Excessive precision

"There is no point in being precise when you don't know what you're talking about." - John Tukey

It's a familiar trope in science fiction that the smartest character will answer questions with excess precision. On Star Trek, Scottie might give a number to one significant figure and Spock will correct him giving the same result to four significant figures.

The trope works on two levels. The innumerate viewer will think "Wow, the smart guy is really smart! He knows a lot more than the other guy." The mathematically savvy viewer will see it as a kind of joke, intentional or unintentional. In the Star Trek series, I assume the writers are winking at the audience when precision is excessive. If Scottie says the ship will blow up in 20 seconds, there's no point in Spock replying 19.81 seconds, because it would take more than 0.19 seconds for him to state his correction.

Excessive precision is not the mark of the expert. Nor is it the mark of the layman. It's the mark of the intern.

If you ask for the circumference of a circle that is about a mile across, the expert and the layman will both say about three miles. The intern will pull out a calculator and say 3.14159265 miles.

When finding a circumference from a diameter, it's obvious that the relative error in each are the same; multiplying by a constant like I doesn't change the relative error. But it's usually harder to tell how input precision and output precision are related. Assessing the accuracy of an answer is often a more sophisticated problem than coming up with an answer.