Figure 1

Calculation of π using Monte Carlo simulation. (a) The representative schematics for π calculation with a different number of dots in a square. Calculated numerical value of π (=π_est, where est stands for estimation) is defined as (N_D-in/N_D) × 4, where N_D-in and N_D stand for the number of dots inside a quadrant of a circle (with a radius of R) and the total number of dots in a square (with a length of R). By definition, π_est with four different N_D, i.e. 10, 50, 100, and 1000 are calculated to be 2.40 (=6/10 × 4), 2.72, 2.88 and 3.09 respectively, showing that roughly larger N_D gives a more accurate known value of π (π_known ≈ 3.14). (b) A flow chart depicting algorithmic steps to obtain π_est with various N_D and total number of trials (n_T). (c) π_est as a function of n_T at a given N_D (e.g. 10, 50, 100, or 1000). In general, π_est approaches to π_known with the increasing n_T at relatively larger N_D values, as expected. (d) π_est as a function of N_D at a fixed n_T (e.g. 1, 5, 10, 50, or 100 marked as a dotted line in (c). From observation, π_est approaches to π_known with the increasing N_D at relatively smaller n_T but π_est is roughly independent with N_D at relatively larger n_T. (e) A representative graph of π_est as a function of N_D. As N_D is increased, fluctuation of π_est from π_known tends to decrease. Insets show tendencies of fluctuation of π_est in the two different ranges of N_D.