# Number and Algebra

> IB Mathematics: Applications and Interpretation SL · Number and Algebra
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u1-overview/
> Weight: 18-20% of overall IB AI SL exam

This is a foundational unit that introduces core number and algebra concepts underpinning all other topics in IB AI SL. You will build skills for calculation, error analysis, sequences, logarithms, and financial math that you will use across the entire course.

**Prerequisites:** Basic secondary school arithmetic and introductory algebra

## Learning objectives

- Perform accurate calculations with numbers in scientific and standard form, and calculate error for approximations
- Identify, find terms, and calculate sums for both arithmetic and geometric sequences and series
- Apply geometric sequence concepts to solve common real-world financial problems
- Manipulate expressions with rational exponents and logarithms using core algebraic rules
- Solve exponential equations of any base using logarithmic methods

## Unit at a Glance

This unit moves from basic number skills through to algebraic tools that you will use for modeling and problem solving throughout the course. We start with working with numbers in scientific notation, before moving to arithmetic sequences, then geometric sequences and their real-world financial applications.

We then build exponent and logarithm rules, learn how to solve exponential equations, and finish with core skills for approximation and error calculation that are required for all practical problem-solving in IB AI SL.

This unit is split into the following sub-topics:
- [Operations with numbers in scientific notation](https://www.owlsprep.com/study/ib-math-ai-sl-u1-operations-with-numbers-in-scientific/) — Perform addition, subtraction, multiplication and division with numbers written in scientific notation.
- [Arithmetic sequences and series](https://www.owlsprep.com/study/ib-math-ai-sl-u1-arithmetic-sequences-and-series/) — Find nth terms, common differences, and sums of finite arithmetic sequences and series.
- [Geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-sl-u1-geometric-sequences-and-series/) — Calculate common ratios, nth terms, and sums of finite and infinite geometric series.
- [Financial applications of geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-sl-u1-financial-applications-of-geometric-sequences/) — Apply geometric sequences to solve compound interest, depreciation, and other common financial problems.
- [Laws of exponents with integer exponents](https://www.owlsprep.com/study/ib-math-ai-sl-u1-laws-of-exponents-with-rational/) — Simplify and manipulate expressions involving integer exponents using core exponent rules.
- [Introduction to logarithms (base 10 and e)](https://www.owlsprep.com/study/ib-math-ai-sl-u1-simplification-of-logarithmic-expressions/) — Understand the concept of a logarithm as the inverse of exponentiation, and evaluate base-10 and base-e logarithms using a GDC.
- [Solving exponential equations using logarithms](https://www.owlsprep.com/study/ib-math-ai-sl-u1-solving-exponential-equations-using-logarithms/) — Use logarithms to solve exponential equations with any base for unknown exponents.
- [Approximation, absolute and percentage error](https://www.owlsprep.com/study/ib-math-ai-sl-u1-approximation-absolute-and-percentage-error/) — Calculate absolute error and percentage error for approximate measurements and calculations.
- [Standard form of numbers](https://www.owlsprep.com/study/ib-math-ai-sl-u1-standard-form-of-numbers/) — Convert between ordinary numbers and standard (scientific) form for large and small values.

## Common pitfalls

- **Wrong:** Forgetting to re-normalize results after adding/subtracting numbers in scientific notation
  - Why it fails: Many students add the coefficients but leave the result with a coefficient outside the required 1-10 range for standard form
  - Correct: After any operation, adjust the coefficient and exponent to meet the standard form definition
- **Wrong:** Confusing nth term formulas with sum formulas for sequences
  - Why it fails: Arithmetic and geometric formulas are similar enough to mix up under exam pressure
  - Correct: Always confirm if the question asks for a specific term or the total sum of a set of terms before selecting a formula
- **Wrong:** Treating regular percentage growth/decay as an arithmetic (linear) pattern
  - Why it fails: Students often assume repeated percentage changes follow a linear sequence
  - Correct: Recognize percentage changes are multiplicative, so they always follow a geometric sequence pattern

## Cheatsheet

| Concept / Formula | Key Use |
| --- | --- |
| Standard form: $a \times 10^k$, $1 \leq a < 10$, $k \in \mathbb{Z}$ | Represent very large or very small numbers |
| Arithmetic nth term: $u_n = u_1 + (n-1)d$ | Find any term in an arithmetic sequence |
| Arithmetic sum: $S_n = \frac{n}{2}(u_1 + u_n)$ | Sum the first n terms of an arithmetic sequence |
| Geometric nth term: $u_n = u_1 r^{n-1}$ | Find any term in a geometric sequence |
| Geometric finite sum: $S_n = \frac{u_1(1 - r^n)}{1 - r}, r \neq 1$ | Sum the first n terms of a geometric sequence |
| Compound interest: $FV = PV(1 + \frac{r}{100k})^{kn}$ | Calculate future value of compound interest investments |
| Exponent rule: $a^m \times a^n = a^{m+n}$, $(a^m)^n = a^{mn}$ | Simplify expressions with rational exponents |
| Logarithm rule: $\log(ab) = \log a + \log b$, $\log a^b = b \log a$ | Simplify logarithmic expressions |
| Percentage error: $\frac{\|v_{approx} - v_{exact}\|}{\|v_{exact}\|} \times 100\%$ | Calculate percentage error for approximations |

## What's next

Begin your study of this unit with the first sub-topic, Operations with numbers in scientific notation, to start building core number skills. After completing all sub-topics in this Number and Algebra unit, you will progress to the next unit on Functions, starting with an introduction to core function concepts that build on your new algebra skills.

- [Operations with numbers in scientific notation](https://www.owlsprep.com/study/ib-math-ai-sl-u1-operations-with-numbers-in-scientific/)
- [Arithmetic sequences and series](https://www.owlsprep.com/study/ib-math-ai-sl-u1-arithmetic-sequences-and-series/)
- [Geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-sl-u1-geometric-sequences-and-series/)

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