According to the following Wikipedia article
https://topospaces.subwiki.org/wiki/Euler_characteristic
the Euler characteristic of a topological space $X$ (with finitely generated homology) is defined as the alternating sum of the Betti's numbers: $$\chi(X)= \sum_{p}(-1)^p ~ b_p(X)~~~~~~~(1) $$ where $b_p(X)$ is the rank of the torsion-free part of $H_p(X)$.
For a CW complex $X$, the definition of Euler charateristic is defined by $$\chi(X)= \sum_{n}(-1)^n ~ c_n$$ where $c_n$ is the number of $n$-cells in $X$.
As the Wikipedia article states, these two definitions should be equivalent. However, Hatcher says at page 146 (http://pi.math.cornell.edu/~hatcher/AT/AT.pdf) that, given a CW complex $X$, one has $$\chi(X)= \sum_{n}(-1)^n ~ H_n(X^n,X^{n-1})= \sum_{n}(-1)^n ~ \text{rank}(H_n(X))~~~~~~~(2)$$ but in general $H_n(X)$ is not torsion free, so I don't understand how are we even allowed to consider $\text{rank}(H_n(X))$ and how (1) and and (2) are related. What am I missing?