Undefined versus Cannot-Be-Defined

Clayton W. Dodge explains as follows:

Note that division by zero is not defined. In fact, division by zero cannot be defined consistent with our definitions of addition and multiplication and equality. Consider  a  0 +  1  1 =  a 1 + 0 1  0 1 =  a + 0  0 =  a  0   by applying Definition 38.4 and the properties of integers. If a/0 is a number x, then it asserts that x + 1 = x, an equation which is impossible, having no solution, for if it were true, then 1 = 0 would immediately follow. Thus the expression "a/0" has just as much meaning in mathematics as the word "glbxw" in English; it is just a meaningless collection of symbols! (refers to Dodge, Clayton W., Numbers and Mathematics, Prindle, Weber, and Schmidt, Boston, 1969, page 185)

 
 
It is interesting to note that Dodge's definition 38.4 [30] applies only to members of Q'. Definition [28] clearly states that Q' is the set of all ordered pairs [a,b] of integers a and b with b  0, or in the above notation, Q' consists of all ordered pairs of the form a/b with b  0. Hence the definition he used does not apply since a/0 is not considered to be a member of Q'.

This is the most frequent fallacy in trying to show why division by zero cannot be defined -- appealing to a definition or theorem which does not apply.

Another reason given many times for not dividing by zero is a consideration of the limit of 1/x as x  0. The argument goes like this:

	lim x  0+	1/x	=	+
	lim x  0-	1/x	=	-

 
Therefore, since the limit from the right does not equal the limit from the left, the limit does not exist and the function 1/x cannot be defined at x = 0.

The fallacy in this argument is that the limit of a function has nothing to do with the value of the function at a given point. All this argument has established is that the function 1/x is not continuous at x = 0. It did not establish that 1/x cannot be defined at x = 0.

Apparently J. Houston Banks noted the difficulty in deriving a contradiction from his definition of division and division by zero since he completely ignores his definition [47] and appeals to a person's intuition:

It is not possible to divide by zero. Let us see why. We normally check the process of division by multiplication. For example 8 2 = 4 is checked by noting that 8 = 2  4. Now consider the example 8  0 = n:   8 = 0  n. However, the product of 0 and any number n is always 0, and thus cannot be 8. Hence, division by 0 is meaningless and undefined. (refers to Banks, J. Houston, Algebra: Its Elements and Structure, McGraw-Hill, St. Louis, 1965, page 116)

 
It should be noted that the truth of a mathematical statement has nothing to do with how we "normally check" our work.

Don K. Enns tries to explain why division by zero is a "no-no."

Suppose 0/0 = k for a specific real number. Then k =  0   0    =  0  a      0 =   (0    1  0 )  a =    k  a         where a is any real number But if k = ka for all real numbers a, the only finite value for k is 0. Note however what happens when both sides of k = ka are divided by k for k = 0:   0/0 = any real number! Is 0/0 = 0 even a possibility? Indeed, division by 0 is a "no-no." (refers to Enns, Don K., "0/0 is a 'no-no'", Journal of the Kansas Association of Teachers of Mathematics, page 21)

 
One need only look carefully at his equation k = ka to discover the fallacy. Dividing k = ka by k produces:

	k  k	=	ka  k
Let k=0:	0  0	=	0a  0

Then, he incorrectly "cancels" the 0's on the right side of the equation, resulting in:

Let k=0:

0  0

=

a, any real number.

Since he let k=0, 0/0=0, not 1 as he assumed when he replaced (0a/0) with a.

One energetic calculus student tried to show the impossibility of division by zero in the following way:

f'(x) =   lim   h  0 f(x+h) - f(x) h If h=0, we have:   f'(x) =   lim   h  0 f(x+0) - f(x) 0   =   lim   h  0 f(x) - f(x) 0   =   lim   h  0  0  0   =  0  0   = c Therefore, all derivatives would have the same constant value, which, of course, is not true.

 
We need to remind this student that we are interested in the limit of a function of h as h approaches zero, not the value when h=0.

