A new definition of the unpaired electrons in a molecule is proposed, which derives from the one-particle reduced density matrix. It yields lower estimates of the number of radical electrons than the widely discussed 'distribution of effectively unpaired electrons', with a maximum possible difference of a factor of two. Unlike the existing definition, the new definition cannot yield numbers of unpaired electrons higher than the total number of electrons, and also recovers the intuitively expected result for the dissociation of O-2. (C) 2003 Elsevier Science B.V. All rights reserved.