Teaching Approximations (was Re: Microcode, which is a no-go for

[Philip Belben]

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...