The imprecise real-time system provides flexibility in scheduling time-critical tasks. Most scheduling problems of satisfying both 0/1 constraint and timing constraints, while the total error is minimized, are NP complete when the optional tasks have arbitrary processing times. Liu suggested a reasonable strategy of scheduling tasks with the 0/1 constraint on uniprocessors for minimizing the total error. Song et al suggested a reasonable strategy of scheduling tasks with the 0/1 constraint on multiprocessors for minimizing the total error. But, these algorithms are all off-line algorithms. On the other hand, in the case of on line scheduling, Shih and Liu proposed the NORA algorithm which can find a schedule with the minimum total error for a task system consisting solely of on-line tasks that are ready upon arrival. But, for the task system with 0/1 constraint, it has not been known whether the NORA algorithm can be optimal or not in the sense that it guarantees all mandatory tasks are completed by their deadlines and the total error is minimized. So, this paper suggests an optimal algorithm to search minimum total error for the imprecise on-line real-time task system with 0/1 constraint. Furthermore, the proposed algorithm has the same complexity, O(N log N), as the NORA algorithm, where N is the number of tasks.