One of the things students seem to have lost after we have transitioned from analog computing via slide rules to hand-held calculators is a sense of the scale of numbers. I'll see student answers that are hundreds, nay, thousands of times too large or too small, because there was no mental checking going on with respect to the reasonableness of the size of the number. It is just so easy to punch the buttons on the box and scribble down the answer.
Now using slide rules the scale of numbers is part of the process (staying on track here, computing, e.g., the points on a parabolic curve for a bridge hammock. Cat curves with slides....are you out of your mind????).
*** grizz mounts lecture podium ***
the use of slide rules is based fundamentally on scientific notation. Every number is expressed as D x 10^e, where D is a real number of the form M.xxxxxxx where M is a single digit from 1 to 9, and the xxxxxx refers to the rest of the number to the right of the decimal point.
To multiply D x 10^e and C x 10^f you take
(D x 10^e) x (C x 10^f) = (DxC) x 10^(e+f)
You use the slide rule to compute DxC, then normalize the result (by hand)
DxC = E x 10^g
where E is again, in the form where there is one non-zero digit to the left of the decimal point. The result is
(D x 10^e) x (C x 10^f) = E x 10^(e+f+g)
**** grizz is thrown off the podium ****
the point here is that the scale of the numbers : e, f, g , are evermore at the front of your mind in the calculation. It's easy enough to be off by a factor of ten by a mistake in arithmetic---my specialty---but you cannot not be thinking about the size of the numbers being manipulated.
that said, I much prefer calculators for my personal use. Actually, for computing the various curves and lengths I do for my DIY gear I use a digital computer, programming in (again showing my age) Perl. Can't be bothered to stop and learn that new-fangled stuff that babies you, like strong type enforcement. Real men write programs in languages that can corrupt memory.