Augustus De Morgan and the Absurdity of Negative Numbers
De Morgan was a very prominent English mathematician of the 19th 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 set-theoretic 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 y-axis 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
- 1 9 by the new arrangement
- 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:
- 1 9
4(-2), i.e. 40-2, 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 8-5=3. Also to say that a exceeds b by c, we use the equation a-b=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 8-3. 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 a-b when b is the greater of two quantities, instead of b-a. 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 a-b and b-a. For example, 8-3 is easily understood; 3 can be taken from 8 and the remainder is 5; but 3-8 is an impossibility; it requires you to take from 3 more than there is in 3, which is absurd. If such an expression as 3-8 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 3-8 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(29-x)=56-x, 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 and-c 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.
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 b2-4ac (in some cases, and of b2+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 √(b2)=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