Definition·D063
The rationals
Equivalence classes of pairs of integers (nonzero second coordinate) under cross-multiplication: each class is a fraction.
Let
be the relation on
(
the integers) with
. Define
the quotient of
by
. Write
for the class of
.
In words
Two pairs of integers, second coordinate nonzero, are declared equivalent when they cross-multiply to equal products. The rationals are the classes of pairs of integers under this equivalence relation: each class is a fraction.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Read
as "
over
": e.g.
since
. By L55,
really is an equivalence relation, which is what makes this quotient meaningful. The integers embed into
via D064; addition, negation and multiplication are defined representative-wise in D065, D066, D067, and every nonzero class has a multiplicative inverse (D069), making
a field (T27). Like
, this is a special case of a general "field of fractions" construction, applicable to any commutative ring with no zero divisors.