A common misconception that is exactly 22/7: origin and remedy
Many of you may have noticed that a large fraction of students are under the impression that the value of is 22/7. Exact. These students have no idea that the rational number 22/7 is just an approximation to the irrational number .
I think the reason lies in the way schools treat : There might be a couple of mentions that 22/7 is an approximation, but it remains under-emphasized. On the other hand, see the “numerical problems”. Notice the multiples of 7 in diameters or radii in almost all the problems, hinting the students to use 22/7 as and get non-fractional answers! With such practice problems for many years, 22/7 is firmly imprinted on students’ minds as the value of .
I think we can do better: Make the students use various approximations. Let them choose their approximation options from many possibilities: 22/7, 355/113, 3.14…, and so on. Also make them aware of the fraction of error with each approximation; possibly ask them to calculate that. When talking about the value of or results of calculations using , make them always say “approximately”. (We have such rituals in other places: for example we make them write an additive constant C when they calculate indefinite integrals! We even cut points for missing that!)
The point is that it is not a problem if students do not know the exact value of , because nobody does! But they should know that the values that they use are approximations.