Arithmetic operations

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• 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.