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How Numbers Will Change in the Future

Date: 10/29/2003 at 19:43:06
From: Pat
Subject: how numbers will change in the future

I'm asking my students to develop new number systems.  When they're 
finished, I'd like them to see what other people think about the way 
numbers will change in the future.  Can you provide me with any 
information?


Date: 10/29/2003 at 22:53:32
From: Doctor Ian
Subject: Re: how numbers will change in the future

Hi Pat,

What do you mean by "new number systems"?  Can you give me an example
of one? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 


Date: 10/30/2003 at 08:02:47
From: Pat
Subject: how numbers will change in the future

Dr. Ian -

My 4th/5th grade high-ability pullout class has been discovering how 
number systems have changed over time.  For example, we recently 
read how Fibonacci brought our current numbers (1, 2, 3, ...) to 
Europe and replaced the Roman Numeral numbers.  

We also discuss "change" and I've asked them to think how numbers as
we know them will change in the future.  Will they be dots, shapes?  

This is a creative activity and there will be no right or wrong
answers.  I was hoping to show them what other people are speculating
about future number systems, but I was unable to find anything.

Thanks so very much for your assistance.

Pat


Date: 10/30/2003 at 11:22:36
From: Doctor Ian
Subject: Re: how numbers will change in the future

Hi Pat, 

Thanks for getting back to me. 

First, note that you're talking about _numerals_, i.e., ways of
representing numbers, as opposed to numbers themselves. The following

   ||||||||||||||         Tallies
   XIV                    Roman
   14                     Base 10 (decimal)
   16                     Base 8 (octal)
   10000                  Base 2 (binary)
   E                      Base 16 (hexadecimal)

all refer to the same _number_.  They just use different _numerals_ to
represent the number.   

Fibonacci didn't change the numbers that anyone used.  He just changed
the way those numbers were written down.  

A change in _numbers_ would be something more like the invention of
negative numbers, or imaginary numbers, or transfinite numbers, where
we extend our notion of what a number actually is.  

At first, numbers just represented magnitudes.  With negative numbers,
we can represent direction as well as magnitude.  With complex
numbers, we can represent direction in more dimensions (which brings
angles into the picture, which introduces periodicity)--which we also
do with vectors, and tensors.  With transfinite numbers, we can
represent various levels of "infinity".  And so on.  

In each of these cases, the definition of "number" itself was extended
to allow us to carry out new kinds of operations, and explore new
kinds of concepts.  

Second, I would guess that inertia will prevent people from
speculating too much about new ways of representing the numbers we
already have. 

Think of it this way.  When Fibonacci came to Europe, how many
mathematics books already existed?  Recall that as late as the 
founding of the American colonies, it was possible for one man to have
read, more or less, everything that had ever been published. 

Now, of course, we have zillions of pages of mathematical textbooks
(to say nothing of journal papers, lecture notes, notebooks, and so
on), so the cost of changing to a new system of representation (i.e,
either we'd have to reprint everything using the new notation; or
after a generation or two, only specialists would be able to go back
and consult the older volumes) would be astronomical.  

This issue sometimes comes up with regard to other notations.  For
example, why are electrons specified to have a negative charge,
instead of a positive one?  It's just a historical accident, and at
one time it might have been possible to say: "Wait!  Everyone's
getting confused.  Now that we know more about what's going on, let's
switch and make the electron's charge positive, since that will make
our calculations easier."  But by now, the amount of confusion that
such a change would create would dwarf any possible contribution it
could make to ease of calculation. 

It's easy to think of change as something that can be arbitrarily
imposed on the world.  That's certainly how politicians seem to think
about it ("If we just raise taxes, people will just keep doing what
they were doing before, only we'll get more money").  But consider
that Congress told us that we were supposed to switch to the metric
system about 25 years ago.  And yet, we haven't.  Why?  

The details will be left to future historians to figure out, but
broadly the answer has to be this:  Because the people upon whom the
new system was to be imposed perceived the cost of changing to be
greater than the cost of remaining with the old system.  Yes, it's a
pain to try to remember how many ounces are in a quart... but we still
have all those measuring utensils that grandma left us, and the
cookbooks and recipe cards we have are all written using teaspoons and
pints, and a football field is still 100 yards, and it's still 60 feet
6 inches from the pitcher's mound to the plate, and ten feet from the
floor to the basket... and we _know_ what a pint of beer is, and what
a gallon of gasoline is, and what they should cost...

In general, if you want people to make a change, you've got to offer
them a strong incentive to do it.  Consider the Grafitti system of
lettering used by the Palm operating system for PDAs.  It's somewhat
awkward to learn, and hard to read on paper... but if offers one
really stunning advantage, which is that the letters are so distinct
from one another than a computer can recognize nearly 100% of the
characters written by _any_ person who learns the system.  Individual
differences between strokes become insignificant. 

But now consider this:  Will we all be switching over to Grafitti for
all our writing?  Only if we give up writing on paper, and do all (or
nearly all) of our writing on pressure sensitive computer screens!  It
solves one particular problem in one particular context (handwriting
recognition in PDAs), so it will be used there.  Similarly, the
Braille alphabet solves one particular problem for one particular
group of people, bar codes solve one particular problem, and so on. 

The thing about creativity is that it's relatively easy to engage in
low-grade creativity, e.g., 

  What if we represent each digit by a polygon with that 
  number of sides?  Then we wouldn't have to memorize arbitrary
  symbols, so that would be good, right?

What's much, much harder to do is to come up with creative solutions
that solve real problems without introducing more problems than they
solve.  (Is it really a big problem to memorize a set of digits?  Just
about every human being is able to pull it off.  On the other hand,
can you imagine trying to tell the difference between a polygon with
7, 8, or 9 sides printed by a copier that's getting a little low on
toner?  When we see a polygon, how will we know whether it's supposed
to represent a number, or whether it's just a polygon?)

Nearly every realistic exercise in real-world problem-solving starts,
not by asking "What could we do differently?", but rather by asking
"What is _wrong_ with what we're doing now?  And what is _good_ about
what we're doing now, that we'd have to give up if we started doing
things a different way?"  

I hope this helps.  Write back if you'd like to talk more about this,
or anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
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