Augustus De Morgan and the Absurdity of
Negative Numbers
De Morgan was a
very prominent English mathematician of the 19^{th} Century. His name survives today mainly in the
"De Morgan Laws" concerning the logical connectives "and"
and "or" and their transpositions under negation; there are
equivalent settheoretic formulations as well. In his own time he was better
known as a newspaper columnist, a popularizer in the tradition carried down to
our time by Martin Gardner, only funnier.
His wife collected and published a collection of his writings under the
title, A Budget of Paradoxes, in which most of the pieces
concerned his hilarious correspondence with people who insisted they had
squared the circle.
Let me also
recommend, though for quite different reasons, De Morgan's much earlier book The
Study of Algebra, written in 1831.
My copy, from the University of Rochester library, is an American
reprint of many years later (Open Court, 1878 I think), but apparently the same
as the original book. De Morgan was
very young, about 25, when he wrote it, but he was of course not stupid. Just the same, he shows enormous ignorance
of mathematical developments of his time, even though the book is not intended
to be a treatise but a text for students of high school age, and their
teachers.
His treatment of
negative numbers is the reason I recommend the book. I believe it is worthwhile for anyone concerned with high school
algebra today to understand the 19th Century attitudes which were carried over
to the teaching here as late as perhaps 1950, and how hard it is to get people,
including teachers, to make better sense of such things as story problems and
quadratic equations. De Morgan on
complex numbers is less startling to us today because most of us were brought
up on such mysteries ourselves before we got them straightened; but on negative
numbers he really is a surprise.
Negative numbers did
contain mysteries, and that not so long ago.
Even today, while we teach children the number line, positives to the
right and negatives to the left (or positives up and negatives down, as the
yaxis is marked in the Cartesian plane), and while we feel quite superior to
those of our ancestors who said you couldn't subtract 9 from 7 (We know the
answer to be 2; don't we?), let us consider our algorithm for the more
difficult subtractions that we teach in the third or fourth grade:
We subtract 19 from
57; how? We can't take 9 from 7 so we
regroup: Instead of subtracting 10+9
from 50+7, we subtract 10+9 from 40+17.
Now 9 from 17 is 8 and 10 from 40 is 30, and our answer is 8+30 or
38. In my day this was called
"borrowing": we borrowed the
"1"  really 10 – from the 5 (really 50), and so on, with a certain
way of placing the borrowed digit on the page.
In effect, we replace the array
5 7

1 9 by the new arrangement
4(17)

1 9
before performing the operation

that produces 3 8 as
the answer.
But this whole scheme
is predicated on the notion that "you can't take 9 from 7", surely
nothing other than the quaint prejudice we have just been priding ourselves on
having overcome! Why not have the
courage of our convictions? Damn the
torpedoes; let us take 9 from 7 and get 2, and then take 10 from 50 and get
40, and then combine 2 with 40 to get 38, by golly, the correct answer! Here
is the layout:
5 7
 1 9

