The Case of The Missing Numbers |
describes the extension of the idea of number to
negative numbers. |
---|

Back | Contents | Comments | Next |
---|

In The Case of Being Digital, we introduced the notion of number in order to discuss notational systems for describing numbers. In this example, we will discuss one extension of the notion of number to include "negative numbers." Just as the zero was introduced to fill a formal model that included counting and addition, negative numbers were used to fill a formal model which includes subtraction.

Subtraction is the reverse of addition. If we add one number to another and then subtract the same number from the result, we are left with whatever we started with. We can think of subtraction as "taking away" the elements of a collection or as measuring a difference in length between two lines. This second definition was used by early geometricians like Euclid and Pythagoras when proving properties of geometric figures based on sums and differences of sides. But because they could not imagine measuring a "negative distance," they did not have the notion of negative numbers.

We can't be too sure about the real history of negative numbers
because they appeared at a time in history when writing was sparse.
However, we can reconstruct two possible ways in which negative
numbers might have been invented. The first involves a common trick
in working with models which mathematicians call *reification*;
the second involves the use of algebra to *realize* an unknown
(this is my own phrase).

*Reification* takes a process or an action and turns it into
an imaginary object. For instance, going from one's home to the bank
is an action or a process. However, we can also take this action and
treat it as an object, chaining it together with other actions (first
the bank, then the store, then work), moving it around in time (I can
go to the bank tonight), or other operations. The value of
reification is that it allows us to treat processes as objects and
imagine combining processes in some of the ways which we combine
objects.

Reifying subtraction produces the negative numbers. Combining a negative number with a positive number is the same as subtracting that number from it. And by using the name "addition" to describe the combination of both positive and negative numbers, we have developed a system of arithmetic that includes both things we can count and things we can only imagine.

Using the term "addition" to describe our new operation cannot be
done recklessly. Consider the addition `9+(-5)=4`. We can
only include negative numbers in our arithmetic if there is no
*x* among the positive numbers such that `9+x=4`. The
requirement we need to satisfy is the *systematicity* of
reference emphasized in Chapter 5.
Systematicity means that once we've come up with one translation, we
cannot translate it another way. If there were already some positive
*x* which would make `9+x=4` true, we could not use the
negative number for that purpose. Fortunately, negative numbers don't
need to take over any of the places which had previously been taken by
counting numbers.

The requirement of systematicity spreads further and gives us the
rules for combining negative numbers with one another and with the
positive counting numbers. Addition of the counting numbers is
commutative, which means that the order of operations doesn't matter.
Adding x to y is the same as adding y to x (e.g. `3+5=5+3` and
`9+11=11+9`). Addition is also associative, which means that
the grouping of operations doesn't matter. Adding z to x and y is the
same as adding z and x to y. Together with our definition of adding a
negative number to a positive number as subtraction, these properties
give us the rest of the addition rules.

Because of commutativity, adding a positive to a negative number
has to be the same as adding the negative number to the positive
number. And because of associativity, adding two negative numbers has
to produce a number which is *more* negative than either number
alone. If adding negative numbers produced a less negative number,
the grouping of operations would matter because combining the two
negative numbers first would produce a different result than doing
their subtractions separately.

The same sort of reasoning gives us the rules for multiplication. Multiplication has the following properties over the counting numbers:

`x*y=y*x`(commutativity, just line addition)`x*(y*z)=(x*y)*z`(associativity, just like addition)`x*(y+z)=(x*y)+(x*z)`(distribution)

which together imply the rules we know for the multiplication of
positive and negative numbers. From the rule `x*y=y*x` we know
that the results of multiplying a negative by a positive has to be the
same as multiplying a positive by a negative. This means that we can
define systematic rules for multiplication by just answering these two
questions:

- what is the product of a positive and a negative number?
- what is the product of two negative numbers?

