The critical transition Reynolds number is the lowest value at which the turbulent flow can hold in real flows. The determination of the critical transition Reynolds number not only is a scientific problem, but also is important for some engineering problems. However, there is no available theoretical method to search the critical value. For the hypersonic boundary layer with significant importance for engineering problems, there is no available experimental method to search the critical value so far. Consequently, it is imperative to take numerical method to search it. In this paper, direct numerical simulations (DNS) method is employed to determine the critical transition Reynolds number for the incompressible flat-plate boundary layer. Firstly, under the assumption of parallel flow, the temporal mode DNS is performed to determine the critical value as $Re_{xpcr} =43767,$ which is quite close to the numerical results of other people. Secondly, under the condition of nonparallel flow, the spatial mode DNS is performed to determine the critical transition Reynolds number as $Re_{xcr}=3×10^5$, which is well consistent with the experimental results. In principle, the proposed method in this paper can be extended to the supersonic/hypersonic boundary layer, and that problem will be discussed in the subsequent papers.