4(2), i.e. 402, or 38.
Is there anything
wrong with this? (There isn't,
actually.)
Yet with no sense of inconsistency,
teachers, who tell children about negative numbers in Grade 2, aver in Grade 3
that "you can't take 9 from 7", to introduce the apparent necessity
for "borrowing". I now quote
from Augustus De Morgan, on negative numbers (1831):
"If we wish to
say that 8 is greater than 5 by the number 3, we write this equation
85=3. Also to say that a exceeds b by
c, we use the equation ab=c. As long
as some numbers whose value we know are subtracted from others equally known,
there is no fear of our attempting to subtract the greater from the less; of
our writing 3 8, for example, instead of 83.
But in prosecuting investigations in which letters occur, we are liable,
sometimes from inattention, sometimes from ignorance as to which is the greater
of two quantities, or from misconception of some of the conditions of a
problem, to reverse the quantities in a subtraction, for example to write ab
when b is the greater of two quantities, instead of ba. Had we done this with the sum of two
quantities, it would have made no difference, because a+b and b+a are the same,
but this is not the case with ab and ba.
For example, 83 is easily understood; 3 can be taken from 8 and the
remainder is 5; but 38 is an impossibility;
it requires you to take from 3 more than there is in 3, which is
absurd. If such an expression as 38
should be the answer to a problem, it would denote either that there was some
absurdity inherent in the problem itself, or in the manner of putting it into
an equation. Nevertheless, as such
answers will occur, the student must be aware what sort of mistakes give rise
to them, and in what manner they affect the process of investigation..."
I caution the reader
here that De Morgan is not naive, and that he is making a philosophical point
from which he wishes to derive the usual rules of algebra as we know and use
them, including "negatives", and that his general idea, as we shall
see, is that playing with absurdities like 38 AS IF they made sense can be
made to lead to correct final conclusions.
It takes him, however, a full chapter to explain this.
De Morgan observes
this himself (that such absurdity can make sense) later in the same chapter. He has set up a problem in which the
answer has turned out to be c, and the surprise is that we suddenly discover
that c is positive. What are we to make
of the absurd answer, c? On page 55 he
gives an simple example:
"A father is 56
and his son 29 years old. When will the
father be twice the age of the son?"
Putting x a time when this will happen,
i.e. in the future, he arrives at the equation 2(29 + x) = 56 + x, i.e. twice
the age of the son x years from now will equal the father's age x years from now. The solution is x = 2. It checks in the equation, but what does it
mean? Can it mean that the problem has
no solution?
Today we would
immediately construe this solution to mean that it was two years ago that the
son was half the age of the father, and we would be done with it. To De Morgan this needed more
explanation. It was a mistake, he
explains, to have begun the algebraic formulation of the problem by putting the
date in the future. The negative sign,
an absurdity, tells us we have made such a mistake and have asked an impossible
problem. We should instead let x be the
number of years into the past that the doubling of age occurred. then
2(29x)=56x, i.e., twice the age of the son x years ago equals the
father's age x years ago. The solution is x=2, and De Morgan is
philosophically satisfied.
Just the same, this
kind of thing happens so often that there must be a simpler way to interpret
what has happened. He announces his
principle, his justification for the use of absurd numbers, on page 121:
"...When such
principles as these have been established, we have no occasion to correct an
erroneous solution by recommencing the whole process, but we may, by means of
the form of the answer [by 'form' he means negative or positive], set the
matter right at the end. The principle
is, that a negative solution indicates that the nature of the answer is the
very reverse of that which it was supposed to be in the solution; for example,
if the solution supposes a line measured in feet in one direction, a negative
answer, such as c, indicates that c feet must be measured in the opposite
direction; if the answer was thought to be a number of days after a certain
epoch, the solution shows that it is c days before that epoch; if we supposed
that A was to receive a certain number of pounds, it denotes that he is to pay
c pounds, and so on. In deducing this
principle we have not made any supposition as to what c is; we have not asserted that it indicates the
subtraction of c from 0; we have derived the result from observations only,
which taught us first to deduce rules for making that alteration in the result
which arises from altering +c into c at the commencement; and secondly, how to
make the solution of one case of a problem serve to determine those of all the
others...reserving all metaphysical discussion upon such quantities as +c andc
to a later stage, when [the pupil] will be better prepared to understand the
difficulties of the subject."
From this point
onwards, De Morgan uses negative numbers without shame, stating for example
that a positive number has two square roots, one of them negative. On the other hand, he still does not use
negatives entirely freely. In discussing the quadratic equation a few pages
later he distinguishes six cases, viz.
ax^{2}+b=0,
ax^{2}b=0,
ax^{2}+bx+c=0,
ax^{2}bx+c=0,
and
ax^{2}bxc=0.
This is to say that
he is loath to permit a, b, or c to be negative, since, after all, there is no
need. Whatever we today might call the
signs of the coefficients is taken care of by letting the letters be positive
but having the equation take on the appropriate one of the six forms
listed. This all leads to an analysis
of the sign of b^{2}4ac (in some cases, and of b^{2}+4ac in
others), all very correct and difficult to remember. But worse is to come:
When the discriminant, as either of these expressions is nowadays
called, is negative, a wholly new problem emerges: imaginary numbers.
The introduction of
negative numbers by means of equivalence classes of ordered pairs of positive
numbers is a satisfactory device for some philosophical purposes, but not
others. The avoidance of negative
numbers as philosophically unsound  ditto.
The combination of the axiomatics or logic of the numbers with their
actual use is a difficult matter, too much neglected by mathematicians when
they become educators.
I know that when I
first broke into this racket I was charmed with Peano's axioms and their
development into the line R by suitable definitions and proofs, and went
immediately on into locally convex infinite dimensional vector spaces with
crazy topologies. After living a few
years with pure mathematics I no longer remembered that the man in the street,
even the freshman in college, often did not make a connection between the use
of negatives in banking and their use in graphing polynomials. I was shocked when a freshman would insist
that √(b^{2})=b. Now I see that De Morgan would have insisted
so as well, since he wouldn't admit a negative number for "b".
I'm not sure what
the lesson in all this might be, except that some appreciation of how we got
here is as necessary to teaching as  sometimes  an appreciation of where we
are. Understanding the difficulties of
the historical development of mathematics enables us to understand better the
corresponding difficulties experienced by a young student. The youth of a person is bound to mimic the
youth of civilization, after all.
Ralph A. Raimi^{}
1996