★★★★⏱ 8 min

Matrices and matrix operations

Perform matrix operations, find determinants and inverses for 2x2 and 3x3 matrices.

★★★⏱ 10 min

Systems of linear equations

Solve linear systems using matrices and row reduction, and identify inconsistent/dependent systems.

★★★⏱ 10 min

2. Common Pitfalls

Wrong move:

Treating $i$ as a regular variable when squaring, forgetting $i^{2} = - 1$

Why:

This leads to incorrect simplification of all complex number expressions

Correct move:

Always substitute $i^{2} = - 1$ after simplifying complex number operations

Wrong move:

Mixing up permutations and combinations in counting problems

Why:

Permutations account for order of selection, combinations do not, leading to wrong counts

Correct move:

Always check if order matters before selecting the appropriate counting formula

Wrong move:

Forgetting to check for extraneous solutions when solving logarithmic equations

Why:

Logarithms are only defined for positive arguments, so some solutions are invalid

Correct move:

Always test solutions against the domain of the original logarithmic equation

3. Quick Reference Cheatsheet

Concept / Formula	Description
$S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}, r \neq = 1$	Sum of first $n$ terms of a geometric series
$\egin{aligned} \log_b(xy) &= \log_b x + \log_b y \\ \log_b\left(\frac{x}{y}\right) &= \log_b x - \log_b y \end{aligned}$	Core logarithm product/quotient rules
$^{n} C_{r} = \frac{n !}{r ! ( n - r )!}$	Combination of $n$ items taken $r$ at a time
$(a + b)^{n} = \sum_{k = 0}^{n} (k n) a^{n - k} b^{k}$	General binomial expansion formula
$z = a + bi = r (cos θ + i sin θ) = r e^{i θ}$	Three standard forms of a complex number
$(cos θ + i sin θ)^{n} = cos n θ + i sin n θ$	De Moivre's theorem for integer powers
$det (A B) = det (A) det (B)$	Key determinant property for matrix products
$A x = b$ has unique solution $⟺ det (A) \neq = 0$	Condition for unique solution of a linear system

What's Next

Begin your study of this unit with the first sub-topic on number representation and number systems, which lays the groundwork for all other algebraic topics in this unit. Once you complete all sub-topics in this Number & Algebra unit, you will progress to the first sub-topic of the next unit on Functions.