Introduction to Number and Algebra
What You Will Learn in Number and Algebra
IB Mathematics: AA SL topic guide
Number and Algebra
Number and Algebra is a core part of IB Mathematics: AA SL. This guide connects the syllabus ideas behind Introduction to Number and Algebra, Sequences and Series, Laws of Exponents and Logarithms, The Binomial Theorem with Positive Integer Exponents, Mathematical Proof and 2 more units, shows how they appear in worked problems, and points you to the formulas and full lessons needed for exam revision.
What you will learn
Number and Algebra syllabus outline
The units below follow the structure used in the full Study to Learn course. Use the outline to identify exactly which idea needs attention, then work through the public example before continuing to the complete lesson path.
Sequences and Series
Arithmetic Sequences & Series · Geometric Sequences & Series · Infinite Geometric Series & Applications
Laws of Exponents and Logarithms
Laws of Exponents & Logarithms · Solving Exponential & Logarithmic Equations
The Binomial Theorem with Positive Integer Exponents
Pascal's Triangle and Binomial Coefficients · The Binomial Theorem — Expansion
Mathematical Proof
Deductive Proof · Proof by Counterexample & Disproof
Counting Principles
Permutations & Combinations · Applications of Counting Principles
Systems of Linear Equations
Systems of Linear Equations · Systems with Parameters and Geometric Interpretation
Free worked preview
What You Will Learn in Number and Algebra
This complete preview comes from the Introduction to Number and Algebra unit. It introduces the core language, shows the method in context, and gives you a real example of the lesson quality before you create an account.
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 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 , and the convergence of infinite geometric series to a finite limit when
- 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 , and the systematic expansion of 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
- 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
How to Approach This SectionAlgebra rewards repetition more than any other branch of mathematics. The formulas for arithmetic and geometric series feel unnatural the first three times you use them; by the twentieth problem, they are automatic. The same holds for logarithm laws and binomial expansions. The single most productive habit is to attempt every worked example yourself before reading the solution — the learning is in the attempt, not in the reading. When you reach induction, remember that the logical structure is always identical (base case, hypothesis, inductive step); only the algebra changes from one problem to the next. If the inductive-step algebra resists, the problem is almost always factoring, not the induction logic itself. For complex numbers, treat i as an ordinary algebraic symbol that follows every rule of algebra with the single extra fact that — addition, multiplication, and conjugation all flow from that one rule.
Reviewed by the Study to Learn editorial team · Updated 2026-07-24