Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

Where does e come from? Compounding more and more often

AA SLAA HL Solid Also in: Pure maths, Calculus

Research question

How does the value of an investment change as interest is compounded yearly, monthly, daily and continuously, and why does (1 + 1/n)ⁿ approach e?

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Why it makes a good exploration

The number e first appeared in a question about interest. Exploring compounding frequency numerically and then proving the limit connects finance with some of the most important ideas in AA.

The mathematics you'll need

Compound interest with n periods
Sequences and limits
The binomial expansion to show convergence
HL: limits using logarithms and L'Hôpital's rule

Where the data comes from

No data needed; optionally compare with a bank's stated AER and nominal rates.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Explore numerically with increasing n.
Conjecture the limit.
Show convergence with the binomial expansion.
Link AER and nominal rates.
Reflect on why continuous compounding is used in models.

Pitfalls that cost marks

Only numerical tables without explanation.
Hand-waving the limit.
Confusing AER with nominal rate.

Showing personal engagement

Compare savings accounts you could open.
Investigate the historical story of Jacob Bernoulli's question.
Estimate how fast the sequence converges.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Compound interest with n periods; Sequences and limits
AA HL	Good fit	Compound interest with n periods; Sequences and limits
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: only numerical tables without explanation; and hand-waving the limit. Say how each one affects your answer.

Use of mathematics (E)

SL: compound interest with n periods; sequences and limits; the binomial expansion to show convergence, each explained in your own words and carried out correctly.

HL adds: limits using logarithms and L'Hôpital's rule, each explained in your own words and carried out correctly.

How finance explorations reach the top bands in general: Finance & economics IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

HL: prove that (1 + 1/n)ⁿ is increasing and bounded.

Extending it for HL

This idea already has HL mathematics in it: limits using logarithms and L'Hôpital's rule. Make that the centre of the exploration and use it with full rigour.

See a complete IA, marked

Should I lease or buy my first car? (AI SL) is a complete finance exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL and AA HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Compound interest with n periods, sequences and limits, the binomial expansion to show convergence and hL: limits using logarithms and L'Hôpital's rule.

Where can I get data for this IA?

No data needed; optionally compare with a bank's stated AER and nominal rates. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Where does e come from? Compounding more and more often” filled in: the question, the maths and the data sources.

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