Which typing errors does a check digit catch? ISBN-10 versus ISBN-13

AA HLAA SL Solid Also in: Pure maths

Research question

Which single-digit errors and swaps of adjacent digits are always detected by the ISBN-10 and ISBN-13 check digits, which slip through, and why does working modulo 11 catch more than working modulo 10?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

Check digits are everywhere (books, bank cards, parcels) and the mathematics is a clean, provable piece of number theory. You can test the theory on real barcodes from your own shelf.

The mathematics you'll need

Modular arithmetic (new: explain it with examples)
Weighted sums and why a prime modulus matters
Proof that ISBN-10 detects every single error and every adjacent swap
Finding the adjacent swaps ISBN-13 misses (digits differing by 5)
Counting: proportion of errors detected

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

No data needed; check the rules on ISBNs from books you own.

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

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Explain modular arithmetic with your own examples.
Verify both check-digit rules on real books.
Prove which errors ISBN-10 always detects.
Find and explain the swaps ISBN-13 misses.
Count the proportion of all possible errors each scheme catches and reflect on the trade-off.

Pitfalls that cost marks

Checking examples instead of proving.
Using mod notation before explaining it.
Forgetting the 'X' check digit in ISBN-10.

Showing personal engagement

Use the books on your own shelf.
Design your own check-digit scheme and test it.
Find a real error the scheme would miss.

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	Modular arithmetic (new: explain it with examples); Weighted sums and why a prime modulus matters
AA HL	Good fit	Modular arithmetic (new: explain it with examples); Weighted sums and why a prime modulus matters
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

Personal engagement (C)

Build and break your own small cipher or code, invent examples to test each result, and record the conjectures you made and the ones that turned out to be false.

Reflection (D)

Reflect on what each result guarantees and what it doesn't: which errors a check digit misses, which attacks a cipher survives, and how the answer depends on the size of the numbers. For this idea, start with: checking examples instead of proving — say how it affects your answer.

Use of mathematics (E)

SL: Counting principles, probability or frequency statistics used correctly; any number theory (modular arithmetic, primes) introduced with your own small worked examples and explained, not quoted.

HL: Rigorous proofs (by contradiction or induction) of the number-theory facts you rely on, counting arguments made general, or a statistical attack tested formally.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Analyse another real scheme (bank card numbers use the Luhn algorithm) or design a scheme that catches two errors.

Extending it for HL

Prove the key result in general (why the check digit catches every single-digit error, why the decryption undoes the encryption) rather than checking examples.

See a complete IA, marked

Our annotated exemplar How far can a stack of books lean over the edge of a table? (AA HL) asks a different question, but shows how a complete cryptography exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).