Question

In high school, some students have been confused to believe that 22/7 is already the actual value of π or an acceptable approximation. Show that 355/113 is a better approximation in terms of absolute and relative errors.

Answer 1

The absolute and relative error of 355/113 compared with π is less than when π is compared with 22/7, that's why 355/113 is a better approximation for the actual value of π.

Explanation:

The absolute error is the difference between a value measured and the real value.

abs = π - approximation of π

The relative error indicates how large the absolute error is when compared with the actual value of π.

Now, let's calculate the absolute an relative error for each approximation of π, for simplicity the calculations will be rounded to 4 decimal digits.

rel = abs / π

• For 22/7

abs = π - 22/7

abs = -0.0013

rel = (π - 22/7) / π

rel = -0.0402 %

• For 355/113

abs = π - 355/113

abs = -2.6676 x10-7

rel = (π - 355/113) / π

rel = -8.4914 x10-6 %

You can see that both the value of the absolute and relative error for the 355/113 approximation are smaller numbers, in conclusion, 355/113 is a better approximation for π.