# Arithmetic operations

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Addition, subtraction, multiplication, and division are a family of basic operations on two quantities within a specified number system and domain.

mko-arithmetic-operations · concept · v0.10.0



## Mathematical statement

Each arithmetic operation forms a result from two inputs. Whether the operation is defined and whether the result remains in the original number system must be stated together with boundaries such as division by zero.





## Representative formula (exact TeX)

a+b





## Assumptions


- a and b belong to a specified number system.
- For subtraction and division, the result range and conditions such as a nonzero divisor must be checked separately.





## Symbol scope


- a — left operand
- b — right operand
- + — addition operation





## Intuition, definition, and misconceptions

Addition combines quantities, multiplication can describe repeated combination of equal groups, subtraction finds a difference, and division finds equal shares or a ratio. Every operation must be interpreted in an explicit number system.

Natural numbers are closed under addition and multiplication, but 3−5 is not a natural number and 3÷2 is not a natural-number quotient. Extending to integers, rationals, or reals changes which results are available.

The expression a+b is a representative addition formula, not a complete listing of all four operations. Multiplication ASTs and exact algebraic laws for each number system remain future work.

A common mistake is to assume that all four operations are unconditionally closed in every number system, or to treat division by zero as an ordinary result.





## Adopted derivation guide

[Arithmetic operations: whether the same inputs give a defined result in ℕ, ℤ and ℚ](https://ocme.evemisslab.com/en/derivations/derivation-mko-arithmetic-operations-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Natural number](https://ocme.evemisslab.com/en/mko/mko-natural-number/) — Natural numbers provide the first arithmetic setting for basic quantities and successor-based arithmetic.





## Verification boundary

This entry is a conceptual candidate for an operation family; complete computational and formal Evidence for all four operations and their number-system boundaries has not yet been registered.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Specify addition, subtraction, multiplication, division, closure conditions, and division-by-zero boundaries for natural numbers and extension systems, then create exact Lean statements.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/arithmetic-operations.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/index.md)