Proving 0/0 ≠ 1: Does Math Really Work This Way?

[AndersHermansson]

It's worse than indeterminate -- it's undefined.

It boggles me why people insist on using x/x = 1 to "prove" 0/0 = 1, but they never accept 0/x = 0 to "prove" 0/0 = 0.

Try asking her to actually mathematically prove it. I imagine she won't even know where to begin.

It is a common misperception that 0/0 is undefined. It is merely indeterminate.

Consider that expression:
\frac{0}{0}=a

is equivalent to:
0=a\cdot 0

which is true for any number a (it is not undefined). Hence 0/0 is indeterminate.

 

[Hurkyl]

It's not a misperception if it's right.

0/0 is infix notation for the application of the binary function '/' to the argument (0, 0) -- a pair that is outside of the domain of '/'. When one writes a string of symbols that denotes the evaluation of a function someplace outside of its domain, one says that string of symbols is undefined.

 

[AndersHermansson]

It's not a misperception if it's right.

0/0 is infix notation for the application of the binary function '/' to the argument (0, 0) -- a pair that is outside of the domain of '/'. When one writes a string of symbols that denotes the evaluation of a function someplace outside of its domain, one says that string of symbols is undefined.

But the statement is clearly true for all a. How can an undefined statement be true? No that doesn't add up my friend!

 

[Hurkyl]

0 = a * 0 is true for all a. 0 / 0 = a is not, because it's undefined.


Leaving formalism behind and just speaking heuristically for a moment, a/b is supposed to be the unique number such that b * (a/b) = a. There is no x that is the unique solution to 0 * x = 0. Thus, there is no x that could satisfy 0/0 = x.

 

[AndersHermansson]

Leaving formalism behind and just speaking heuristically for a moment, a/b is supposed to be the unique number such that b * (a/b) = a. There is no x that is the unique solution to 0 * x = 0. Thus, there is no x that could satisfy 0/0 = x.

Which is why i would call it indeterminate.

An expression in mathematics which does not have meaning and so which is not assigned an interpretation. For example, division by zero is undefined in the field of real numbers.

Since a/b=c/d iff a*d=c*b, the expression 0/0=a isn't meaningless, hence I wouldn't call it undefined.

I guess it blows down to what you want undefined to mean, exactly. I get your point about uniqueness. Only I wouldn't call it undefined, since something undefined is just something that lacks definition. And this particular expression can be defined this way and not cause any trouble ...

 

[Zurtex]

If it causes no particular trouble and 0/0 = a, for some a we don't know (one might some undefined a but nevermind). Then how can we be sure 0/0 = 0/0? And if equality isn't so simple then surely it does cause trouble!

 

[shmoe]

Since a/b=c/d iff a*d=c*b,

Have you ever looked into a rigorous construction of the rationals from the integers? You take all pairs of integers (a,b) where b is non-zero, (a,b) intended to represent a/b, and define the equivalence relation you have above, (a,b)~(c,d) iff ad=bc. Key here is b is by definition non-zero. If you allowed b to be zero you'll run into many problems, so (a,0) is not allowed to represent a fraction for any a and we leave it undefined.

If you want to allow 0/0=a for all a (which would happen if we allowed (0,0) above) then -1=0/0=1, in fact everything ends up equal. Or you would have to admit that your = relation is no longer transitive, which will put you in a world of hurt.

 

[CTS]

http://mathworld.wolfram.com/Indeterminate.html

 

[Hurkyl]

For example, a limit of the form 0/0 ...

I think this is the biggest reason for confusion -- people hear "indeterminate form" for this kind of limit form, but make the mistake of thinking that the label is referring to this arithmetic expression, rather than to the form of a limit.

 

[JonF]

0/0 turns up pretty regularly in almost all levels of math. I was helping a trig student verify some trig identities and they were asked to show that: sin(x)/tan(x) = cos(x). But that isn’t true because sin(x)/tan(x) isn’t defined at 0 and cos(x) is.

 

[master_coda]

I guess it blows down to what you want undefined to mean, exactly. I get your point about uniqueness. Only I wouldn't call it undefined, since something undefined is just something that lacks definition. And this particular expression can be defined this way and not cause any trouble ...

Something isn't undefined because it can't be defined; it's undefined because it's not defined. Since 0/0 has no defined value, it is therefore undefined. The fact that defining it doesn't break as many rules as defining 1/0 would doesn't really change that.

 

[AndersHermansson]

Ok, thanks for all the interresting responses. I will delve deeper into this!

 

[AndersHermansson]

I think this is the biggest reason for confusion -- people hear "indeterminate form" for this kind of limit form, but make the mistake of thinking that the label is referring to this arithmetic expression, rather than to the form of a limit.

Yes! It seems I have to think about this some more. Thank you =)