What Does 1/0 Mean and Why is It Undefined?

[deda]

Originally posted by suyver
Where did you learn this? I certainly never saw it in a textbook on conventional math.

it's simple logic:
$$0\infty=n<=>\frac{n}{0}=\infty$$
if n<>0 and n<>infinity.

 

[HallsofIvy]

Another result of simple logic:

"Toledo is a nation in South America" <=> "The sun will rise in the west tomorrow".

$$0\infty$$ is not equal to n and
$$\frac{n}{0}$$ is not equal to $$\infty$$

$$0\infty$$ and $$\frac{n}{0}$$ are not defined.

 

[suyver]

And what about $0/\infty$ ?

Would you agree that $0/\infty=0$ ?

 

[HallsofIvy]

The point that has been made repeatedly is that $\infty$
is not a standard real number. Before it is possible to answer that question, you have to specify which of the several extensions to the real number system you are working in.

 

[suyver]

(Sorry to keep bugging you, but I just want to understand this.)

So, you are saying that, depending on the extension to the real number system I am working in, $0/\infty$ could mean different things?

Just out of curiousity, can you show a kind of extension that would give $0/\infty\neq 0$ ?

 

[oen_maclaude]

as of the moment, the debate is on how to define infinity.

as far as I'm concerned, infinity is just a simple description on the behavior of the function as the value of the variable goes too big (going to the largest value on the number line) or too small ( in this case the smallest value which can be seen on the left part of the number line).

if infinity were to be defined as a variable, then 0 times infinity is defined and it is equal to 0. however, if otherwise defined as a behavior, then we cannot give an exact value for the problem 0 times infinity ( ie it is not defined).

 

[deda]

Originally posted by oen_maclaude
as of the moment, the debate is on how to define infinity.

as far as I'm concerned, infinity is just a simple description on the behavior of the function as the value of the variable goes too big (going to the largest value on the number line) or too small ( in this case the smallest value which can be seen on the left part of the number line).

if infinity were to be defined as a variable, then 0 times infinity is defined and it is equal to 0. however, if otherwise defined as a behavior, then we cannot give an exact value for the problem 0 times infinity ( ie it is not defined).

zero times infinity is not undefined nor undecided.
it's a whole set of values, almost the entire set of real numbers (excluding zero and infinity). any number of the solition set is solution.

as for zero times infinity being zero it doesn't hold because it's zero times any thing but infinity is zero.

try it on concrete cases.
0*1=0 1 is solution to 0n=0
0*10=0 10 is solution to 0n=0
0*100=0 100 is solution to 0n=0
...
0*infinity=real

nevermind

 

[suyver]

Originally posted by deda
nevermind

yep...