Chern classes from the projective-bundle relation

Assume AC. Let $E\to B$ be a numerable complex rank-$n$ bundle with $n\ge 1$ over a path-connected CW complex $B$ (or over one of the CW-type bases of Integral complex projective bundle theorem). By that theorem there are unique classes $a_i\in H^{2i}(B;\mathbb{Z})$, $1\le i\le n$, such that $x_E^n-a_1{x}_E^{n-1}+a_2{x}_E^{n-2}-\cdots +(-1{)}^n{a}_n=0\text{in }{H}^{2n}(P(E);\mathbb{Z}),$ where $x_E\in H^2(P(E);\mathbb{Z})$ is the Euler class of the underlying real bundle of the tautological line (Complex projective bundle and tautological complex line). The Chern classes of $E$ are these coefficients: $c_i(E):=a_i\in H^{2i}(B;\mathbb{Z})(1\le i\le n),$ completed by the conventions $c_0(E):=1\in H^0(B;\mathbb{Z})$ and $c_i(E):=0$ for $i>n$, and the total Chern class is the finite sum $c(E):={\sum }_{i\ge 0}{c}_i(E)=1+c_1(E)+\cdots +c_n(E)\in H^{\ast }(B;\mathbb{Z})$.

For the zero bundle of rank $0$ the conventions give $c(0_B)=1$. The definition depends only on the isomorphism class of $E$, because an isomorphism of bundles induces an isomorphism of projective bundles pulling $x$ back to $x$ and hence preserves the unique relation. For a complex line bundle $L\to B$ one has $P(L)\cong B$ over $B$ with ${\gamma }_L$ corresponding to $L$ under that identification, so the relation is $x_L-c_1(L)=0$; since $x_L=e(L_{\mathbb{R}})$ under the identification, this gives $c_1(L)=e(L_{\mathbb{R}})\mathrm{.}$ This is the normalization used throughout the page: the first Chern class of a complex line is the Euler class of its underlying real bundle in the complex orientation, not a separately chosen normalization.