THE FALLACY IN THE FALLACY A philosophical essay on the plausibility of dividing by zero By Jeryl W. Lafon DATAMATION, Nov. 15, 1971 It happens to virtually all computer programmers. If it has not yet happened to you, the odds dictate that someday it will. Your program has been running smoothly for weeks, months, maybe even years. But suddenly one day, out pops a tell-tale message on the operator's console: ZERO DIVISOR! Have you ever noticed, when the type-out results from a limitation in the hardware or software, the message comes out in the form of some obscure code, like "ERR NO. X-15-P2A"? (When you look up the meaning of Error No. X-15-P2A, it probably says something like "UNDERFLOW EXCEEDS FIXED-POINT BOUND IN MIDDLE CURTATE OF REGISTER Q" which is a deliberately ambiguous way of admitting that Register Q is not adequate to handle numbers beyond the range of +-999. But when the message results from some microscopic oversight on the part of the programmer, the message comes out in plain English--DIVISION BY ZERO--so that all of the computer operators will know you goofed. ("Old Berkenheimer has been dividing by zero again--har,har,har!") If you are lucky, the software has been designed to ignore the zero divisor and give you a correct answer in spite of the error message. If you are not lucky, you will have to modify your program to test for a zero divisor, and branch around the offending instruction in case the zero divisor occurs. My question is, how much longer do we intend to put up with this indignity? Since it might be asking a bit much of the compiler writers, software experts, and hardware designers to handle such problems automatically, I am formally prepared to advance a plucky new hypothesis that could do away with this zero-divisor nonsense forever. The mathematicians assure us that division by zero is not permitted. The mathematicians say (and not without cause) that when we divide by zero, mysterious things happen. Examples they are fond of using to illustrate the problem frequently start out by asking the student to assume that x=y. From this, a series of apparently innocent equations are developed (e.g. x(squared)=xy, or x(cubed)=y(cubed)), which end up by offering seemingly incontrovertible proof that 2=1 (or similar anomaly). After the student has been given time to ponder the enigma, the mathematicians will explain that the fallacy lies in some step where both members of the equation were surreptitiously divided by x-y, which must be zero since x and y are equal. But it seems to me that there is a fallacy in the fallacy. (One of my computer associates, Mr. L. Eisenzimmer, refers to this as a nested fallacy.) Granted that the mathematicians have been operating under this premise for quite a while now, I nevertheless resent being told that I am not PERMITTED to divide by zero, on such tenuous grounds as displayed in the usual examples. Therefore, I will attempt to show that dividing by zero is logical, desireable, and practical. In order to clarify the ensuing discussion, I am introducing a new word to describe such concepts as zero and nothing. The word is pragmadox. A pragmadox (pragmatic + paradox) is defined as a concept which is inherently meaningless or self-contradictory, but which nevertheless, has practical application. Imaginary numbers (e.g. the square root of -1) and geometrical points are classical examples of a pragmadox. Geometrical points, for instance, have been defined as locations in space, without size or shape. Now if a point has no dimensions, it cannot be properly said to exist, except as a figment of the imagination. But without such concepts we could never have reached the moon. The word "nothing" is another example of a pragmadox. Nothing can only be defined with respect to something--i.e. as the opposite of something, the absence of something, or as that which does not exist. Yet even nothing must seemingly be something, in order to be an "opposite" or a "that which." As the poet Wallace Stevens once put it, "There is not nothing; no, no, never nothing." Well, maybe there isn't and maybe there is. But in either case, we definitely need the concept. Okay. If we select a quantity, x, and refuse to divide it by anything, the quantity (I think any competent mathematician will agree) remains the same. In slightly more mathematical terms, x(not divided by anything)=x -or- x(divided by nothing)=x -or- x/ =x Now if zero may be considered to be the mathematical equivalent of nothing, I see no great harm in expressing the foregoing bit of philosophy in an equation of the form, x/0=x. But here, some mathematician will pounce gleefully on the fact that x/1 (is also)=x, and will accuse me of saying, in effect, that zero equals 1. (These mathematicians really know how to hurt a guy.) Well, in a peculiar sort of way, that may be exactly what I am saying--as I will attempt to demonstrate in a moment. First, however, let's examine the problem from a slightly different angle. When we divide a number, x, into another number, y, we are trying to determine the minimum number of x's contained in y, without exceeding the value of y. If the numerator, y, happens to be zero, and the denominator happens to be 2, we are asking how many pairs of units are needed to equal zero. And the answer, of course, is no pairs. We need a total absence of units, and the answer is therefore zero. Conversely, when we divide a positive number by zero, we are asking how many non-units are needed in order to equal or closely approximate a given number of units. But