[Numeracy 507] Another perspective on numbers, operations, and negatives

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Ladnor Geissinger ladnor at email.unc.edu
Fri Aug 20 16:44:26 EDT 2010

I think that some of the comments about equality, negative numbers,
and operations such as those in Numeracy 505 and predecessors in that
thread are a bit skewed and in some ways ask unreasonable things of math
tools. My intent here is to give a slightly different perspective to
the discussion.

Most of us have seen in print the brief description of Mathematics as
the Study of Patterns. That is, math tools are distillations of
patterns that people have drawn out of (i.e., abstracted from) the
study of a great variety of physical settings. They are mindtools
constructed to aid in analysis of phenomena and prediction of outcomes.
When we come upon some situation where our previous math tools don't
seem to apply directly or do so but only quite laboriously, then we
either generalize the old methods to a new class of settings, or if even
that doesn't give us efficient analytical methods then we invent
something new. This may lead to hubris, to thinking that the new tool
should apply everywhere: invent a hammer and everything looks like a
nail. Negative numbers were invented for some specific purpose, and
then it turned out they are very convenient to use for many other
purposes, but we shouldn't expect them to be useful in all settings
involving measurement or finance.

The idea of natural numbers gradually condensed over a very long time as
a way to record how many things there are in a collection of objects,
especially large collections where simple tallies are not efficient.
Numbers don't have to be imbued with any fancy metaphysical existence
to make them a useful tool. All we need is a simple way to generate and
write down numbers that we have all agreed we will use to label sets of
items at the end of a standard "counting process". That is, we start
with the ordered list of digits and we learn how the "next number
algorithm" works and so we get the strictly ordered sequence of numbers:
1,2,3,4,5,6,7,8,9,10,11,12,13,... .
Now every other set S of objects is compared to the numbers by the
counting process -- arrange the items in S in order one after the other
at the same time pairing each new item with the next number. When you
run out of items, label the set S with the last paired number and call
it "size of S" or the "number of items in S". Great, so now we can
count and record our results. But actually we can do more. It is easy
to decide which of two numbers s and t is the smaller, that is, comes
earlier in the number sequence. So when we count two sets S and T and
get sizes s and t respectively, we can decide which of the sets has the
smaller number of items in it. If I need to keep records of some
standard inventories of different types of objects, I will also find it
convenient to use 0 to indicate having none of some type -- so I've
effectively added 0 to the counting numbers.

But suppose I am the king's accountant and two people bring in big bags
S and T of coins which they have independently counted, and they tell me
the bags have s and t coins. I need to be able to record the total
number of coins. Can I do no better than to "count on", essentially
count the coins in T but beginning the pairing with the "number after
s", and so effectively count the whole combined collection of coins.
Now is the time to invent "addition" of numbers, which we indicate
briefly by using + and call the result the sum. From basic counting
principles we can prove the elementary properties of addition, and then
we can begin to work out practical algorithms for computing sums. So in
effect addition is fast counting -- it allows us to replace actual
counting of the combination of two sets by the operation of addition of
numbers, which we expect will be more efficient.

One could give a similar description of multiplication as another form
of fast counting when there is regularity ( What is the total count if
we have n bags each having k items? n*k).
Then maybe for measuring quantities more precisely we may find it useful
to invent fractions and fill in spaces between counting numbers. And
very soon people will wonder if they can generalize addition and then
multiplication to these "rational numbers", and will the results be
useful in some settings?

I have already described in an earlier email note [Numeracy 478?] a
geometric situation that could lead to introducing negative numbers.
Then various uses of this idea would lead to investigating how to
generalize addition and multiplication to the negative rational
numbers. In the history of development of mathematics we are now at
about the time of Vieta and Descartes (1637) and the promulgation of the
idea that geometry could be done very efficiently using real numbers and
coordinate systems [analytic geometry]. Soon it became clear that
efficiency and ease of use required the usual rule of signs (-1)*(-1) = 1.

I'll end there and hope others will comment on this and further the

Ladnor Geissinger