I am a physics student trying to self-learn Chern numbers and Chern class. The book I am learning (Nakahara) introduces the total Chern class as an invariant polynomial of local curvature form $F$
$P(F) = \det (I + t\frac{{iF}}{{2\pi }}) = \sum\limits_{r = 0}^k {{t^r}{P_r}(F)} $
and each ${P_j}(F)$ defines the j-th Chern class ${c_j}(F) \in {H^{2j}}(M)$
The book didn't mention anything about the Chern number. According to some other material I found (may be wrong), the Chern number is defined as an integral over 2$r$-cycle,
$\int_\sigma {{c_{{j_1}}}(F)} \wedge {c_{{j_2}}}(F) \cdots {c_{{j_l}}}(F) $
where ${j_1} + {j_2} + \cdots {j_l} = r$
The material also said that this integral is always an integer. Due to my limited knowlege, I cannot see how is this proved and I cannot find some reference that is easy enough to me. So can anybody help ?