Definition·D065
Addition of rationals
Add rationals by the schoolbook fraction rule: a/b + c/d = (ad+cb)/(bd).
For
, write
and
with
,
, and define
(using integer addition and multiplication). Since
, also
, so the result is a valid representative pair; this does not depend on the choice of representatives, so
is a well-defined binary operation on
.
In words
With b, d nonzero, the sum x+y is the cross-multiplied fraction (ad+cb)/(bd).
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).