Re: TI-89 virtue email needed
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Re: TI-89 virtue email needed
Just a few more words on this topic. I'm sure that some readers have
begun simple deleting messages with this title, but there is perhaps just a
little bit of juice left to squeeze!
Tom Lake says:
> Does anyone do square roots with pencil and paper anymore? Does
> anyone even remember how? I learned in 1968 and still use it from time
> to time when I am in the field without a calc. It is slow and
> cumbersome to be sure, but how else can a square root be found without
> a calc when it has to be done. I was very suprised when I discovered
> that my son wasn't even taught how to do square roots by hand.
Back when I was in junior high school (some time before 1968) I ran across, and
taught myself, the analaogous pencil-and-paper algorithm for extracting _cube_
roots, mainly because it was "neat" (it was a heck of a lot of work, though!).
I later went on to learn other stuff that was even more neat, and forgot the
details of the algorithm. Despite some effort over the years I have not been
able to re-create the steps, nor do I know a source where I could look this up.
What has been lost? If you gave me a pencil and paper and offered me $100 if I
extracted a particular cube root to four places, I bet I could do it, somehow,
just by playing around. In practice, this challenge doesn't come up (in my
experience), but it is often useful to be able to come up with the square root
of a given number to a couple of places of precision by direct computation
(without a calculator), and I encourage my students to develop this skill.
Say you want the square root of 75. By inspection, this should be 5 times the
square root of 3 (everyone should be able to see this immediately) and by
recollection the square root of 3 is approximately 1.732 (this is one of the
handful of approximations that it is useful to keep in your mental arsenal).
Therefore the square root of 75 is about 8.66 (we can't be more precise and say
8.660 -- this is also something to be aware of).
Oh, was it the square root of 76 you wanted? Must be between 8 and 9 even if we
don't run through the thinking in the last paragraph. If 8.5 squared is too
small, adjust and try again. Shouldn't take very long to get three places,
particularly if you use the idea of weighted averages.
The formal method would be to divide 76 by 8.5, then average the result in with
8.5 to get a second estimate. Repeat once or twice. This happens to be
equivalent to Newton's method for approximating a zero, involving tangents to
the nurve, and is quadratically convergent -- the number of correct decimal
places in the result doubles at each step. This is even better in many ways
that the other traditional square root algorithm which is based on the binomial
expansion of (a+b)^2. I guess I could work out Newton's method for x^(1/3) to
get a better formal process than the one I forgot so many years ago.
Sure this kind of playing around is worthwhile even in the days of calculators.
Anyone who has been keeping up with this list knows that there are few stronger
proponents than myself of advancing the use of calculator power in the
mathematics curriculum. But you need to keep a balance, and practicing mental
math (together with informal scratching on paper) helps you to know when _not_
to use a calculator. And to relieve fears of others that you are somehow
dependent on your calculator. By the way, the next time someone asks you how
much math you could do if they took away your calculator, why don't you ask
them just how much math they could do if you took away their pencil? :-)}
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<
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