it seems intuitively obvious that if Dick had no apples, Jane had no apples, and Spot had no apples,--in fact, if everybody in the universe had no apples and pooled them all together--there still wouldn't be enough non-apples to make even one tiny fraction of a real apple. The answer, therefore, is "no" amount of non-apples (or non-units (or zeros)). Thus the true quotient of x/0 is zero. This answer may come as a shock to some readers. Being conditioned to the idea that the smaller the divisor the larger the quotient, such readers might suppose the quotient of x/0 to be of infinite magnitude --which is, in fact, what the calculus textbooks attempt to teach. But as will be seen later, this is true only under limited conditions--one of which is that the dividend, x, also be of infinite magnitude. Readers who are familiar with the surface of the Moebius strip (or the shape of Klein's bottle) shouldn't have too much difficulty in grasping the reasons underlying my theory that the quotient of x/0=0. (In fact, anyone who finishes reading the article may find that his mind has been bent into approximately the same shape.) The average computer programmer may find the idea easier to grasp if he associates it mentally with a wrap-around computer memory. And, for my mathematician friend, I will express the concept symbolically as follows: WRAP-AROUND INFINITY (also courtesy of Mr. L. Eisenzimmer) Absolute value(x/a) approaches infinity as a approaches 0; and x/a=0 when a=0 But there is still a third way of approaching the problem of x/0, and I'm sure my mathematician won't overlook it. He may wish to interject, at this point, that when we divide x into y we are actually attempting to determine how many times we can subtract x from y without going negative. "Surely you can see," he will argue, "that we can subtract zero from any positive number an infinite number of times without going negative?" Yes, I can see that. The trouble is, I can also see the foolishness of it. To my way of thinking, the question is not how many times we can subtract, but rather how many times we can subtract successfully. And I submit that we do not add or subtract successfully unless we succeed in increasing or decreasing the original quantity. For example, when we add 5 to zero we have done something meaningful, because we have altered the original amount. But if we attempt to add zero to 5, we accomplish nothing. (We can alleviate the embarrassment of this dilemma by saying that we are adding zero "and" 5, rather than zero "to" 5.) The mathematician, however, adds zero to 5 with a flourish, smacks his lips in satisfaction, and deludes himself that he has obtained a constructive result. In actuality, he has merely gone through an exercise in futility, and obtained an inevitable result. If he has done anything constructive at all, it is to demonstrate the utter impossibility of adding zero to anything. Therefore, although we subtract zero from x an infinite number of times, we subtract successfully exactly zero times (the true quotient). If my mathematician is still around, he will probably want to ask me how I propose to reconcile my original proposition (x/0=x) with the statement I just made (x/0=0). In order to bridge this seemingly impossible chasm, I must touch briefly on a subject which has gone too-long neglected--namely, the relativity of numbers. Obviously, numbers are relative, and the usual practice is to define them as either positive or negative with respect to zero. But we showed earlier that the word "nothing" can only be defined with respect to "something", and the same is true here--i.e., zero itself can only be defined with respect to some other number (or numbers). If our hypothesis is correct that the true quotient of x/0 is zero, then the immediate problem is to isolate the relative value of zero on the imaginary mathematical scale (Cartesian horizontal axis). Since we know that zero lies exactly halfway between +n and -n, we can express the relative quotient of n/0 by the following equation: (n/0)relative=[n-(-n)]/2=(n+n)/2=2n/2=(tilde)n In other words, by halving the difference between +n and -n, we have found that the relative quotient of n/0 is a neutral n--i.e., it lies n units in a negative direction from +n and n units in a positive direction from -n. (I have stressed the neutrality of n in this case by using the Spanish letter (tilde)n, which is doubly appropriate because the neutral n was discovered in New Mexico.) The practical application of this pragmadox is manifested in the fact that it satisfies the mathematician's craving for a unique result--i.e., it is not the same n that we would have obtained if we had divided n by 1 instead of by zero. But my mathematician loves consistent results as well as unique results, and he won't overlook the apparent fact that my answer still doesn't check. He will be quick to point out that if my neutral n had a value, say, of 5, then 5 zeroes wouldn't make 5, and zero fives wouldn't make 5 either. Well, I absolutely agree that zero fives wouldn't make five, but I'm not so sure about the first proposition. If we start out with one zero, then multiply that zero by 5, it seems fairly reasonable to me that we should end up with five zeroes. In fact, I am gripped by an urge to place a string of five zeroes right here on the printed page, then ask my mathematician to count them for himself and see if they don't add up to 5. His immediate response, naturally, would be: "Ah, but that is mere word-trickery. You are treating zeroes as if they were