IB Mathematics: AA HL topic guide

Number and Algebra

Number and Algebra is a core part of IB Mathematics: AA HL. 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 7 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.

1.0

Introduction to Number and Algebra

What You Will Learn in Number and Algebra

1.1

Sequences and Series

Arithmetic Sequences & Series · Geometric Sequences & Series · Infinite Geometric Series & Applications

1.2a

Laws of Exponents and Logarithms

Laws of Exponents & Logarithms · Solving Exponential & Logarithmic Equations

1.2b

The Binomial Theorem with Positive Integer Exponents

Pascal's Triangle and Binomial Coefficients · The Binomial Theorem — Expansion

1.3

Mathematical Proof

Deductive Proof · Proof by Counterexample & Disproof

1.4

Counting Principles

Permutations & Combinations · Applications of Counting Principles

1.5

Extended Binomial Expansion

Extended Binomial Expansion

1.6

Proof by Mathematical Induction

Induction with Sums & Series · Induction with Divisibility & Inequalities

1.7

Complex Numbers: Cartesian Form

Introduction to Complex Numbers · The Complex Conjugate & Division

1.8

Complex Numbers: Polar Form and De Moivre's Theorem

Polar (Modulus-Argument) & Euler Form · De Moivre's Theorem & Roots of Unity

1.9

Roots of Polynomials & Partial Fractions

Sum & Product of Roots of Polynomials · Partial Fractions

1.10

Systems of Linear Equations

Systems of Linear Equations · Systems with Parameters and Geometric Interpretation

Free worked preview

Extended Binomial Expansion

This complete preview comes from the Extended Binomial Expansion 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.

Binomial Expansion for Negative & Fractional Exponents

When the exponent n in (1+x)n(1+x)^n is not a positive integer — perhaps it is negative, or a fraction like 12\frac{1}{2} — the binomial expansion no longer terminates; it becomes an infinite series. This is the HL extension of the binomial theorem you learned earlier, and it opens the door to approximating functions like 1+x\sqrt{1+x} or 11−x\frac{1}{1-x} as power series. By the end you will be able to expand expressions of the form (1+x)n(1+x)^n for any real n, find the first few terms, and state the values of x for which the expansion is valid.

When n is not a positive integer (e.g. negative or fractional), the binomial expansion becomes an infinite series — it no longer terminates, and only converges for restricted values of x.

Extended Binomial Series
(1+x)n=1+nx+n(n−1)2!x2+n(n−1)(n−2)3!x3+…(∣x∣<1)(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \ldots \qquad (|x| < 1)
where
nnany real number (not necessarily a positive integer)
xxthe variable term, requires |x| < 1 for convergence

Unlike the standard binomial theorem, factorial notation (nr)\binom{n}{r} isn't directly usable when n isn't a positive integer — the pattern above (numerator as a product of decreasing factors) is used instead.

Common ErrorThe expansion is only valid for ∣x∣<1|x| < 1. If the expression isn't already in the form (1+x)ⁿ, it must be manipulated into that form first — e.g. (2+x)−1=2−1(1+x/2)−1(2+x)^{-1} = 2^{-1}(1+x/2)^{-1}, factoring out the constant.
Worked Example Find the first three terms of the expansion of (1+x)−2(1+x)^{-2}
1Identify nn=−2n = -2
2Apply the extended series formula1+(−2)x+(−2)(−3)2!x21 + (-2)x + \frac{(-2)(-3)}{2!}x^2
1−2x+3x2+…1 - 2x + 3x^2 + \ldots
Practice QuestionFind the first three terms of the expansion of (1−2x)1/2(1-2x)^{1/2}, and state the values of x for which it is valid.

Use the formulas

Keep the formula booklet beside you

Revision becomes faster when you practise locating the right equation and understand what every variable means. The full reference is public and searchable on any device.

Open IB Math AA HL Formula Booklet →

Reviewed by the Study to Learn editorial team · Updated 2026-07-24