The following essay was published in two parts, in issues #17 and #18
of the Humanistic Mathematics Network Journal (1998), edited by Professor
Alvin White of Harvey Mudd College, Claremont, CA 91711, and
supported by a grant from the Exxon Education Foundation. Printed copies
of old issues of this journal are sometimes available; write to Professor
White, or to email@example.com
On Solving Equations, Negative Numbers, and Other Absurdities
by Ralph A. Raimi
University of Rochester
Let me be clear about the development in school algebra I wish to track here. It is exemplified by the "solution of equations", as I learned it in my childhood around 1938. Before the era of The New Math, the setting up of linear and quadratic equations to model stories of perimeters and areas, and ages of fathers and sons, was generally done in the 9th grade. Having got an equation with x the unknown -- this having been the hardest part -- we would apply some rules such as that "Equals added to equals are equal", or maybe some rituals called "transposition" and "dividing through", to obtain one or more numbers we called the "solution", which we then "checked" by substitution. If the answer didn't check, one would look back for some miscalculation; otherwise we were done.
Most books -- and many teachers, including my own -- made little effort to put into English what we were doing. Algebra, it appeared was a language and literature of its own, unconnected with words like "if", "then", or "but". Its pronouncements did not begin with capital letters or end in periods. It was no wonder that routine calculations like factoring were easy for us and "story problems" very hard. What can stories have to do with algebra?
It will be the purpose of most of what follows to work through a rather simple problem such as should be understandable to any beginner in high school algebra, to show how putting it into English makes all the difference between a ritual and an epiphany. Not that I advocate a lesson along these lines (it would take some weeks, I should think, and in part would have to stretch over years), but that I advocate a curriculum along these lines, or if not a curriculum a continuing conversation in algebra classes, that conveys the lesson I hope to illustrate with this example.
The last few sections concern more sophisticated interpretations of this problem and a very similar one, which illustrate how mathematics of an unreal sort can get used in a real world, and how understanding of such usage would be impossible without full appreciation of the logic of the simpler versions.
1. A simple problem in 9th Grade Algebra
Here is a typical "story problem" such as might have been found in high school algebras in 1840 as easily as in 1997. Very likely this problem was known (and solved) in 1997 B.C. as well, in ancient Babylonia. No calculators are needed, and only the simplest arithmetic and algebraic
Problem: A rectangular garden is to have an area of 600 square yards, and its length is to be fifty yards greater than its width. What are its dimensions?
Solution: Let x be the length; then x-50 is the width, and x(x-50) therefore the area. So:
x(x-50) = 600
x²-50x-600 = 0
(x-60)(x+10) = 0
x = 60
x = -10
Now what? Well, -10 can't be the length of a garden, so the answer must be 60. We put a circle around the '60' and wrote, if we were meticulous, and the year was 1938, something like this:
"Check: Length x = 60
w = x-50 = 10
60X10 = 600, check."
In my day we got 10 points for this. What more is there to say?
A thoughtful student might wonder where that -10 came from and where it went so suddenly. "Length x = 60" we wrote; why not "Length x = -10"? If asked, the teacher might reply, "Well, that's not a length, is it?" Or, "The length can't be negative." Somewhere else in the book (fifty years ago; today it is no longer so) there might have turned up "extraneous roots," i.e. apparent answers to algebraic equations that for some reason didn't work; maybe this was such a case. Perhaps that was why we had to go through "checking the answer". We will come back to this later on. For the moment, let us consider the language in which the above solution was written. The standard format appears to be a string of equations without punctuation. Let us look again at the model "solution" as it typically appears (verbatim) in the student's notebook, and even as printed in many typical textbooks:
Solution: Let x be the length, so
w = x-50
After the setting-up, "Let x be the length, so...", there are no commas, no periods, no words. The meaning of "x" was established at the beginning, and the rest seems to be equations, i.e. sentences so simple that periods aren't needed. But actually it is not clear that the meaning of x has been established, for all that it was said to be "the" length, since we seem to end with a sudden appearance of two answers, one of which (the negative number) gave little pleasure to either the textbook or the teacher. One might ask the teacher, of these last two lines in the student's notebook, does this mean "x=-10 OR x=60" -- or does it mean to say "x=-10 AND x=60."? Neither interpretation seems to fit the idea that "x" had a definition, that is, a single meaning.
Yet that was the way it began: "x" was supposed to represent "length," a very definite length, a very definite number: the length of the garden wall, perhaps, or a prescription for the purchase of lumber. But isn't x also a "variable"? Maybe it is an "unknown". Is a variable or an unknown different from a number? This isn't funny. The amount of nonsense that has been written about "variables" has not only filled volumes, but has confused generations of both students and their teachers. One begins to suspect that the lack of punctuation and connectives such as "or" and "and", in the traditional way of writing the solution to this problem, are not just abbreviations, but evasions.
2. Playing With False Statements
Now, what was that definition of x? "The length" is what was written above, as if "the length" of a garden with the given description necessarily existed, or was unique. But this is exactly what we are trying to discover: Is there such a length? Maybe there isn't. For example, one might ask for the length of the side of a rectangular garden whose perimeter is 100 yards and whose area is 1000 square yards. We can write equations until blue in the face; we can call its length x as above, so that 50-x is the width and 50x-x² its area, but it should be plain that there is no such rectangle even before trying to solve the equation 50x-x² = 1000. You simply can't enclose 1000 square yards in a rectangle with only a 100 yard perimeter (try a few guesses). Calling the length of such a rectangle "x" doesn't make x the name of anything real. This impossibility was undoubtedly known in ancient Babylonia, and most elaborately analyzed in geometric language by Euclid.
