Division by Zero

Understanding division by zero is widely acknowledged as difficult for both students and teachers (e.g., Ball, 1990; Blake & Verhille, 1985; Grouws & Reys 1975; Reys, 1974; Reys & Grouws, 1975; Tsamir, 1995; Wheeler & Feghali, 1983). Extensive experience with mathematical operations that consist of performing ,manipulations to arrive at numerical solutions inevitably leads to the development of an intuitive belief that every mathematical operation must result in a numerical answer. Division by zero, however, does not result in a number, violating the numerical-answer belief. Therefore, students and teachers often perceive division by zero either as a somewhat scary instance of mathematics "misbehaving" or as an indication of their own poor comprehension of mathematics (Reys, 1974; Tirosh, 1993; Tsamir, 1996).

The complicated case of division by zero provides mathematics educators with an opportunity to study several central issues regarding the learning and teaching of mathematics. In this paper we refer to two such issues: the relationship between intuitive, numerical-answer beliefs and formal mathematical definitions; and the question of what counts as an acceptable mathematical argument. Studying the responses of students in different grades to division-by-zero tasks can contribute to our understanding of the complex relationship between students' intuitive beliefs about mathematical operations and their responses to concrete and formal arguments involving these operations. A search for studies that have dealt with this issue revealed several articles that focused on students' and teachers' common responses to division-by-zero tasks. These studies reported that middle school students and preservice primary teachers tend to assign numerical values to such expressions, results that illustrate the coercive effect of the intuitive, numerical-answer belief mentioned above (Ball, 1990; Blake & Verhille, 1985; Grouws & Reys 1975; Reys, 1974; Reys & Grouws, 1975; Wheeler & Feghali, 1983). Secondary school teachers, however, know that division by zero is undefined (Ball, 1990; Tirosh, 1993; Tsamir, 1996). 

 A second major issue in mathematics education is what counts as an acceptable mathematical argument (Dreyfus, 1999). In the case of division by zero, two types of arguments are often used: concrete and formal. While a concrete argument uses real-world experience to "give meaning" to the mathematical expression, a formal argument uses only mathematical definitions and theorems. It is often advocated that in elementary schools, concrete arguments should be used to show the impossibility of division by zero (Duncan, 1971; Henry, 1969; Sundar, 1990; Watson, 1991). On the secondary level, however, students are expected to use , formal arguments to support the non-definition of division by zero (Reys, 1974; Tirosh, 1993; Tsamir, 1996). Similar recommendations have been made with respect to many other mathematical concepts and operations (Fischbein, 1987; Freudenthal, 1973; Orton & Frobisher, 1996).

Several questions arise concerning the nature of arguments that students consider as appropriate for the non-definition of division-by-zero expressions: Do secondary school students use concrete arguments? Do they use formal arguments? What are the effects of grade and achievement level on students' use of these different types of arguments? The second aim of the present study is to examine secondary school students' arguments for the non-definition of division by zero as well as their evaluations of suggested concrete and formal arguments. Before describing the study, we present a brief review of prevalent concrete and formal arguments for the non-definition of division by zero.

Common Arguments for the Impossibility of Division by Zero

Arguments for the impossibility of division by zero vary slightly depending on whether the dividend (the number being divided by zero) is itself zero. 

Division of a Non-Zero Number by Zero

 Concrete arguments. Several concrete arguments have been suggested to help students understand why division by zero is undefined (Duncan, 1971; Knifong & Burton, 1980; Watson, 1991). These explanations often refer to division as "sharing" or "dividing evenly" and to zero as "nothing". The following two examples are taken from Watson (1991, p. 375).

Example 1: Given six apples to be divided evenly among zero children (i.e., no children arrive to collect them), how many apples will each child get? It is clear that there are no sets into which partition can take place. Hence the operation is impossible to perform. The impossibility of distributing the apples allows us to be justified in declaring the operation undefined.

Example 2: Given a whole apple, can we cut it into zero (no) pieces? If so, how do we refer to each piece? It is likely that the suggestion will provoke some laughter among students. The impossibility of the process is again evident, and the fraction which would have occurred, 1/0 can be declared to be undefined as a result.

Another concrete argument is based on relating the "repeated subtraction" model of division to daily experience. Duncan (1971, p. 381) suggests the use of the "emptying a basket of eggs" context to explain why 6 / 0 is undefined.  Here we have 6 eggs in the basket. Each time we reach in, we don't take out any. We shall be reaching into the basket quite a while if we have intentions of emptying it. In fact, there is no ordinary number of times we can reach into the basket that will suffice for us to empty it. The validity of concrete arguments such as the above is debatable (Dreyfus, 2000). These arguments seek to extend familiar relationships involving whole numbers of objects to rather fanciful situations where no objects are present. Quite convincing concrete arguments can be constructed which lead to different conclusions. For example, Mitchelmore (personal communication, 4 October 2000) reports the following argument which his students recently advanced: "Dividing by nothing means you do not divide at all; therefore 5 / 0 = 5". We might therefore expect more sophisticated students to reject concrete arguments in favour of more formal arguments.

Formal arguments. The most common formal argument relies on the definition of division as the inverse of multiplication (Reys, 1974; Watson, 1991). For example, if 4 / 0 =c then c x 0 = 4; but c x 0 =0 for every number c, therefore 4 / 0 cannot be an ordinary number. Theoretically, the expression a / 0 (a:t 0) could be defined as a new number but only if one is willing to violate either the definition of division as the inverse of multiplication or the zero-multiple theorem. Since these are regarded as fundamental properties of numbers, mathematicians have decided to leave a / 0 (a :t 0) undefined. A second formal argument is based on a version of the repeated subtraction model of division. Knifong & Burton (1980, p. 180) exemplify this approach, advancing from the "friendly", defined case of 15 7 3 ( "How many times should one subtract 3 from 15 to reach O?") via 0 7 3 ("How many times should one subtract 3 from 0 to reach 0?") to the case of 3 7 0 ("How many times should one subtract 0 from 3 to reach 0?"). Since there is no answer to this last question, in order to preserve the repeated subtraction model of division 3 / 0 must be left undefined.

Another argument applies an intuitive notion of limit (Maor, 1991; Watson, 1991). Watson (1991, p. 376), for instance, suggests that students "graph the sequence of values, say, {..., 1/6, 1/5, 1/4, 1/3, 1/2, 1, 2, 3, 4, 5, 6, ...} against their reciprocals ... leading to the conclusion about what happens to the reciprocals as n becomes very large or 1/n becomes very small". The temptation is to deduce that 1/0 is a number, larger than any normal number, commonly called "infinity". However, there is a big difference between division by zero and the limit of 1/x as x tends to 0. As Maor (1991, p. 7) points out, "[to say] that the limit of 1/x ; as x tends to o(through positive values) is infinite ... is just another way of saying that 1/x grows without bound as x tends to zero". This argument must therefore be regarded as suggestive rather than definitive.