Every Operation Has a Reverse

Definition

Carson's "Third Rule of Maths": "You Can Always Do The Reverse." Every operation you learn has an opposite that undoes it — addition undoes subtraction, multiplication undoes division, squaring undoes square-rooting, expanding a bracket undoes factorising — and nothing in maths is a dead end, because whatever you can do forwards, you will eventually need to do backwards too.

In the Book

The rule is introduced right before "So-Called Long Division," after the book has already used the idea implicitly several times. It lists the pairs explicitly: subtraction reverses addition, division reverses multiplication, square-rooting reverses squaring, and factorising reverses expanding a bracket. The book then uses the specific case of division questions that give answers "less than 1" (e.g. 2/8) versus "more than 1" (e.g. 8/2) to introduce the reciprocal — two numbers are reciprocals when one is the "flip" of the other (4 and 1/4) — and frames the reciprocal itself as a direct instance of the Third Rule. From here on the book repeatedly invokes the Third Rule as a fallback: when working with negative-number multiplication tables, when values in geometry problems don't behave as expected, and whenever a "forward" method stalls, the text tells the reader to ask what the reverse operation would look like instead.

Why It Matters

Naming reversibility as a general property — rather than re-deriving "how do I undo this" separately for every new operation you meet — gives you a standing question to ask whenever you're stuck: what's the opposite move, and does running it backward solve the problem forward? It converts "I don't know how to solve this type of question" into "I already know how to solve the reverse of this, so let me invert it," which is a transferable diagnostic habit well beyond arithmetic — anywhere a system or process has a describable inverse (encoding/decoding, building up/tearing down, cause/effect).