# Number & Algebra

> IB Mathematics: Analysis and Approaches HL · IB AA HL
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u1-overview/
> Weight: 18-20% of total IB AA HL exam score

This foundational unit covers all core number and algebraic concepts for IB AA HL, from basic number systems to advanced topics like complex numbers, matrices, and proof. It forms the base for all other topics including calculus, functions, and statistics.

**Prerequisites:** Standard high school algebra and basic number concepts

## Learning objectives

- Master core number and algebraic structures required for all other IB AA HL topics
- Solve practical and abstract problems involving sequences, series, logarithms, and combinatorics
- Apply properties of complex numbers and matrices to algebraic and geometric problems
- Construct rigorous mathematical proofs for HL-specific algebraic statements

## 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](https://www.owlsprep.com/study/ib-math-aa-hl-u1-number-representation-and-number-systems/) — Learn to represent numbers in different bases, classify number sets, and work with approximations and error.
- [Sequences and series](https://www.owlsprep.com/study/ib-math-aa-hl-u1-sequences-and-series/) — Master arithmetic, geometric, and infinite series, including applications to compound interest and recurrence relations.
- [Exponents and logarithms](https://www.owlsprep.com/study/ib-math-aa-hl-u1-exponents-and-logarithms/) — Explore properties of exponents and logarithms, and solve exponential and logarithmic equations.
- [Counting principles](https://www.owlsprep.com/study/ib-math-aa-hl-u1-counting-principles/) — Apply addition, multiplication principles, permutations, and combinations to solve combinatorics problems.
- [Binomial theorem](https://www.owlsprep.com/study/ib-math-aa-hl-u1-binomial-theorem/) — Expand binomial expressions and find specific terms in expansions using the binomial theorem.
- [Proof techniques (HL only)](https://www.owlsprep.com/study/ib-math-aa-hl-u1-proof-techniques/) — Learn induction, proof by contradiction, and counterexample for constructing rigorous algebraic proofs.
- [Complex numbers fundamentals](https://www.owlsprep.com/study/ib-math-aa-hl-u1-complex-numbers-fundamentals/) — Introduce complex numbers, perform operations in Cartesian form, and solve polynomials with complex roots.
- [Polar and exponential form of complex numbers](https://www.owlsprep.com/study/ib-math-aa-hl-u1-polar-and-exponential-form-of/) — Convert between forms of complex numbers and perform multiplication/division in polar form.
- [De Moivre's theorem](https://www.owlsprep.com/study/ib-math-aa-hl-u1-de-moivre-s-theorem/) — Apply De Moivre's theorem to find powers and roots of complex numbers for algebraic and geometric problems.
- [Matrices and matrix operations](https://www.owlsprep.com/study/ib-math-aa-hl-u1-matrices-and-matrix-operations/) — Perform matrix operations, find determinants and inverses for 2x2 and 3x3 matrices.
- [Systems of linear equations](https://www.owlsprep.com/study/ib-math-aa-hl-u1-systems-of-linear-equations/) — Solve linear systems using matrices and row reduction, and identify inconsistent/dependent systems.

## Common pitfalls

- **Wrong:** Treating $i$ as a regular variable when squaring, forgetting $i^2 = -1$
  - Why it fails: This leads to incorrect simplification of all complex number expressions
  - Correct: Always substitute $i^2 = -1$ after simplifying complex number operations
- **Wrong:** Mixing up permutations and combinations in counting problems
  - Why it fails: Permutations account for order of selection, combinations do not, leading to wrong counts
  - Correct: Always check if order matters before selecting the appropriate counting formula
- **Wrong:** Forgetting to check for extraneous solutions when solving logarithmic equations
  - Why it fails: Logarithms are only defined for positive arguments, so some solutions are invalid
  - Correct: Always test solutions against the domain of the original logarithmic equation

## 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 |
| $\begin{aligned} \log_b(xy) &= \log_b x + \log_b y \\ \log_b\left(\frac{x}{y}\right) &= \log_b x - \log_b y \end{aligned}$ | Core logarithm product/quotient rules |
| $^nC_r = \frac{n!}{r!(n-r)!}$ | Combination of $n$ items taken $r$ at a time |
| $(a+b)^n = \sum_{k=0}^n \binom{n}{k}a^{n-k}b^k$ | General binomial expansion formula |
| $z = a+bi = r(\cos\theta + i\sin\theta) = re^{i\theta}$ | Three standard forms of a complex number |
| $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$ | De Moivre's theorem for integer powers |
| $\det(AB) = \det(A)\det(B)$ | Key determinant property for matrix products |
| $A\mathbf{x} = \mathbf{b}$ has unique solution $\iff \det(A) \neq 0$ | 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.

- [Number representation and number systems](https://www.owlsprep.com/study/ib-math-aa-hl-u1-number-representation-and-number-systems/)
- [Sequences and Series](https://www.owlsprep.com/study/ib-math-aa-hl-u1-sequences-and-series/)
- [Exponents and logarithms](https://www.owlsprep.com/study/ib-math-aa-hl-u1-exponents-and-logarithms/)

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