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The Chern number is defined in terms of the Chern Class of a Manifold as follows. For any collection Chern Classes such that their cup product has the same Dimension as the Manifold, this cup product can be evaluated on the Manifold's Fundamental Class. The resulting number is called the Chern number for that combination of Chern classes. The most important aspect of Chern numbers is that they are Cobordism invariant.
See also Pontryagin Number, Stiefel-Whitney Number