Multiplying by Zero

The explanation for why division by zero is undefined often goes like this; To say that 6/3=2 is to say that 3*2=6. Now take 6/0=x we would have to find some number that when multiplied by zero gave us 6 (x*0=6). This we can’t do. So, since division and multiplication are inter-defined in this way (generally a/b=c if and only if c*b=a) we can’t divide by 0.

But another way of talking about the interdefinition is to start from multiplication and work back to division; so we can say that a*b=c if and only if c/b=a and c/a=b (e.g. 3*2=6 if and only if 6/3=2 amd 6/2=3). But this will not work for 5*0=0. One way works fine since 0/5=0, but the other fails since it tells us that 0/0=5. But 0/0 s indeterminate and so cannot equal 5. Therefore 0*5 is indeterminate.

Now it is true in a sense that 0/0=5 since according to our interdefinition this just means that 0*5=0 which is of course true. But the problem is that this will be true for any answer. So, suppose that you thought that 0/0=120. This is a perfectly good answer since 0*120=0. But since any number will do as an answer for 0/0 this is definied as indeterminate and the above argument should go through.

What’s the right answer to this problem?

4 thoughts on “Multiplying by Zero”

1. In the field axioms, the multiplicative inverse is defined for all numbers except 0 (where 0 is the number such that a+0=a for all a in the field). Division is then defined as follows: x/y = x * 1/y, where 1/y is the multiplicative inverse of y. In other words, division is defined in terms of multiplication by a multiplicative inverse. if we start from the field axioms, then division by 0 is undefined because 0 just has no multiplicative inverse. For that reason 0/0 = 120 is not at all a good answer — the rule that x/y=z means x=y*z works only if y is not 0, since 0 simply has no multiplicative inverse and cannot be the divisor of anything.

2. Hi Ponder Stibbons, thanks for the comment.

Right, I get that the field axioms rule by fiat that there is no multiplicative inverse for zero, but other than teh fact that we want to rule thsi out there is no reason that there shouldn’t be. Since the multiplicative inverse is simply the number that when multiplied by the priginal number equals 1, the MI of 0 should be 1/0, which would give us 0/0=1…bthis is also a perfectly good answer (as per above). As far as I know this is why 0/0 is indeterminate NOT undefined…so the above argument should go through…

3. It is pretty much by fiat. As far as I see it, the field axioms are a result of mathematicians inventing an abstract structure that they hope captures the important aspects of some more… everyday concepts.

0/0 is an indeterminate form in the context of limits in calculus, because 3x/x and x/x converge to different numbers as x->0 (even though in both cases the numerator and denominator converge to 0). It is inaccurate, however, to say that 0/0 has an indeterminate value — it doesn’t have a value, and we call it an indeterminate form when we get it while taking limits in calculus, even if the limit itself has a definite value (as in 3 and 1 for 3x/x and x/x respectively). In the context of the field axioms alone (sans calculus) it is simply undefined.

Of course, as you point out, the field axiom which claims 0 has no multiplicative inverse is there just because we want it that way, but that goes for most mathematical structures, at least at the axiomatic level! You could start with the set of field axioms but without that axiom — you’d just get a different mathematical structure, then. Perhaps someone has done this, and perhaps it even turned out to be useful.

4. Hey Ponder,

I have to say I pretty much agree with everything you say here…