High school students have often been misled to believe that 22/7 is either the actual value of π or an acceptable approximation to π. Here, I show that 355/113 is a better approximation of π in terms of both absolute and relative errors.
By definition, the absolute error is a quantitative representation of the difference between an exact (S) and an approximated (S0) solution values; that is,
On the other hand, the relative error is a qualitative measure of the amount of absolute error (Eabs) with respect to the exact solution value,
Using a handheld calculator with an accuracy of nine decimal digits, we obtain the following approximate values and use them in subsequent computations
22/7 ≈ 3.142857143
355/113 ≈ 3.141592920
Therefore, using equations (1) and (2), we obtain the following absolute and relative error values using the 22/7 approximation
Similarly, the following are the absolute and relative error values using the 355/133 approximation
Figure 1 summarizes the results, which show that 335/113 is a better approximation to the value of π in terms of both absolute and relative errors.
Figure 1: Summary of absolute and relative errors of π approximations