Fraction

Mathematical statement

For integers p and q with q≠0, p/q represents a rational number. Two valid representations p/q and r/s have the same value exactly when ps=rq; reduction and positive-denominator normalization change the representation without changing the value.

Representative formula (exact TeX)

\frac{p}{q}

Assumptions

• p, q, r, and s are integers, and every number used as a denominator is nonzero.

Symbol scope

• p — numerator
• q — nonzero denominator

Intuition, definition, and misconceptions

When q>0 and 0≤p≤q, p/q may describe dividing one unit into q equal parts and taking p of them. A representation with p>q extends across more than one unit, while p<0 represents the additive inverse of |p|/q.

A zero numerator with a nonzero denominator represents zero. Multiplying numerator and denominator by the same nonzero integer gives an equivalent representation. A customary canonical form is reduced with a positive denominator, and zero is written 0/1.

For nonzero q and s, p/q+r/s=(ps+rq)/(qs). This is an exact operation in the stated rational-number context; adding numerators and denominators separately is not fraction addition.

A written fraction is not unique: different denominators do not imply different values, and a zero denominator does not represent zero.

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

• Rational number — A fraction represents a rational number, so the representation must remain distinct from the represented value.
• Arithmetic operations — Equivalence, reduction, and addition use integer multiplication, addition, and division by a nonzero value.

Verification boundary

Integer ratios and finite additions can be recomputed exactly, but those instances do not replace a universal proof of the quotient construction of rational numbers or the well-definedness of its operations.

not_formalized

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

Next obligation

Bind the quotient representation of Rat, nonzero denominators, gcd normalization, equivalence, and addition to exact formal 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.