How can such a problem, as it eventuates in an equation, be understood in the first place, then? What right do we have to say "Let the length of the field be x," before we even know there is such an x? Without a more careful statement of what we are trying to find out, and how, no amount of "subtracting the same thing from both sides" and the like will do us a bit of good, except maybe on multiple-choice exams. Both sides of what, for goodness sakes? An equality between symbols involving a possibly non-existent number -- or maybe variable -- named "x"? (In my second example here, 1000 for area and 100 for perimeter, x is a certainly non-existent length, or maybe variable, or place-holder, or unknown, yet it still seems to have a name, "x", and an equation to describe its properties. Are we permitted to debate the physiology of unicorns?
One reason for a more careful statement of the problem is that it will explain some of the wordless, comma-less, period-less "algebra" that appears in the middle of the typical solution. Consider: The textbook says we have an "axiom" stating that if A and B are numbers, and if A = B, and if C is some other number, then A-C = B-C, i.e. "subtracting the same number from equals yields equals." In the solution to the original rectangular garden problem above this fact was used in the following way:
From x(x-50)=600 we derived x(x-50)-600=0 by "subtracting 600 from both sides". Both sides of what? An equation? Yes, the equation x(x-5)=600. We call it an equation because it has an equals sign in the middle, but does that make it true? And if it isn't true, does our axiom still hold? Indeed, the equation in question is usually false. When x=14.7 it is false; when x=435 it is false. What right have we to subtract 600 from two sides of an "equation" that is usually false; and then call the result a consequence of some axiom cribbed from Euclid?
(How that "axiom" got from Euclid into 19th Century algebra books is a story of its own. Euclid in his axioms did not mean "equals" in the algebraic sense at all, but was talking about geometric figures, where by "equal" he meant "congruent" in the first instance, and then decomposable into pieces congruent piece by piece, and ultimately even more sophisticated equivalences than that. There is also the equality of ratios to be found in Euclid’s Book V, with a definition of "ratio" hardly anyone remembers today. Furthermore, the "added to" and "subtracted from" phrases used by Euclid in his Postulates did not refer to anything numerical at all. Modern algebra textbooks tend to forget the origin of these axioms, and they list them along with corresponding rules for division and multiplication, too, something that would have been quite meaningless to Euclid, and which cannot be made to have meaning in his geometric context. [footnote: See, e.g., Dressler and Keenan's Integrated Mathematics, Course 1, New York, Amsco School Publications 1980, p.108: "Postulate 7: Multiplication Property of Equality".]
In their present-day 9th Grade use these statements may be called axioms, because of a long tradition culminating in the author's ignorance, but they are no more axioms than any other properties of the arithmetic operations construed as functions or operators. One might as well call an "axiom" the statement that if x(x-50)=600 then log[x(x-50)] = log(600), or cos[x(x-50)] = cos(600). These statements are, as applied to "equality of numbers", nothing more than the recognition that taking cosines, subtracting 600, etc. are well-defined operations with unambiguous results. It isn't that two numbers are equal, in these applications, as that the two algebraic expressions are intended to be different names for a single number. In Euclid, "equality" denoted not a mere renaming of a number, but an equivalence between two genuinely different geometric entities.)
But this is a digression. Axiom or not, it is true that if two symbols represent the same number, subtracting 600 from each will yield two new symbols also representing the same number, i.e. the original number diminished by 600. Now let us return to the equation "x(x-5) = 600", which is almost always a false statement, and see why we have a right to subtract 600 from both sides of it and somehow use the result for a worthy
3. Inductive and deductive reasoning
To understand all this we must return to the origins of algebra, which was brought to Europe in the Middle Ages by Arabs who themselves had been influenced by Indians, Babylonians and Greeks many centuries before that. The Greeks in the three hundred years between Socrates and Appollonius of Perga, and mainly in the unparalleled age of Plato's Academy, 2400 years ago, had perfected what is now called the synthetic method in geometry (and a bit of number theory as well), showing the world how to proceed from axioms and other known truths to more complicated statements by means of a sequence of airtight deductions, going from each known truth to the next by a step whose validity can no more be denied than the plain evidence of our senses -- and even more so, in that Plato had some doubts about our senses that he did not entertain about geometry.
Most of human life goes in the other direction: We humans use experience more than logic. This use of experience we call inductive, as opposed to the deductive, or synthetic method. We see a thing happen and we look for its cause; if its apparent cause is consistent with what we see, we call that connection a theory. And then we use the connection, the theory, as if deductively (for we can never be as certain of our scientifically postulated causes as we are of the axioms of geometry) until or unless we find out it was wrong or not useful.
This method is certainly not Euclidean mathematics, but it is natural to mankind, and while it has led to many mistakes it has also given us science. The use of experience has been most fruitful of all, as Galileo explained, when the hypothetical "cause" is linked to observation, both past and future, and both real and imagined, by a deductive mathematical argument. Hence Galileo's insistence that experience be reducible to quantities amenable to mathematical method, to number and figure, as in Euclid.
