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

Staying Consistent

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:

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:

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=negneg*pos=pos
neg*neg=pos5 = 55 ≠ 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/xn). The reason is that among the counting numbers, multiplying exponentials of the same number is the same as adding the exponents, so that 23*25=23+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 25*2-2=23 or (32*2-2=8, so that we would have to say 2-2=1/22=1/4).

Another Path

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:

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.

Uses of Negative Numbers

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.

Copyright (C) 1997, 1998 by Kenneth Haase
Draft, not for citation or circulation
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