I learned to
do square roots on paper, probably something over 70 years
ago, but today I'd have to use a calculator AND the answer would have to
make sense
You'd be surprised to know the percentage of people that would
accept *any* result from a calculator, even if it doesn't make
sense at all.
When I was in high school most math or physics teachers would
accept an error in the calculations for an exam problem if the
logic of the solution was right. But I had one who didn't. His
reasoning was that if you make a stupid calculation error as an
engineer, the result would be as useless as if you didn't grasp
the problem at all. The bridge would collapse or the airplane
would fall out of the sky. And he was right. Remember the 10^8
dollar NASA Mars probe that got lost because JPL was using
imperial units while NASA expected metric ones ?
maybe the engineers payed too much attention on getting all the
computations right :)
errors happen, most of us are humans. the question is mainly how to
catch and avoid them: programming languages can easily do dimensional
analysis at compile-time if the dimensions are encoded into the type system.