The inventors of algebra were faced with problems that had no counterpart in Euclid's scheme. We want a rectangle whose sides do this and that; how do we find it? Can we begin with the 'known', as in Euclid, whose assertions begin with a given circle, a given length, a given point, or some hypothesis, and go on from there? In the problem of the rectangular garden we haven't been given anything! We don't know the longer side, the shorter side, or even if there can be such sides! It looks as if we have been given an area; hmm, some "gift"! Area of what? Can there even be an area such as we hope to have been 'given'? Where do we begin?
4. The Analytic Method: Induction followed by deduction
It begins by guessing. Nobody can stop us from guessing, after all, and if we guess right we can easily show the answer is right, by "checking". If by some miracle I could think, "Eureka! "60 by 10 will do it!" I then could convince any child that this is surely a rectangle of the desired type. Of course, I couldn't convince anyone immediately that this is the only rectangle that would do it, and it is hard to see at first how one could show such a thing, but in fact algebra formalizes the "method of guessing" in such a way as to answer both sorts of questions:
(1) Find a number or numbers that answer the problem, and
(2) Show that these are the only answers there are.
The Arabic method of "algebra" (a word itself of Arabic origin, having to do with taking apart and putting together) is entirely systematic and convincing, but accomplished the goals (1) and (2) in the opposite order. It first finds out the only (possible) answers there are (or, rather, can be), and then shows them, or it, to answer the problem in fact:
Instead of starting with the known, as in Euclidean geometry, let us start with the unknown, BUT PRETEND IT IS KNOWN! What is unknown? The length of the rectangle, for one thing -- and, for that matter, the very existence of such a rectangle. O.K., we pretend there is such a rectangle and that we know its length: we give it a name, "x". But remember now, x is really the pretend length of the pretend rectangle that we are pretending to know all about, that solves the problem, if the problem can be solved. Maybe it can't. We are not entitled yet to guarantee the problem can be solved -- we have earlier, above, seen an apparently similar one, that can't be solved -- but we can pretend this one has a solution.
If the pretend length is x then the pretend width is x-50; that's what the problem demands. Some children have trouble with a number like x-50, which looks more like a 'problem' than like a 'number'. We can explain, though, that this is because we don't actually know what number x is. If x were 258 then the width would be 208, which also could be written 258-50; if the length were 111 the width would be 61, which also could be written 111-50. So, if L is the length, the width is L-50. In our case we called the length x; so... "x-50" is the width. The pretend width. Then the pretend area of this pretend rectangle is the product x(x-50), which can be written in the ‘expanded' form x2-50x if we like, because that's what the distributive law says we can do with numbers -- and remember, we are pretending that x and x-50 are numbers, maybe not known to us, but, we hope, known to God at least. Notice that x does not have to be called a "variable", or anything else with mystical import. It is a number -- well -- a pretend number.
Now if this pretend rectangle is to be a real one as demanded in the problem, it must be that its area is 600 square yards, or, to put our pretenses into an English sentence:
x is the length of a rectangle that can solve our problem,
This is the key to the whole analytic method, and it is meaningless if it is not written (or understood) as a whole sentence, with a very strong "if" at the beginning and a very strong "then" in the middle. The mere equation, x²-50x=600, is not the statement of the problem. It is not even a restatement of the problem; it is only a part of a longer statement, the one that begins with "if" and ends with "then". In the language of English grammar, the equation "x²-50x=600" is but a clause in a complex sentence.
A clause, in English grammar, is a statement that sounds a bit like a sentence itself, since it has a subject and predicate of its own, but within a sentence it doesn't actually say what it sounds as if it is saying. In a true sentence a clause can nonetheless be false. "If pigs could fly, then they would have wings." This sentence is true, even though both its clauses happen to be false. Such is often the case with sentences of the "if... then ..." form, which is what most mathematical sentences sound like. Of course, some of the clauses might be true, too. But we must not confuse the truth of the sentence with the truth of the clauses. We can even know certain sentences to be true while we have no idea whatever whether the clauses in it are true or not. We don't even need to care if the clauses are true (when taken as if they were sentences of their own) or not. Try this one: "If John is 6 feet tall and Jim is 5.9 feet tall, then John is taller than Jim." Who John? Who Jim? Doesn't matter; the sentence is true, even though it says nothing at all about John or Jim, or even whether they exist.
Thus in our restatement of our problem one need not ask whether "x²-50x=600" is true or false. It is an equation, to be sure, a statement that a couple of things are equal, but, like "John is six feet tall" and "John is taller than Jim", it is just part of a true sentence, having no truth value of its own, except the knowledge that IF the opening clause or clauses are true, this one is, too.
Despite these uncertainties, we have got somewhere; we have narrowed down the problem. IF the problem can be solved, THEN x will have to satisfy the equation x²-50x = 600. Very well; next question: Are there any numbers x which do in fact satisfy "x²-50x=600."? To answer this, we go on with "if...then..." sentences.
If x²-50x=600 then x²-50x-600=0. Why? Because x²-50x Really and truly = 600? NO! Don't let a student believe this for A minute! We don't know if that equation is true (it usually isn't, remember), or that there exists even one value of x which would make it true. What we know is that IF it were true, THEN the second statement would also be true. Subtracting 600 from a certain number, whether it is called x²-50x or is called 600, can produce only one result, and since we happen to know the result is 0 when the "certain number" is called 600, so we also know the result is 0 when that "certain number" is called x²-50x. Provided x²-50x is another name for 600.
