  We describe a modification of the usual definition of astronomical
magnitudes, replacing the usual logarithm with an inverse hyperbolic sine
function; we call these modified magnitudes `asinh magnitudes'. For objects
detected at signal-to-noise ratios of greater than about five, our modified
definition is essentially identical to the traditional one; for fainter objects
(including those with a formally negative flux) our definition is well behaved,
tending to a definite value with finite errors as the flux goes to zero.
  This new definition is especially useful when considering the colors of faint
objects, as the difference of two `asinh' magnitudes measures the usual flux
ratio for bright objects, while avoiding the problems caused by dividing two
very uncertain values for faint objects.
  The Sloan Digital Sky Survey (SDSS) data products will use this scheme to
express all magnitudes in their catalogs.