Other people have approached the problem of division by zero, among them, Patrick Suppes, in his book entitled Introduction to Logic. (refers to Suppes, Patrick, Introduction to Logic, D. Van Nostrand Co., Princeton, NJ, 1957, pages 163-9)

He examines five approaches to division by zero:

The first approach differs from the others in that it recommends a change in the basic logic to deny meaning to expressions like: 1/0=1/0 . . . The first objection is that it is undesirable to complicate the basic rules of logic unless it is absolutely necessary. Second, if such a change were adopted, the very meaningfulness of expressions would sometimes be difficult if not impossible to decide. For example, . . . consider now the expression:   For every natural number n, 1/n* = 1/n*, where n* is the unary operation defined as follows: n* = 1 if n is an odd integer or n is an even integer which is the sum of two prime numbers. n* = 0 if n is an even integer which is not the sum of two primes. The problem of the existence of even integers which are not the sum of two primes is a famous problem of mathematics which is still unsolved (Goldbach's hypothesis.) Thus on the basis of the first approach the meaningfulness (not the truth or falsity) of [the statement] is an open question. The second approach is to let x/0 be a real number, but to define division by the conditional definition:   if y 0, then x/y=z if and only if x = y  z. In this case, for every number x, x/0 is a real number, but we are not able to prove what number it is. In fact, we cannot even decide on the truth or falsity of the simple assertion: 1/0 = 2/0. The third approach makes x/0=0 for all x . . . It eliminates the undecidability of statements like: 1/0=2/0. An advantage of this approach is that it permits the definition of division by a straightforward proper definition fully satisfying the criteria of eliminability and non-creativity. The main disadvantage of this approach is the one mentioned in 8.5: many mathematicians feel uneasy with the identity: x/0=0. The fourth approach requires no basic change in logic; it differs in placing the object x/0 outside the domain of real numbers; more precisely, it differs in not making it possible to prove that x/0 is a realy number . . . The situation can be improved by introducing into our system a primitive symbol for some object which is not a real number. Without saying what the object is, let us designate it by . This has the virtue of making x/0=y/0 where x and y are real numbers, thus eliminating a vast proliferation of odd mathematical entities. On the other hand, we cannot decide whether or not  /  is a real number. A fifth approach should be mentioned which is of considerable theoretical importance but does not correspond at all to ordinary mathematical practice. The idea is simple:   banish operations symbols and individual constants, and use only relation symbols. Thus "0" is replaced by the primitive one-place predicate "Z", where it is intended that Z(x) means that x is an identity element with respect to addition. The ternary relation symbol "A" is used to replace the addition operation symbol: A(x,y,z)  x + y = z Similarly, the ternary relation symbol "M" is used for multiplication: M(x,y,z)  x  y = z With this apparatus we may easily give a proper definition of the division relation symbol: D(x,y,z)  -Z(y) & M(y,z,x) In this approach, there is no need for unusual mathematical entities, but it is extraordinarily awkward to work continually with relation symbols rather than operation symbols. For example, the associativity of addition has to be expressed in some manner like the following: A(x,y,w) & A(w,z,s1) & A(y,z,v) & A(x,v,s2) s1 = s2 I have considered these five approaches to division by zero along with their advantages and disadvantages. In most of these approaches, the disadvantages outnumber the advantages and we should no longer consider those approaches as practical. However, the third approach is very fascinating. Advantages were mentioned by Suppes. I found other advantages as I worked with this approach. The only disadvantage mentioned by Suppes is the statement, "many mathematicians feel uneasy with the identity: x/0=0." First, it should be obvious that defining x/0=0 does not violate any of the field postulates. Since the field postulates hold for the real number 0, they should hold for x/0 by substitution. The long and tedious verification can be made in the following manner: a/0 + b = 0 + b = b + 0 = b + a/0 a/0 + b/0 = 0 + 0 = b/0 + a/0 This verifies the commutative property for addition for terms such as a/0. The remaining properties can be verified in a similar manner. Since no contradiction can be found in the field postulates, where might we derive a contradiction to the definition x/0=0 in the system of rational or real numbers? One might say that using the definition of adding rational numbers leads to a contradiction:  a  b +  c  d =  ad + bc  bd   Let a=1, b=0, c=1, d=1      1  0 +  1  1 =  1  1 + 0  1  0  1 =  1  0   This may seem to be the contradiction we are looking for, but we incorrectly used the definition for addition. It does not apply to numbers of the form a/0. What definitions and theorems do apply to a/0?   Actually, there are none, since all statements which may involve division by zero have a build-in exclusion such as "if x  0, then . . . "   Therefore we are required to prove those theorems which may be stated without exclusion.