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From: mayne@delta.cs.fsu.edu (William Mayne)
Newsgroups: comp.edu,sci.math,sci.misc
Subject: Re: Subtle Math Questions
Message-ID: <1991Apr23.144230.14500@mailer.cc.fsu.edu>
Date: 23 Apr 91 19:26:41 GMT
References: <1991Apr22.165415.9843@contact.uucp> <1991Apr22.235606.10856@ms.uky.edu> <1991Apr23.163231.27780@beaver.cs.washington.edu>
Reply-To: mayne@cs.fsu.edu
Organization: Florida State University Dept. of Computer Science
Lines: 79

First:

I think one subtle question which should definitely go on the list
of things math teachers should know well (but often don't) is one
seen recently here (and answered to death):

Why is 0!=1?

I recall asking my teachers this when I was first exposed to it,
and not getting a satisfactory answer. It is actually a pretty
interesting question, leads into some real math, and helps show
how some things aren't as arbitrary as they seem (contrary to
what I was told when I asked this question way back when.)

Second:

Why are radians used as the preferred measure for angles
(in some situations)? Similarly, why is e so important?
How are trig tables figured?

Granted, the real answer (as I see it) requires going into
a little calculus and might be beyond most students, but teachers
ought to know it and maybe be able to explain in general terms
to students sufficiently advanced to be studying trig.

Third:

If they still teach how to find square roots by hand, using the
algorithm which produces the digits one after another working
on two digit chunks of the argument, not the methods in which
successive approximation is obvious, teachers should know the
justification for it.

Fourth:

Why isn't 1 considered a prime number?

As far as I know (I am not a mathematician) this really is somewhat
arbitrary. It makes the definition of a prime arguably easier, or
at least the question of the least prime factor of a number more
useful, but is this a good reason? Here is a problem, copied from
rec.humor, which illustrates possible confusion when people don't
remember that 1 is not a prime. Perhaps showing it to students
would reinforce that lesson:

In article <8098@utacfd5.UUCP> slh@utacfd5.UUCP (Scot Haire) writes:
>
>    One drab day when Perce and Eve were reduced to thumb-twiddling, Perce
>  suddenly brightened and said, "I'll think of a positive number of 75 or
>  less.  Ask me yes-or-no questions and see how quickly you can guess the
>  number."  Eve, who had never been known to ask an irrelevant question,
>  plunged in thusly:
>
>              1.  Is it a prime number?
>
>              2.  Is it divisible by 2?
>
>              3.  Is it divisible by 3?
>
>              4.  Is it divisible by 5?
>
>              5.  Is it less than 25?
>
>    The questions are given in the order Eve asked them.  After the fifth
>  question was answered - and not before - Eve had ferreted out Perce's
>  number.  Can you find the number and the answers Perce gave to the five
>  questions?

I'll bet most people who solve this (at least most non-mathematicians)
say 15. The first answer I thought of was 1. Verifying that leads to
the other possiblity, 49. I don't know if there are more possible answers
besides these three. Actually since there are at least three numbers
and corresponding answer sets which satisfy the requirements and the
question was "Can you find the number" the answer should be "no."

How many would accept "no" to a "Can you find..." question on test
(assuming someone who knows enough could)?

Bill Mayne