One can say here that "the same quantity subtracted from equals produce equals", and that is a common way to remember the drill, but in logic it doesn't say very much, for x²-50x and 600 are not just "equals" in the sense of Euclid. x²-50x and 600 are here assumed to be the same THING, a supposedly "certain number", except that one of the descriptions of that number is more complicated than the other. OF COURSE subtracting 600 from a thing is the same as subtracting 600 from that thing! Only the names are different. And don't forget, it is only a pretend equality to begin with, in that we are assuming we are dealing with a number x, for the moment, that does make x²-50x that real thing, 600. Who knows but that we might not someday find out that there really cannot be any such number x?
5. A Chain of Implications Without Truth
Now we can apply a rule of logic called "the transitivity of implication." There was a time when textbooks made much of this idea, which is really only common sense which we use every day. The rule is this: If A implies B and if B implies C, then A implies C. What are A, B, and C here? They are not numbers, they are statements. The clause "A implies B" is mathematical shorthand for the statement "If A, then B," and it is sometimes more convenient to use the word "imply" and its allies than to go through the entire "if...then..." routine.
In the present case our statements A, B and C are as follows:
A. "x is the length of a 600 square yard rectangular field whose width is 50 yards less than its length";
Remember, these are merely statements, clauses, things that look like assertions but are really only parts of assertions we intend to make seriously. We have already established that A implies B, though we wrote it down in the "If A, then B." format. "Subtracting 600 from both sides" is the most usual language we use to justify, in this problem, "B implies C." So the transitivity of implication, combining the twoassertions, tells us "A implies C", or "If A, then C"; that is,
IF there is a rectangle answering the conditions of the problem and x is its length, THEN x²-50x-600=0.
Well, now that the idea is plain, that at each step we are faced with a hypothetical statement and not an absolute statement, we can speed things up a little, making our explanations briefer. We continually use the transitivity of implication to permit us to "forget" the intermediate stages of our argument. Knowing A implies C permits us to forget all about B from now on. B has served its purpose. Similarly we will soon be able to forget C, as follows; consider the two clauses:
D. "For any number x whatsoever, x²-50x-600 = (x-60)(x+10)"; and
E. "(x-60)(x+10) = 0".
Statement D is simply a true statement, as everyone knows and anyone can check using the elementary rules of arithmetic (the distributive law, etc.). That D is true for all real numbers x is not trivial, of course, and it demands a careful definition of "real number" before it can be asserted.
(Actually, D is true not only for real numbers, but also for complex numbers and for many things that are not numbers at all, provided addition and multiplication are suitably defined for these things among themselves and between these things and ordinary numbers. Square matrices of size 7X7 are an example, but this is by the way.)
Is E a true statement, like D? Of course not. For most values of x it is false. What is true is this: If C is true (for a certain x), then E is true. Why? Because D assures us that the left hand side of C is the same as the left-hand side of E even though we do not know what x is, and so if C is true, then E, known to be the same statement, is also true.
Here is where we stand now: A implies E. If there is a length x that does our job, x satisfies the equation in E. From here it is easy. The product of two numbers can be zero only if one or both of the numbers is zero. So, if E is true, then so is F:
F. "x-60 is 0 or x+10 is zero, or both."
Finally, if x-60 is true, then x=60 (I won't repeat the details about doing the same thing to both sides), and if x+10 is true then x= -10. We can discard the "or both" because we know a single number named x cannot be both 60 and -10. But we do have to pay attention to the "or". In other words, F implies G, the statement
G. "x = 60 or x = -10."
Combining all the implications in a sort of chain, A implies B implies C implies E implies F implies G (remembering D was merely a truth we used along the way) we end up with the statement
"A implies G"
worded as follows:
"If x is the length of a 600 square yard rectangular field whose width is 50 yards less than x, then x = 60 or x = -10."
We see from this statement that we do not yet have the solution, if any, of the problem; all we know is that any number which is not 60, and is not -10, will not solve the problem. This is rather a limited result, but I does clear away the underbrush. (Notice that we have now answered one of the questions about the original way I quoted a typical solution of this problem: The word is "or", not "and".) And the actual solution is now not far away. With only two possible answers, we don't have to have a flash of inspiration and shout "Eureka!" We can systematically try out the two possible solutions. Try 60: Then the width is 10, and since 60X10 is indeed 600 we have a solution. Put a circle around it. Ten points? Not yet; there might be another answer, since we haven't yet excluded -10 by all those implications. But any fool can see that -10 can't be the width of a rectangle of area 600, so we reject -10, as the book said. There is one answer, and the answer is 60.
6. Checking the "Solution"
This last part of the argument, the actual multiplying out of our candidate answer (x = 60) by the number fifty less than x, to see if it indeed gives us our area of 600, is called "checking the answer" in most schoolbooks, and students and sometimes teachers tend to consider this part a check on whether or not one has made a numerical error somewhere along the way. [Footnote: The Dressler and Keenan Integrated Mathematics mentioned earlier is but one among many texts containing no logical explanation of why one has to check an answer. Their typical instruction is "solve, and check, ...", and they make it appear that if the "solve" part contains no errors the "check" is supererogatory.] It is true, of course, that if one has made a numerical error the "checking" step will very likely uncover it, and this already makes the step valuable, but the logical function of the "check" is not often mentioned.
