# Number & Algebra

> IB Mathematics: Analysis and Approaches SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u1-overview/
> Weight: 15-18% of overall examination

This foundational unit covers core number and algebra concepts that are used across all other units of IB AA SL, from measurement to logarithms to combinatorics.

**Prerequisites:** Basic middle school arithmetic and algebraic manipulation

## Learning objectives

- Apply approximation, error estimation, and scientific notation to real-world measurement problems
- Model discrete patterns and growth/decay using arithmetic and geometric sequences and series
- Manipulate exponential and logarithmic expressions and solve related algebraic equations
- Calculate permutations, combinations, and expand binomials for combinatorics and probability applications

## Unit at a Glance

This unit progresses from practical number skills to advanced algebraic manipulation and combinatorics, building the foundational tools you will use for functions, calculus, and statistics later in the course. We start with measurement and error handling, move to discrete sequence patterns, then explore exponential and logarithmic relationships, and finish with combinatorial counting needed for probability and binomials.

This unit is split into the following 8 sub-topics:
- [Approximation, significant figures and scientific notation](https://www.owlsprep.com/study/ib-math-aa-sl-u1-approximation-significant-figures-and-scientific/) — Learn to round correctly, write values in scientific notation, and identify significant figures for calculations.
- [Error bounds and absolute/relative error](https://www.owlsprep.com/study/ib-math-aa-sl-u1-error-bounds-and-absolute-relative/) — Calculate upper and lower error bounds, and find absolute and relative error for measured values.
- [Arithmetic sequences and series](https://www.owlsprep.com/study/ib-math-aa-sl-u1-arithmetic-sequences-and-series/) — Derive and use formulas for the nth term and sum of finite arithmetic sequences and series.
- [Geometric sequences and infinite geometric series](https://www.owlsprep.com/study/ib-math-aa-sl-u1-geometric-sequences-and-infinite-geometric/) — Work with geometric sequences, and find sums of finite and infinite convergent geometric series.
- [Laws of exponents and logarithms](https://www.owlsprep.com/study/ib-math-aa-sl-u1-laws-of-exponents-and-logarithms/) — Simplify expressions using exponent rules and the core laws of logarithms.
- [Solving exponential and logarithmic equations](https://www.owlsprep.com/study/ib-math-aa-sl-u1-solving-exponential-and-logarithmic-equations/) — Solve exponential and logarithmic equations using algebraic and GDC methods.
- [Permutations and combinations](https://www.owlsprep.com/study/ib-math-aa-sl-u1-permutations-and-combinations/) — Distinguish between permutations and combinations, and count arrangements of items.
- [Binomial expansion and binomial theorem](https://www.owlsprep.com/study/ib-math-aa-sl-u1-binomial-expansion-and-binomial-theorem/) — Apply the binomial theorem to expand binomials and find specific terms in expansions.

## Common pitfalls

- **Wrong:** Mixing up permutations and combinations when counting arrangements
  - Why it fails: Permutations count ordered arrangements, while combinations count unordered groups
  - Correct: Always ask whether order matters before selecting the right counting formula
- **Wrong:** Forgetting the convergence condition for infinite geometric series
  - Why it fails: Infinite sums only exist if the common ratio satisfies $|r| < 1$
  - Correct: Check the value of $|r|$ before calculating an infinite geometric sum
- **Wrong:** Premature rounding of intermediate results in multi-step calculations
  - Why it fails: Rounding early leads to accumulated error that shifts your final answer away from the correct value
  - Correct: Keep full precision on your calculator until the final step, then round to the required significant figures

## Cheatsheet

| Concept | Key Formula/Expression |
| --- | --- |
| nth term of arithmetic sequence | $u_n = u_1 + (n-1)d$ |
| Sum of finite arithmetic series | $S_n = \frac{n}{2}(u_1 + u_n)$ |
| nth term of geometric sequence | $u_n = u_1 r^{n-1}$ |
| Sum of infinite convergent geometric series | $S_\infty = \frac{u_1}{1-r}, \quad \|r\|<1$ |
| Logarithm product rule | $\log_b(xy) = \log_b x + \log_b y$ |
| Logarithm power rule | $\log_b(x^n) = n \log_b x$ |
| Permutation of n items taken k at a time | $^n P_k = \frac{n!}{(n-k)!}$ |
| Binomial coefficient | $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ |

## What's next

Start your study of this unit with the first sub-topic on approximation, significant figures, and scientific notation to build core number skills you will use for all future calculations. Once you complete all 8 sub-topics in this unit, you will be ready to move on to the next unit on functions, which builds directly on the algebra skills you learn here.

- [Approximation, significant figures and scientific notation](https://www.owlsprep.com/study/ib-math-aa-sl-u1-approximation-significant-figures-and-scientific/)
- [Arithmetic sequences and series](https://www.owlsprep.com/study/ib-math-aa-sl-u1-arithmetic-sequences-and-series/)

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