Xref: utzoo comp.edu:4206 sci.math:16948 sci.misc:4929 ut.general:1513 uw.general:3326
Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!samsung!rex!ukma!ghot
From: ghot@ms.uky.edu (Allan Adler)
Newsgroups: comp.edu,sci.math,sci.misc,ut.general,uw.general,uw.math.grad,york.general
Subject: Re: Subtle Math Questions
Message-ID: <1991Apr22.235606.10856@ms.uky.edu>
Date: 22 Apr 91 23:56:06 GMT
References: <2729@ttardis.UUCP> <1991Apr22.165415.9843@contact.uucp>
Organization: University Of Kentucky, Dept. of Math Sciences
Lines: 70


Now that Roy Wood has explained that he has no hidden agenda, I would
like to contribute some "subtle" questions off the top of my head.



(1) Any positive real number can be represented as an infinite
   decimal (e.g.3.14159265358979323846...), possibly ending in
   all zeroes or all ones. We teach students how to add decimals.
   How do we add positive real numbers represented as infinitely
   long decimals ? How do we subtract or multiply or divide them ?

(2) We routinely allow students to use calculators. We do not
    normally teach them how to know how much confidence they
    can have in the answer the calculator gives. Of course,
    that depends to some extent on the calculator and on the
    problem it is given.
  (a) What are some simple tests we can give to a calculator to
      determine the nature of the errors it will give us ?
  (b) Take a calculator, take the square root of 2, square the
      answer, take the square root of the answer, square the result,
      and repeat this a dozen times or more. Explain to your
      weakest student why this is happening and how much confidence
      this student should have in the device he/she is using in
      view of this.

(3) You will need a Friden desl calculator for this: how many
    interesting rhythms can one play on this device ? How many
    can one play on a modern calculator ?

(4) Is i greater than 0 or less than 0 ? (i is the square root of -1).
(5) Galileo gives constructions for regular pentagons and regular
    7-gons somewhere in his collected works. How accurate are his
    constructions ? (Yes, look them up. That's where I found them.)
(6) What are the last 4 digits of 5 to the 7777th power ?
    (YOu are not allowed to use a calculator. Anyone who uses a
    calculator will be expelled, their reputation tarnished, their
    future ruined and their children left to fend for themselves
    in a cold and hostile world.)
(7) Are any of the telephone numbers (7 digits, or 10 with the area
    code) at your school perfect cubes ?
(8) Once I was in the Science Center at Harvard on the 5th floor and
    passed someone who was frantically trying to get into the men's
    room but did not know the combination. Figuring that at Harvard
    one could expect someone to figure it out with a little hint,
    I told the person that the number is the sum of the cubes of
    its digits and walked away. Question: how many solutions would
    a person have to try before finding the right combination, in
    the worst case ?
(9) We can reduce the fraction 95/19 to lowest terms by cancelling
    the 9's, right? When is it safe to use this rule ?
(10) When we teach children to reduce fractions to lowest terms,
     we teach them to do it by factoring. We often teach them to
     factor by trying to divide by primes. We teach them to decide
     whether a number is prime by telling them that it is divisible
     only by itself and 1 (which, naively means that we have to try
     all numbers less than the number), presumably because they
     are not scheduled to learn square roots for several years.
     Question: Why don't we teach them to use the Euclidean algorithm
     to reduce fractions to lowest terms ?
(11) True or false: x^2-x+41 is always prime ? This is a good exercise
     because lots of students ignore general statements and guess
     the general rules based on examples. This example shows that
     statements can be false in spite of "overwhelming" numberical
     evidence.

Please don't send me the answers. I already know them.

Allan Adler
ghot@ms.uky.edu