For another example, the book The Teaching of Junior High School Algebra, by David Eugene Smith and William David Reeve (Ginn & Co. 1927) was written by two of the most prominent mathematics educators of the time, both professors of the teaching of mathematics and each the author of numerous books on the subject. On page 191, a paragraph headed The Value of Checking contains this instruction for future teachers of algebra:
"On the whole, however, it is usually better for a pupil to solve one problem and check the result than to solve two and not check at all... (1) he does a piece of work that is ordinarily quite as good an exercise as the original solution; and (2) he has the pleasure of being certain of his result and of his mastery of the whole situation."
Smith and Reeve thus consider checking to be good for the student; what they fail to mention, and probably don't even have in mind, is that "checking" is in fact the only genuine proof of the "result" they think was already in hand.
For in truth, the "result" they refer to (or the results, in the present case 60 and -10) is only hypothetical until the checking, the real proof, is done. Otherwise, -10 is just as good a "result", having been obtained by the same means as the 60. But 60 "checks" in the problem, while -10 -- which solves the equation, to be sure -- fails any check imaginable concerning the area of a rectangle with such a side length.
The so-called check, simple as it might appear, is really the deductive proof in the sense of the ancient Greeks, that our answer is right. What is a deduction? It is an argument that proceeds from something given to something else we then deduce from it.
In the present case we are now (after all that analysis) given a length 60 yards to study. We can actually build parallel garden wall 60 yards long and the other walls 10 yards long, i.e. fifty less than the length, and compute the area. Behold! (The word "Theorem" is ancient Greek for the English word "Behold".) Behold, the area is 600. No ifs or buts here. This particular "theorem" is pretty trivial, but it is a theorem nonetheless: What is this theorem? It says that if a field is 60 yards long, and 50 yards less than that in width, then its area is 600. That's all the problem asked us to show, isn't it? And in truth, we didn't really know that before we got to the so-called "check" of the answer; all we knew earlier was that IF a field did this and that, its length had to be -- if anything! -- either 60 or -10. The "check" is in fact the solution, while what is usually called the "solution" is nothing but the narrowing-down of possibilities.
Yet the traditional "solution" did tell us something else, perhaps equally valuable. It told us that the number we did check out by multiplication was the only one. Or the only positive one, anyhow; and since we didn't want a negative one we now know our solution is unique. There is only one set of dimensions for a garden with the properties demanded. The "theorem" given us by our check tells us that 60X10 worked; the other "theorem", given us by the preceding analysis, told us that only 60X10 could work, unless we wanted to get into negative "lengths", whatever that might mean.
For there does remain a nagging question about that -10. Where did it come from? Of course it can't be the solution to the problem, but it was a solution to the equation that somehow got into the problem. If you diminish -10 by 50 you get -60, and (-10)X(-60) = 600. If -10 checks in the equation, and the equation expresses the conditions of the problem, maybe there is some reason for its having turned up there. Why do we reject it? Because we know something about gardens? What have gardening facts to do with mathematics?
Suppose we hadn't been talking about gardens, but about something we didn't have so much advance information about? How would we have known to reject the "wrong" solution? How wrong is it? It checks in the equation, doesn't it?
Well, there was a slippery phrase two paragraphs back: "...the equation expresses the conditions of the problem..." That isn't quite true. The conditions of the problem were two: First that x be a positive number, since we are looking for the length of the side of a real garden, built of real fencing in a real city; and second, that the equation be satisfied. This is how we know to reject the -10. Had we been more careful in stating the problem, we might have put it thus at the very outset: "Find the (positive) length of the side of a garden... Then at each step of the narrowing down part of the solution, well before the "check", we would repeat "positive number" before the symbol "x", e.g. "Let x be the positive number of the pretend length of ..." and so on. We would end, "Then the positive number x must be either 60 or -10", and it is clear that our final statement would be "Then x must be 60" (if such an x exists). There would be no need to worry further about the -10, but the check that 60 works would still be needed as before.
7. Negative Numbers
Yet the uneasiness won't go away. The history of mathematics is full of "impossible" objects that later became common, so much so that we wonder at the blindness of our ancestors. Irrational ratios horrified the Pythagoreans, but were quite understandable to the school of Plato. Imaginary numbers were simply not there at all, that is, there were no "numbers" x such that x²+1 = 0, until such numbers somehow turned up "temporarily" in some of the cases of Cardano's sixteenth century solution of the cubic equation. Proper combination of such imaginary numbers turned out to deliver genuine, "real" answers, that "checked" in every detail.
Negative numbers, too, have had such a history, 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
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 a minute earlier been priding ourselves on having overcome! We can take 9 from 7 if we 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?