We can describe the answers to these questions as a table and look
at a concrete example of combining addition and multiplication
(e.g. `-5*(3+-2) =?= (-5*3)+(-5*-2)`) which should be equal
under the distribution property. The table of choices looks like
this:

-5*(3+-2) =?= (-5*3)+(-5*-2) neg*pos=neg neg*pos=pos neg*neg=pos 5 = 5 5 ≠ 25 neg*neg=neg -5 ≠ -25 -5 ≠ 5

where it is clear that the only rules which systematically preserve distribution are the ones we know: multiplying positive times negative (or vice versa) yields a negative number and multiplying negative by negative yields a positive number. The rules for the addition and multiplication of negatives all derive from the need for systematicity in the expanded model

We can see systematicity further applied in the rules for negative
powers. Raising a number to a negative power is the same as dividing
1 by the same number raised to the corresponding *positive*
power (Or in math-ese, x^{-n}=1/x^{n}). The reason is
that among the counting numbers, multiplying exponentials of the same
number is the same as adding the exponents, so that
2^{3}*2^{5}=2^{3+5} (or, 8*32=256). In order
for this rule to remain true with negative exponents and the rules for
adding negative numbers, 2^{-2} (for instance) would have to
satisfy the equation 2^{5}*2^{-2}=2^{3} or
(32*2^{-2}=8, so that we would have to say
2^{-2}=1/2^{2}=1/4).

Another way to define the negative numbers is through algebra.
Algebra is a set of techniques for "filling in" the unknowns in
equations without considering all of the possibilities. We could
answer the question "for what values of x does `x+2=5`?"
either:

**exhaustively**by trying x=1, x=2, x=3*or***algebraically**by saying if`x+2=5`, then`x=5-2`, then`x=3`

For the problem `x+2=5`, each of these methods takes the
same number of steps, much as analog addition in the previous chapter
was sometimes just as efficient as digital addition. But the
algebraic method, like the digital model,
**scales**, so that "for what values of x does
`x+2=1999`?" can also be solved with three algebraic steps but
would require 1,996 exhaustive (and exhausting) guesses.

We can define negative numbers using algebra by asking questions of the form "for what values of x does x+5=2?" Using the same algebraic methods, we get "x=2-5" but this number doesn't exist in the counting numbers. But, says the mathematician, as long as we are consistent, that doesn't need to stop us!

To define negative numbers, we just say that `2-5=-3`, where
`-3` is different from any counting number. Given this
definition and a requirement that the things we know about numbers
stay true, we can determine how to work with these sorts of negative
numbers exactly as we determined how we could work with the reified
negative numbers. In fact, reified negative numbers and "realized"
negative numbers work exactly the same.

Combining algebra with negative numbers, we can solve equations
more easily than we could otherwise. For example, suppose we are told
that `x+(y-z)=5` and that `y=z-3`. Given this, we can
figure that `y-z=-3` and by substitution that
`x+(-3)=5`, so `x=8`. Even though none of x, y, or z is
negative, representing an intermediate fact in terms of a negative
number makes the algebraic problem easier to solve.

In both reification and realization, we introduce something
entirely artificial based on the **form** of our
descriptions. This introduction is not a problem as long as we remain
consistent and systematic. Just as introducing the placeholder zero
enabled a model which made arithmetic operations easier, introducing
negative numbers makes algebraic operations easier. However, the
first uses of negative numbers predates the invention of algebra.

As early as the fifth century, Indian accountants were using negative numbers to represent debts. We can look at ancient accounting records and see that a different syntax (not the negative sign we are used to today) is used to describe account debits. (However, we don't know whether or not the accountants routinely added these positive and negative numbers to one another since they did not show their work.)

The use of negative numbers to describe debts is interesting because it characterizes a common way in which models develop. Often, the logic of the model creates or involves "imaginary" elements of some sort. But as the model evolves, reference relations may be defined "underneath" these imaginary elements. Negative numbers, as far as counting sheep or coins goes, do not exist. But using negative numbers to describe debts incurred turns out to be a powerful principle allowing all sorts of stable economic constructions which would otherwise be difficult or impossible to manage.

We sometimes use the term "imaginative accounting" to describe something slightly underhanded. But accounting itself, in its rigorous and systematic way, involves imagination in the best of senses. Without it, we would not be able to make and maintain financial promises, when some number of coins momentarily becomes some quantity of goods.

Interestingly, though negative numbers in accounting preceded the
development of algebra, early algebra did not include negative
numbers. In the earliest algebra texts, we see the solution to
`x*x=4` as being only the value `x=2`, not the modern
solutions of either `x=2` or `x=-2`. Eventually,
negative numbers entered algebra and this combination led to yet
another feat of arithmetical imagination.

Draft, not for citation or circulation

Back | Contents | Comments | Next |
---|