# Number and algebra

> IB Mathematics Applications and Interpretation HL · IB AI HL
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u1-overview/
> Weight: 15-20% of total IB AI HL examination

This unit builds foundational number and algebra skills required for all subsequent topics in IB AI HL, from modeling growth to statistical analysis and calculus. It covers both pure and applied core algebraic concepts.

**Prerequisites:** Foundational knowledge of middle years algebra and arithmetic

## Learning objectives

- Perform accurate arithmetic and algebraic operations with different number types including scientific notation and complex numbers
- Apply sequence and series concepts to solve real-world financial problems common to AI applications
- Manipulate logarithms and solve exponential equations for modeling growth and decay

## 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 logarithms, exponential equations, and an introduction to complex numbers, 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](https://www.owlsprep.com/study/ib-math-ai-hl-u1-operations-with-numbers-in-scientific/) — Perform addition, subtraction, multiplication and division with numbers in scientific notation.
- [Approximation, error and estimation](https://www.owlsprep.com/study/ib-math-ai-hl-u1-approximation-error-and-estimation/) — Calculate absolute error, percentage error, and apply rounding and significant figure rules.
- [Arithmetic and geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-hl-u1-arithmetic-and-geometric-sequences-and/) — Find nth terms and sums of finite and infinite arithmetic and geometric sequences.
- [Financial applications of geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-hl-u1-financial-applications-of-geometric-sequences/) — Calculate compound interest, depreciation, and the value of annuities using geometric sequences.
- [Laws of exponents](https://www.owlsprep.com/study/ib-math-ai-hl-u1-laws-of-exponents/) — Combine, simplify and evaluate expressions using integer and rational exponents and radicals.
- [Laws of logarithms](https://www.owlsprep.com/study/ib-math-ai-hl-u1-laws-of-logarithms/) — Simplify and rearrange logarithmic expressions using product, quotient and power laws.
- [Solving exponential equations with logarithms](https://www.owlsprep.com/study/ib-math-ai-hl-u1-solving-exponential-equations-with-logarithms/) — Use logarithms to solve exponential equations from real-world modeling problems.
- [Complex numbers: definition, arithmetic and polar form](https://www.owlsprep.com/study/ib-math-ai-hl-u1-complex-numbers-definition-arithmetic-and/) — Perform arithmetic operations on complex numbers in both Cartesian and polar form.

## Common pitfalls

- **Wrong:** Forgetting to align powers of 10 when adding or subtracting scientific notation
  - Why it fails: This leads to incorrect order of magnitude and wrong final results
  - Correct: Always rewrite all terms to use the same exponent of 10 before adding or subtracting

## 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 | $\log_b(xy) = \log_b x + \log_b y$ |
| Change of Base Formula | $\log_b a = \frac{\ln a}{\ln b}$ |
| Modulus of a complex number $x+iy$ | $\|x+iy\| = \sqrt{x^2 + y^2}$ |
| 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 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.

- [Operations with numbers in scientific notation](https://www.owlsprep.com/study/ib-math-ai-hl-u1-operations-with-numbers-in-scientific/)
- [Approximation, error and estimation](https://www.owlsprep.com/study/ib-math-ai-hl-u1-approximation-error-and-estimation/)
- [Arithmetic and geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-hl-u1-arithmetic-and-geometric-sequences-and/)

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From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-ai-hl-u1-overview/