Yet with no sense of inconsistency, teachers who tell children about negative numbers on the number line in Grade 2 aver that "you can't take 9 from 7" in Grade 3, to introduce the apparent necessity for "borrowing". One can give a good reason for all this, in that the "regrouping" or "borrowing" scheme can be chained in a convenient manner for a longish problem, while combining the positive and negative differences over a multi-digit subtraction might prove more tedious, but this is probably not why we have the algorithm we do. Our 'regrouping' scheme as written above was invented four or five hundred years ago, in the early years of the European adoption of the decimal system, and in that era subtracting large from small numbers was suspect. Try it on an abacus, for example, which historically preceded the written algorithm but uses the same idea. In fact, there is not an arithmetic book in the Western world that shows how to subtract 866 from 541 by placing the figures in this form
and then going through the "borrowing" ritual to the bitter end. Where on earth is the last "loan" to come from? Our schoolbooks even today evade the question by merely announcing that the difference b-a is the negative of the difference a-b, and telling us to solve the subtraction problem printed above by computing the opposite difference
in the approved manner, finally changing the sign of the positive result, to -275, to answer the original problem.
8. De Morgan's Reservations about Negatives
Now, so recent and illustrious a mathematician as Augustus De Morgan, while willing to go so far as to make temporary use of so ridiculous a notion as "-2" as we did in the course of performing the earlier subtraction of 19 from 57, was still unwilling to grant a negative number a real final existence. In his 1831 book, On the Study of Mathematics (reprinted in 1898 by the Open Court publishing company in La Salle, Illinois), Chapter IX is named On the Negative Sign, etc. Here (p. 103) De Morgan cautions the beginner in algebra to beware of negatives:
"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 a full chapter to explain this.
It would be wise for present-day teachers to have some appreciation of the philosophical problem involved here, and its clever modern solution by "negative numbers" defined as equivalent pairs of such "impossible" subtraction pairings. But this process, which mathematicians call "embedding a commutative semigroup in a group", while logically satisfying and consistent, does not really attack the problem of what the new numbers mean in applications to the world of apples and gardens. However, this is not something for the 9th grade to elucidate. The mere representation of negative numbers as they appear in practical life, debts as against credits, past as against future, and so on, will usually do the job without needing such sophistication.
De Morgan observes this himself 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. De Morgan ends with the absurd conclusion that x+2=0. (We would say that the solution is x = -2). The absurd statement “x+2=0” does, if the usual rules of arithmetic apply to so bizarre a number x, check with the original equation, but what does it mean? Unlike the problem of the rectangular garden above, this problem allows the negative number as the only answer. Can it mean that the problem has no solution? De Morgan suggested that this would be the case: the problem was impossible. There could be no such father and son. And, as the problem is stated, the problem in fact has no answer. (De Morgan did have an “end run” around this conclusion, as we will see below.)
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 (postive) 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. De Morgan 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."
8. De Morgan's Reservations as to Imaginary Numbers
From this point onwards, De Morgan uses negative numbers without much 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 always represent positive numbers but having the equation take on the appropriate one of the six forms listed. This all leads to an analysis of the sign of the discriminant, b²-4ac in some cases and of b²+4ac in others, all very correct and difficult to remember. (In many American school algebra books of a hundred years ago students were asked to memorize the analysis of all six cases, and whether the roots in each case would be positive, negative, etc.) But worse is to come: When the discriminant is negative, a wholly new problem emerges: imaginary numbers.
De Morgan was writing in 1831, but in an insular England that was largely ignorant of recent developments in Continental mathematics. The Argand diagram for complex numbers had been known for 35 years, and Gauss and Cauchy had developed a science of complex numbers almost to the point of view taken today; but De Morgan makes no attempt in his book to develop a philosophy of their interpretation equivalent to what he has done for negatives. Perhaps he understood more than he was saying, but in this book, designed for teachers of children, he refrained from its elaboration. On page 151 he writes:
"We have shown the symbol √(-a) to be void of meaning, or rather self-contradictory and absurd. Nevertheless, by means of such symbols, a part of algebra is established which is of great utility. It depends upon the fact, which must be verified by experience, that the common rules of algebra may be applied to these expressions without leading to any false results...."
Despite these pleasant features, he denies them any sense. He proposes two problems to distinguish his meanings: The first is the problem of the ages of father and son described above, where a negative answer can be made to yield up some sense, either as a guide to a restatement of the problem, or by the device of interpreting such a number as the same as its positive opposite, taken in an opposite direction. The equivalence of the two devices is of algebraic and practical importance. But his second example, he thinks, yields no such practical interpretation. Here it is:
"It is required to divide “a” into two parts, whose product is “b”. The resulting equation is x²-ax+b = 0..., the roots of which are imaginary when b is greater than a²/4." Try as he may, he cannot get out of this one. If he replaces x by -x in the problem the roots are still imaginary when a is too small. (For De Morgan, "imaginary" means what we call complex.) He concludes that there is an essential difference between mere negative numbers, which can be repaired by a reinterpretation of the problem, and imaginary numbers, which for all that they obey the usual algebraic rules, cannot be made to represent anything sensible.
Of course, he has a physical prejudice in the back of his mind here. The problem of dividing a into two parts whose product is b is an ancient one, Babylonian but put into geometric form in Euclid, where it is construed as asking for a segment of length a to be partitioned into two segments which are sides of a rectangle of given area. (We would say "of given area", whereas Euclid remains purely geometric, and exhibits as the datum "b" a triangle to which he wants the resulting rectangle to be equivalent in his own sense of "equals". There are no numbers at all, hence no "areas" in our sense, in Euclid's formulation of such problems.)
