Check digits: a built-in arithmetic guard

When a shop assistant types a long number, one digit can easily go wrong. Compare the typed code with the printed one and the mistake is obvious — but nobody does that in a supermarket queue.

A check digit is the machine's version of that comparison. The last digit of the code is not extra information: it is calculated from all the other digits.

The computer recalculates it and compares, so a changed or swapped digit makes the two disagree.

You meet check digits on barcodes, ISBNs on books, bank account numbers, national insurance numbers and courier tracking codes.

Here is one common rule, used by ISBN-10. Each digit is multiplied by a weight, the products are added, and the check digit is whatever makes that total a multiple of 11.

The weights run from 10 down to 2 across the first nine digits, so earlier digits matter more.

Worked example: code 978 030640 (nine digits), check digit missing.

1. Write the digits with their weights.

2. Add the products:

$90 + 63 + 64 + 0 + 18 + 0 + 24 + 12 + 0 = 271$

3. Divide by 11 and take the remainder: $271 = 24 \times 11 + 7$, so the remainder is 7.
4. The check digit is $11 - 7 = 4$.

The full code is therefore 9780306404. Check it: $271 + 4 = 275 = 25 \times 11$, a multiple of 11.

If the remainder had been 0, the check digit would be 0. If the answer came out as 10, the usual convention is to write the letter X (Roman numeral for 10), because no single digit can represent 10.

The other common family uses weights of 3 and 1 alternately, starting with 3, and then works modulo 10. This is the idea behind barcodes such as EAN-13.

Worked example: code 509876543, check digit missing.

1. Multiply alternately by 3 and 1.

2. Add the products:

$15 + 0 + 27 + 8 + 21 + 6 + 15 + 4 + 9 = 105$

3. Take the remainder modulo 10: $105 \bmod 10 = 5$.
4. Subtract from 10 and take the units digit: $10 - 5 = 5$.

The full code is 5098765435. Checking: the new total is $105 + 5 = 110$, which is a multiple of 10.

If the subtraction gives 10, the check digit is 0 — you cannot store two digits. The "take the units digit" step handles that: $10 \bmod 10 = 0$.

A single wrong digit always changes the weighted total, so the remainder changes and the recalculated check digit no longer matches.

In the modulus 11 example, mistyping the seventh digit 6 as 7 gives a total of 275 instead of 271; the remainder becomes 0 instead of 7, so the expected check digit is 0, not 4. The code is rejected.

Two swapped digits are usually caught too, because the two digits carry different weights: swapping them changes the total by the difference between the digits multiplied by the difference between the weights.

What a check digit cannot do:

• It cannot tell you which digit is wrong, only that something is.
• It cannot catch a transposition that happens to leave the weighted total unchanged. With weights of 3 and 1, swapping two digits that differ by 5 changes the total by 10, which is 0 modulo 10 — so that swap slips through.
• It cannot catch a value that was wrong at the source. If the person issuing the code made it up wrongly, the check digit agrees with the wrong digits.
• A single check digit catches most, but not all, random errors.