Unit Overview

Number and algebra

IB Mathematics Applications and Interpretation HL· 5 min read 📊 15-20% of total IB AI HL examination

1. Unit at a glance

We start with core number skills including scientific notation and error estimation that underpin all quantitative work in this course. Next we move to sequences and series, with a focus on applied financial problems that are a key part of the AI syllabus. We then cover logarithms and exponential equations, which are critical for modeling exponential growth and decay in real-world contexts.

The unit concludes with combinatorics, binomial expansion, partial fractions, complex numbers, and mathematical induction, all of which support later topics in calculus, probability, and statistical modeling. Each concept builds on previous skills, so it is important to master sub-topics in order.

This unit includes the following sub-topics:

Operations with numbers in scientific notation

Perform addition, subtraction, multiplication and division with numbers in scientific notation.

★⏱ 10 min

Approximation, error and estimation

Calculate absolute error, percentage error, and apply rounding and significant figure rules.

★★⏱ 15 min

Arithmetic and geometric sequences and series

Find nth terms and sums of finite and infinite arithmetic and geometric sequences.

★★⏱ 20 min

Financial applications of geometric sequences and series

Calculate compound interest, depreciation, and the value of annuities using geometric sequences.

★★★⏱ 25 min

Laws of logarithms

Simplify and rearrange logarithmic expressions using product, quotient and power laws.

★★★⏱ 15 min

Solving exponential equations with logarithms

Use logarithms to solve exponential equations from real-world modeling problems.

★★★⏱ 20 min

Counting principles, permutations and combinations

Count possible outcomes using the addition rule, product rule, permutations and combinations.

★★★⏱ 20 min

Binomial expansion

Expand binomial expressions using Pascal's triangle and the binomial theorem.

★★⏱ 15 min

Partial fractions

Decompose rational functions into partial fractions for use in later integration topics.

★★★★⏱ 20 min

Complex numbers: definition, arithmetic and polar form

Perform arithmetic operations on complex numbers in both Cartesian and polar form.

★★★★⏱ 25 min

Complex numbers as roots of polynomials

Apply the conjugate root theorem for polynomials with real coefficients.

★★★★⏱ 15 min

Proof by mathematical induction

Prove statements about sequences, divisibility and inequalities using induction.

★★★★★⏱ 25 min

2. Common Pitfalls

Wrong move:

Forgetting to align powers of 10 when adding or subtracting scientific notation

Why:

This leads to incorrect order of magnitude and wrong final results

Correct move:

Always rewrite all terms to use the same exponent of 10 before adding or subtracting

Wrong move:

Mixing up permutations and combinations for ordered vs unordered selection

Why:

This leads to incorrect counting of outcomes for probability problems

Correct move:

If order matters, use permutations; if order does not matter, use combinations

Wrong move:

Skipping explicit verification of the base case in proof by induction

Why:

Examiners require the base case to be verified, and skipping it loses full marks

Correct move:

Always explicitly test and confirm the base case before starting the inductive step

3. Quick Reference Cheatsheet

Concept	Key Formula / Rule
nth term of arithmetic sequence	$u_{n} = u_{1} + (n - 1) d$
Sum of finite geometric series	$S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}, r \neq = 1$
Compound interest	$A = P (1 + \frac{r}{100})^{n}$
Product Law of Logarithms	$lo g_{b} (x y) = lo g_{b} x + lo g_{b} y$
Change of Base Formula	$lo g_{b} a = \frac{l n a}{l n b}$
Combinations of n items taken r at a time	$^{n} C_{r} = \frac{n !}{r ! ( n - r )!}$
Binomial Theorem	$(a + b)^{n} = \sum_{r = 0}^{n} (r n) a^{n - r} b^{r}$
Modulus of a complex number $x + i y$	$∣ x + i y ∣ = x^{2} + y^{2}$
Conjugate Root Theorem	Complex roots of real polynomials come in conjugate pairs
Sum to infinity of geometric series	$S_{\infty} = \frac{a}{1 - r}, ∣ r ∣ < 1$

What's Next

Start this unit with the first sub-topic on operations with numbers in scientific notation to build your foundational number skills that will be used across all IB AI HL topics. After completing all 12 sub-topics in this unit, you will be ready to move on to the next unit, which covers functions, the core tool for real-world mathematical modeling.