Euclid's theorems provide a construction by which the point of partition may be found, but he notes a limitation: If the triangle b is larger than the square built on a/2 (i.e. half the segment a), then the necessary point of partition cannot be found. And that's the end of it: impossible. Euclid's "impossibility condition" is precisely our criterion concerning the discriminant, as it turns out. It says that the given length a is simply too short to accomplish the asked-for job, no matter where you divide it.
Neither Euclid nor De Morgan construes this problem in any other way; it is plain that the number a, which is to be partitioned in De Morgan's problem, looks to him like a line segment, and that there is plainly no solution, not even one that can be reinterpreted as an "opposite" when it turns out negative, when b is larger than the square upon a/2. Yet today, we often take a different point of view.
To us, to "divide a into two parts" when a is a number, means nothing other than to find two numbers whose sum is a, and this can be done in such a way that the product is any given number (not area) b is easy, when complex numbers are allowed as answers. Complex numbers are absurd if construed as line segments -- or are they? Remember,
-10 was also absurd, when construed as a length.
9. The Negative Root is Not Absurd!
But this is not the only interpretation of the number -10 that turns up in our gardening problem. Ah, how much wiser we are, or think we are, than our forefathers! Let us return to the problem of the garden, whose area is to be 600 square yards, and one side of which is 50 more than the other. We put x for the "length" of the garden, and found that x had to be 60 or -10, if anything. We rejected -10 as absurd, and solved the problem: 60 was the length, 10 the width.
Now where is this garden to be located? Here: One corner of it is under my feet, and the length is to be taken to the east, the width to the north. We can walk around the garden by walking 60 yards east, ten yards north, 60 back to the west and 10 south again and here we are. What about -10? Suppose we use that absurd solution as De Morgan, poor, simple De Morgan, suggested. We now surround what piece of land? Well, x = -10 and the "width" is 50 yards less, or -60, so: We walk -10 miles east, i.e. +10 miles west, then -60 yards north, i.e. 60 yards south, then back ten yards and back 60 yards and here we are at the origin (original corner). It is a totally different piece of land, to be sure, lying in the fourth (Cartesian) quadrant rather than the first. Its east-west dimension is of absolute value 10 rather than 60, so that "length" might be considered a strange description of that part of the boundary; but it, with the "width" of absolute value 60, satisfies all criteria of the problem. Its length is -- as a number -- indeed fifty more than its width (-10 is greater than -60 by 50, is it not?), and its area is 600, if "area" is the product of the numbers that describe the sides.
The answer the teacher expected was then 60 yards east by 10 yards north. But the stupid kid who insisted on "checking" the impossible answer x = -10, and got it to "check" at an area of 600 had just as good an answer, only his garden had a different orientation and position. I wonder what a Babylonian would have said to that.
One lesson that comes from all this is summarized by the title of a famous paper by the physicist Eugene P. Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences (Comm. in Pure & Appl. Math. v.13 (1960), 1-14). The present example, interpreting the 'absurd' second solution of the quadratic equation, is trivial compared to the sort of thing Wigner mainly had in mind, but it is of the same nature: The equations arrived at by scientists to describe some part of the physical world often seem to contain more information than the inventors thought they had put into it; and that it is one of the wonders of the life of science to discover such a thing in practice. But one also has to know how to look.
10. The Analytic Geometry of the Garden
How did anyone ever think of that second solution to the garden problem? It sounds like a stretching of the meaning of "-10 " to suddenly start talking east and west, north and south, but in truth we do talk that way in the 20th Century all the time. Here is a reformulation of the garden problem which will automatically make sense of the "absurd" solution as well as the usual one. The word "analysis" was used above to describe the process of algebra we were using; well, the reformulation has to do with analytic geometry. (Euclid’s geometry is called “synthetic”.) Any child can do it:
Problem: Let a rectangle in the plane have one corner at (0,0) and the opposite corner at (x,y), where y = x-50. Find all the corners if the area is to be 600.
Answer: Notice the problem does not insist on (x,y) being in the first quadrant. The area is clearly the absolute value of xy, whatever quadrant (x,y) is in. Since y = x-50, we set |x(x-50)| = 600 and hope such an x can be found, as above. Notice: the absolute value of x(x-50) will be the area of the genuine rectangle in the x,y plane with opposite corners at (0,0) and (x,y), no matter what quadrant (x,y) is in. Then either x(x-50) = 600 or -x(x-50) = 600, according to whether the number inside the absolute value signs turns out to be positive or negative. The first of the two equations gives x = 60 or x = -10, as earlier, and produces the corners (60,10) and (-10,-60) to define two rectangles (whose opposite corners are at the origin) that do the job.
How easy! Of course x = -10 has a meaning, once we set the thing up on the coordinate plane. But wait, what about the other equation, "-x(x-50) = 600"? This one has solutions, too, and they are x = 30 and x = 20, producing opposite-corner points (30,-20) and (20,-30), either of which, with the origin, sure enough forms a rectangle of area 600. Goodness, the more we want to make sense of the problem, the more answers turn up! But if you look at these last two "solutions", do they "check" when we try to prove they satisfy the conditions of the problem? They do: They give the correct area, and y = x-50 as demanded. The trouble here is that we probably have stated the problem badly.
If all we wanted was that the number that is the y-coordinate of the
opposite-corner point should be 50 less than the number that is the x-coordinate of that point, these last two solutions check out in every detail. But surely this is a poor statement of the original problem, where the architect doubtless intended the length of one side of the garden to be 50 more than the length of the other side. The condition "y = x-50" is not a statement of that condition, while |y| = |x|-50 is the point (either that, or |x| = |y|-50).
