Good evening.
I used to post here a lot; now I mainly lurk, but this subject is one I
feel strongly about...
About every other semester, I would have a student who had been taught
"exactly 22/7"! One guy admitted that he had just never bothered to
divide it out. Once he did, he understood the concept of
"approximation", did his homework, and found better ones, like 355/113.
For me, the one that bugs me is sqr(3), which comes up in electrical
engineering a lot in 3-phase circuits.
What bugs me is seeing people type "1.73" into their calculator when
they mean sqr(3). I know other people disagree with me on this - some
say it doesn't matter, others say it shows the person is thinking about
the problem rather than just using a textbook formula - but here is my take.
Firstly, I don't know about many modern calculators, but on the
calculators I grew up with, and on my HP-48, sqr(3) takes _fewer_
keypresses than 1.73, so why would anyone want to type that?
Secondly, how accurate is 1.73? I am willing to bet these people don't
know. A more precise figure is 1.73205... so 1.73 is about 0.12% too small.
In other words, in most applications the difference is negligible. But
when they are calibrating (for example) the tarriff metering system,
where the system is expected to be accurate to 0.1%, using current and
voltage transformers of nominally 0.1% precision (you need to measure
the error under various conditions and program a correction into the
meter), the error in the figure they are using for sqr(3) will swamp the
errors they are measuring.
My worry is that these people will go on using 1.73, and getting away
with it, until they calibrate the metering CTs, and then will have no
ideas why everything is so far out of spec. If they are lucky, the
error will show a new piece of kit to be out of spec when it is actually
fine but marginal, and the supplier will gently correct them...
And all because, as people here have been saying, people don't know the
validity of the approximations they are using.
For sqr(3) without a calculator, "7/4 less 1%" is actually pretty good.
It comes out at 1.7325, or 0.026% high. If I just need a rough figure,
7/4 is fine. And most importantly, I know how much precision I need for
most applications.
Philip.
PS for pi, I once saw 710/226 in a slide rule manual. This is the same
as 355/113, of course, but owing to the discontinuous nature of the
scale precision, it is easier to set up. On that slide rule, anyway -
others may have the discontinuities in different places...
