Zero as Denominator
Date: 10/22/97 at 15:02:18 From: V. Amburgey Subject: Zero as denominator Why can't zero be in the denominator for rational numbers? My students, preservice elementary education teachers, have asked that this question be submitted to you. We have discussed this using the models of "parts of a whole region" model and the "division model." In discussing models for rational numbers, we have also discussed the "parts of a set" model for rational numbers. My students want to know why 0/0 can't be another name for one when using an empty set as their whole for the "part of a set" model. We look forward to your response. Thanks.
Date: 10/22/97 at 19:55:06 From: Doctor Tom Subject: Re: Zero as denominator The way formal mathematics works is that you make up rules for a system and then study that system. In principle, you can study ANY system this way, but in practice, useful systems get a lot more study than non-useful systems. For the sake of this discussion, let's just consider the set of rational numbers with +, -, x and /. The standard definition says that the operation of division is simply undefined if the denominator is zero. There's nothing on earth to prevent you from using mathematical methods to look at systems where division by zero makes sense, either sometimes (like only when the numerator is zero) or always. What I'll try to show below is that when you do this, you lose so much that adding the definition for division by zero is not worth it. For example, how shall I define 1/0? If you say "infinity," then there's suddenly a new object, "infinity," in your system, and you'd better tell me how it behaves. What's 1+infinity, for example? If it's infinity, then you can no longer use the rule that if you subtract the same thing from both sides of an equation the results will be equal: 1 + infinity = infinity Subtract infinity from both sides, and 1 = 0. So now if I ask you to solve this: x + 1 = 7 you can't just subtract 1 from each side and get x = 6, because there is no general rule. If you modify the rule to apply only to subtracting infinity, it's still no good. Is infinity-infinity equal to zero? If not, what is it? Is subtraction now only sometimes defined? If it is zero, then since 1+infinity=infinity, 1+infinity-infinity = infinity-infinity - I just replace one of the infinities in infinity-infinity=infinity-infinity by 1+infinity, and again I get 1 = 0. I can go on and on, but no matter how you decide to define 1/0, you make one or many of the basic laws of arithmetic fail, and you suddenly find that your system has lost the power to do almost anything. People have looked for good ways to add 1/0 to the rationals (or even just 0/0), and in every case, the new system's rules are so weakened that it is not good for anything useful. Geometers have had better luck. If you look at the line of real numbers and add an imaginary infinity to sort of "close the ends," you get a very servicable system called linear projective geometry. But this geometry has none of the addition or multiplication properties, and only deals with "neighborhoods" of points. It can be very instructional to try various ways to define 1/0 to the rationals (leaving all the other operations as-is), and see what basic laws of arithmetic have to be re-written. Be sure to look at the commutative, associative, and distributive laws, the existence of inverses, closure of operations, et cetera. I think division by zero has always been confusing because there is a lot to it. The books simply weasel out of it by saying, "Division by zero is not defined," and the instant knee-jerk response is, "Well, just define it." It's in trying to find a "reasonable" definition that all the ugly problems come up. -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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