Bret Taylor notes a query from a student on the TI listserver:
> I have a questiion with regards to (a graphing calculator). When
> solving f(x) = 6-2x. When f(x) is 2/3 I get the answer of 4.6667.
> I know by looking in the back of the book that the answer is 4 2/3.
> How can I get the calculator to show me that .6667 is 2/3? I knew
> it had to be 4 and a fraction and I guess I could have solved it by
> hand but 2/3 = .6666666667 etc.
An honest struggle with the question of _representing_ a rational
number in a floating point, limited-precision format. When
computers came on the scene in the 50's (and even more when
pocket calculators arrived in the 70's) questions like these became
highly important to wonder about. We never worried much about
representations of fractions in the old slide-rule days. (Maybe
we should have, though. Old "shop-math" books give five-place
decimal equivalents of 32nds of an inch -- clearly inappropriate
for measured quantities).
But I don't think that this issue has received the attention it is due
right up to the present day. No textbook I have ever seen has
addressed this issue head on. When I discuss this in class I have to
make up my own examples and problems.
The TI-92 can be set to store rational numbers as true fractions
(separately-stored numerator and denominator) and to work with
them in that way. Or, if the user wishes, to display the results
as an (approximate) decimal equivalent. This is probably a good
model of how we will calculate from now on. But there are still
issues of which mode to use, when and why. There are judgments
to be made here, issues which we need to discuss in the classroom,
Geoff Hagopian writes:
> I really enoyed the divisibility thread. Another question has occured
> to me - is there a quick way to check whether a given decimal is an
> integer multiple of some algebraic irrational, say sqrt(2)? Probably not -
> stick to the MAA Monthly problems, eh?
What is a "given decimal"? If you give me a terminating decimal
representation I will _guarantee_ that it is not an integer multiple
of an irrational number. :-)}
In fact, there is no way to _communicate_ an arbitrary irrational number.
When we say sqrt(2) -- even if we use a proper radical sign -- we are
_characterizing_ the number we mean, by saying it has the property
that its square is exactly equal to the integer 2. There are methods
for us then to churn out successively-precise rational approximations
of this number, but we can never _display_ it. And we can only do
even this much for an infinitesmally-tiny portion of irrational numbers.
Because simple multiples of radicals come up often in the sort of
calculations that are standard fare in inermediate math courses, there
is often a hunger to _spot_ a decimal representation on the calculator
as a number in this class, and report its origin. Contributors to the
calculator list sometimes undertake writing such programs. But the
effort is really misguided, because other simple transformations of
the basic radicals may also be involved. Just to recognize the
decimal 0.242640687119... as a simple expression involving integers
and the square root of a small integer is already probably beyond
the value of anyone's time, for example....
What _is_ a number? This is a deep question, one faced again and
again in ever more fundamental ways by the best mathematicians
of each succeeding generation (at the moment John Horton Conway
seems to have the best answer). We can certainly use our wonderful
new technologies to bring our students to see the importance of this
question.
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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