--- In Modular Arithmetic , the residue of an integer ''n'' to base ''b'' is the remainder ''r'' after the largest multiple ''mb'' of ''b'' no greater than ''n'' has been subtracted from ''n''. (If ''n'' < 0, one adds multiples of ''b'' just sufficient to make the result non-negative.) The residues modulo (to the base) ''b'' form a Ring .
--- In a Locally Ringed Space (''X'', O''X''), the residue of a section ''f'' in O''X''(''U'') at a point ''p'' of ''X'' is the reduction of the germ of ''f'' at ''p'' modulo the unique maximal ideal of the stalk of O''X'' at ''p''. When O''X'' is a sheaf of functions on ''X'', the residue of a section ''f'' at a point ''p'' is the value ''f''(''p'').
In general Chemistry , a residue is what is left behind by a Reaction .
In PetroleumRefining the residue comprises the heavier Fractions that fail to vaporize. It can be used as Fuel Oil or Cracked to produce lighter fractions. The volume and degree of cracking carried out is determined by the price difference between the light and heavy fractions and the cost of the cracking process.