units, which isn't cricket at all." (Back to the old 0=1 pragmadox.) Very well. For the time being, I'm prepared to let my mathematician have his way. We will treat zeroes strictly as non-units, and we will assume that there is no distinction in magnitude between 1 non-unit and 5 non-units. (To do otherwise would be to equate non-units with negative numbers.) Under these restrictions, I confess that my answer doesn't check. I can only say, by way of defense, that when my mathematician has a value, x, and doesn't divide it by anything (i.e., divides it by nothing), he is left with a value of x. And if then he divides that x by 1, he is still left with a value of x. But do I run around accusing him of saying that 1 is equal to nothing?!? It would seem that my neutral x, as a quotient for x/0, is valid for all practical purposes, since it is basically the same answer that my mathematician gets when he doesn't divide x by zero. In any case, if x is the relative quotient of x/0, the true quotient may be expressed by taking the algebraic sum of +x and -x, then dividing by 2 in order to obtain the average: (x/0)true=[x+(-x)]/2=(x-x)/2=0/2=0 But here again my mathematician will attempt to pounce, tearing his hair and screaming that, in the first place, x/0 (can't be)=0, because 0/x (is also) =0, and in the second place, how can x/0 be equal to x and zero at the same time (why don't I make up my mind, etc.), and in the third place, even if five zeroes do add up to 5, zero zeroes certainly wouldn't, because zero time zero is ZERO! (You know how these mathematicians always get in a lather about everything.) Okay, In spite of the fact that this particular mathematician has been harrassing me ever since I began the article, I've grown somewhat attached to him. I think he is a good fellow at heart, and it gives me no great pleasure to stick another pin in his balloon. But I must gently point out that zero times zero, at least from a semantic point of view, does NOT equal zero. When we say that we have zero zeroes, we are actually saying that we have no non-units. And an absence of non-units implies the presence of an indefinite number of units. (In this case, my answer doesn't exactly check, but it doesn't exactly not check, either.) My mathematician is not going to be happy about this at all. But please remember that we agreed to play the game according to his own rules. It was he who insisted that we treat zeroes as non-units. In fact, I think this conclusively proves that it is the mathematician who has furtively been treating zeroes as units. And at long last we have reached the crux of the matter. The old nitty-gritty. The fallacy in the fallacy. Mathematicians have, for lo these many years, been harboring a mental image of zero as a non-unit, while simultaneously attempting to treat it as though it were a unit--a neutral unit, to be sure, but nevertheless as a unit. Well, we pays our nickel and we takes our choice. We are free to regard zero as a kind of neutral pseudo-unit, or we may treat it as a non-unit. But not both. If we elect to treat zeroes as non-units, we promptly deprive them of whatever neutrality they might have had, and they become essentially negative in character. (Hence the term non-unit or no-thing or nothing.) Therefore, we cannot apply the same rules to a non-unit that we apply to true units, and expect the non-unit to meekly conform. As the mathematicians are fond of saying (or were up to now), we simply cannot mix apples with oranges. Now for a quick analytical summary of everything we've postulated: 1. If we treat zeroes as pseudo-units, then n/0=n. (This is safe because, as previously noted, it is the same result that mathematicians get when they refuse to divide the number, n, by zero.) 2. If we treat zeroes as non-units, then n/0=0. 3. If we treat zeroes as pseudo-units, n x 0 =n. But we cannon mix pseudo-units with true units any easier than we can mix non-units with true units; therefore, to avoid confusion and stay on the safe side, we must express the product of zero and n as zero with zero in this case being understood as representing n pseudo-units, distinguished from true units and non-units. 4. If we treat zeroes as non-units, then n x 0 = 0, provided n is not equal to zero; otherwise, the product is indeterminate. 5. The same reasoning applies when we divide zero by zero--i.e., the answer is 1 (necessarily expressed as zero) if we treat zeroes as pseudo-units, and indeterminate if we treat zeroes as non-units. Conclusions: Plainly, we computer people are going to be in serious trouble if the mathematicians persist in regarding zeroes as non-units. We have already seen that multiplying one non-unit by another non-unit generates an indeterminate number of real units. There is nothing implausible about this, but it is equivalent to making something out of nothing, and we certainly don't want to be accused of that. Therefore, the only sane course of action is to treat zeroes as pseudo-units, whereby we common folk can more or less follow the conventional rules of mathematics. Yes, that is the only path to follow, short of giving zero back to the Arabs; and I heartily recommend that we follow it. (Unless, of course, there is a fallacy in the (fallacy in the fallacy).) -------------- Mr. Lafon is a management analyst for the Bureau of Indian affairs. He was previously ADP coordinator for the Albuquerque district of the Corps of Engineers. He has had 10 years of experience in data processing and now specializes in DP standards and procedures. (1971)