With this restatement we can go back over the whole problem and find that the third and fourth "solutions" do not check. On the other hand, the new conditions on length, expressed in terms of absolute value, give rise to some new possibilities, and it will perhaps surprise nobody that there are eight solutions, with the origin at one corner and the "opposite-corners" at (10,60), (60,10), (10,-60), (60,-10), (-10,60), (-60,10), (-10,-60), and (-60,-10), that is, all the possible ways you can place a sixty-by-ten rectangle with one corner at the origin and sides parallel to the axes.
Pandora's Box is now open: What if the rectangles are not parallel to the axes? There are answers to that one, too, but they go beyond simple algebraic equations and their meaning. It were best now to cut our losses and go back to the beginning: "Sixty by ten" is doubtless the best answer. But intellectually we have found something out: negative numbers, just as De Morgan said, can be made to mean something valid. We have found something else out, too, just as De Morgan said, which is that we must understand that we are making them mean something, and that the process of associating these invented numbers with some scientific or architectural use is not as simple or obvious as it might seem when they are presented axiomatically. Logic is not only a matter of reasoning from axioms for a field, it is also a matter of reasoning from life.
11. Even Imaginary Solutions Are Not Necessarily Absurd
Finally, let us return to the partition of a segment of (positive) length a into two pieces forming adjacent sides of a rectangle of area b. (This discussion will be rather condensed, compared to what has gone before.)
We suppose x is a length that does the job, i.e. x and a-x are the two side-lengths. We blindly set up the quadratic equation x(a-x)=b and find two solutions (both of which check in the equation if not the problem): they are
(a/2) + √[(a/2)² - b], and
(a/2) - √[(a/2)² - b].
So, if there is a solution it has to be one of these two numbers. (Actually, since these solutions add up to a, this pair of numbers is the only possible solution, i.e. if x is the first, a-x is the second, and if x is the second, a-x is the first.)
When (a/2)² > b all is well; we get two positive numbers which add to a and which solve the problem. We can draw a picture of the resulting rectangle, and we have no negative solution to have to interpret. But what happens when (a/2)² < b? Can we, with Wigner, discover the "unreasonable effectiveness of mathematics" by finding that there really is a genuine visible rectangle that solves the problem even when a is too small to partition properly, i.e. to produce the sides of a rectangle with desired area b? Sure.
Let a four dimensional Euclidean space have its axes labeled x,y,u,v, with the point (x,y,0,0) representing the number x+yi when this is the solution of a quadratic equation using the sign "+" in the quadratic formula, and the point (0,0,u,v) representing the number u+vi where this is the solution of the same quadratic equation using the "-" sign in the quadratic formula. Observe that in our problem, where a and b are positive, the numbers x,y,u and v obtained from our quadratic will always be positive when the discriminant forces us into complex roots. Thus x+yi can be pictured as a vector, or rather an arrow with tail at the origin and arrowhead in the first quadrant of the xy plane, and similarly for u+iv in its uv plane, which is perpendicular to the xy plane. The vectors are (when you disregard the frill of the arrowhead) perpendicular segments in 4-space, and the area of the rectangle they subtend -- a genuine, visible rectangle (a square, actually) -- is equal to b.
How come? Let's work it out. In this problem, a was "too small" to admit such a partition, or to put it in other terms, b was "too big" (in area) for a rod of length a to be broken into two pieces of real lengths for the purpose of making a rectangle that big. But what are the lengths of the vectors that made the sides of our rectangle in 4-space? They are √(x²+y²) and (u²+v²), or √(b) in each case; hey! -- we've even got a square, not just a rectangle! Those are pretty long segment lengths, big enough so that the square they build in 4-space is sure enough of area b. But we found earlier that long things like that can't partition a segment of length a. Indeed, the sum of these two lengths is 2√(b), which is certainly not a.
Well then, what was the problem? Did we ask for a to be partitioned into two pieces whose lengths add to a, or did we ask for a to be partitioned into two numbers whose sum was a? We solved the latter problem, by finding complex numbers whose complex) sum was a but whose lengths were big enough to make a square of size b.
Do I hear someone cry fraud?
"Fraud!" cried the maddened thousands, and echo answered fraud;
But one scornful look from Casey and the audience was awed.
Partitioning a into complex pieces that make, in a suitable geometric interpretation of complex numbers, a suitable real rectangle is no more fraudulent than interpreting the garden problem as one of finding coordinates of a point in the Cartesian plane, rather than lengths of wall, or using negative numbers in the manner of De Morgan to represent the past instead of a putative future.
We all know there is no date at which the son will be half the age of the father; it's too late for that already. In De Morgan's time it was still questionable whether using a negative answer amounted to a swindle. Unfortunately, "hardly a man is now alive", (to quote from another narrative poet) who still appreciates the intellectual effort it took to overcome this natural disinclination to treat mathematical artifices as if they had real significance; and it is a rare teacher who recognizes there is a even a problem.
A garden plot with negative sides is really every bit as silly, at first glance, as a square with complex sides. But you can get used to these things after a while. The important thing is to understand just what it is you are getting used to.
Ralph A. Raimi
5 May 1997