The point is that you should just memorize whatever precision decimal you need. There's no shortcut like 22/7 that will magically give you more accuracy and be easy to memorize.
(pi - 22/7) < (pi - 3.14). 22/7 is closer to pi than 3.14.
It would have been better if I had specified absolute values in the formula. I hope the point came through. (It appears I'm unable to edit this particular comment.)
Sure, but it's not closer than 3.142. You get less than one digit of bonus accuracy.