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Zero Divided by Zero INTRODUCTION Current implementations of APL give one as the result of zero divided by zero. This paper gives the reasons for proposing that this quotient be changed to zero. Two conventions introduced by K.E. Iverson [1] for the exposition of mathematical topics are used here. The first convention asserts the identity of two or more expressions by listing them one above the other. Thus:
a+b
b+a
asserts the commutativity of addition. The second convention allows the definition of a function to be given by listing the function name, a colon, and the expression defining the function, all on one line. The function argument for a monadic function is given by either ⍺ or ⍵ . The left argument for a dyadic function is given by ⍺ , and the right argument by ⍵ . Thus:
hyp:((⍺*2)+⍵*2)*.5
defines a function which gives the
hypotenuse of a right triangle having
legs ⍺ and ⍵ .
Divisors and Quotients The number d
is a divisor of a number m
if there is an integer k
for which m=k×d . k
is called the quotient
of m divided by d .
Since 0=k×0
for all integers k ,
zero is a divisor of zero.
Indeed, any number may be substituted
for k ,
so that the choice of quotient
for zero divided by zero is arbitrary.
The quotient is said to be indeterminate.
Choosing a Quotient for Zero Divided by Zero Although zero divided by zero is indeterminate, it is desirable to provide a fixed quotient for it in a programming environment, in order to reduce the number of circumstances when it is necessary to interrupt the execution of a problem. The current choice in APL for this quotient is one, based on the preservation of the identity:
z÷z
1
The behavior of z÷z as z
approaches zero from any direction suggests
that 1 is indeed a suitable quotient.
This argument has the defect
that an equally valid argument suggests that
the limit of (a×z)÷b×z as z
approaches zero is a÷b .
In this sense, one has no better standing
than any other number as the quotient.
In fact, it is a poorer choice than zero,
as the rest of this paper demonstrates.
THE CHOICE OF ZERO AS QUOTIENT OF ZERO DIVIDED BY ZERO There is a conflict between the two following:
z÷z 0÷z
1 0
They cannot both be identities
for all z .
If the first is preserved, the second is
lost, and vice-versa. Behind the first is
the tautology z=z
(since 1 is absorbed in multiplication).
Behind the second is the more interesting
equation (z×0)=0 ,
which expresses the fundamental multiplicative
property of zero.
The Direction and Magnitude Functions Consider the definition of a number as a directed magnitude. This implies the definition z=(×z)×|z , and gives the definitions for the direction (that is, signum) and magnitude functions as follows:
direction:⍵÷|⍵
magnitude:⍵÷×⍵
Observe that both of these definitions
fail if 1 is the quotient
of zero divided by zero,
but both are true in the entire
complex plane if zero is the quotient.
The Least Common Multiple Function The definition of the least common multiple function is:
lcm:(⍺×⍵)÷⍺∨⍵
where ⍺∨⍵ is the greatest common divisor
of ⍺ and ⍵
[2].
Since 0×0 is 0 ,
and 0∨0 is 0 ,
it is clear that the least common multiple of zero
and zero is zero (in fact,
zero is the only multiple of zero).
If zero is the quotient of zero divided by
zero, the definition for lcm holds universally.
The current quotient causes the definition to fail.
Distributivity of Division over Addition Consider the identity
(a÷c)+b÷c
(a+b)÷c
which expresses the distributivity of division over addition. The identity can be generalized to vectors:
+/v÷c
(+/v)÷c
Both identities fail using the present quotient, and succeed using the proposed quotient, when a , b , c , and the elements of v are all zero. Similarly, the function for the computation of the average of a vector v yields the identity:
+/v÷⍴v
(+/v)÷⍴v
which fails for v an empty vector, using
the present quotient, but is preserved for
the proposed quotient.
The Arc Function In the complex plane, the arc function (whose result is known variously as the phase, amplitude, argument, or angle) of a number is:
arc:¯3○(im⍺)÷re⍺
where im and re
are functions giving the
real and the imaginary parts of a complex number.
The arc function gives the
implausible answer 0.7854 when ⍺ is zero,
using the current quotient,
but gives the more satisfactory answer zero,
using the proposed quotient.
The Effect on the Dyadic Logarithm Function The proposed definition requires a correlative change in the dyadic logarithm function:
dlog:(⍟⍵)÷⍟⍺
when ⍺ and ⍵ are both 1. This points another conflict:
b⍟b b⍟1
1 0
These cannot both be identities.
The first is true universally under the
current definition,
even in the case 1⍟1 ,
but the second fails when b is 1.
With the proposed change in the quotient of
zero divided by zero,
this situation will be reversed.
The Argument from Symmetry With the real number line in view, the
choice of some number other than zero as
the quotient of zero divided by zero is
not too obviously eccentric.
With the complex plane in view, however,
zero is seen to be the only symmetrical choice.
CONCLUSION It is desirable to have a quotient for
zero divided by zero. There are many
persuasive arguments for wishing that
quotient to be changed from its current
definition of 1 to a new definition of zero.
ACKNOWLEDGMENTS Jim Brown and Alex Morrow of IBM called
my attention to the identity conflict
concerning the average of an empty vector,
and Hans Haegi of the Literargymnasium
Zürich and Al Rose of Scientific Time Sharing
both noted the identity conflict
regarding the distributivity
of division over addition.
REFERENCES
First appeared in the APL76 Conference Proceedings, 1976-09-22.
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