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π = 22/7


By
Nasir Alfaraj

February 25, 2017 - Posted in Discussion
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​

π ≈ 3.141592654​
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
 
 
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