What is the Difference Between Undefined and Indeterminate?

[Didd]

0^0,means multiplying zero it self zero times. It is undefined.

 

[chroot]

Didd,

No, it is not undefined. It is indeterminate. The two terms have quite different meanings in mathematics.

- Warren

 

[rayjohn01]

To Didd
I like your way of putting it -- and I think it is defined -- if you do nothing ( i.e. multiply zero times) the result is for sure -- no result.

 

[master_coda]

Didd,

No, it is not undefined. It is indeterminate. The two terms have quite different meanings in mathematics.

- Warren

Actually 0^0 is undefined, not indeterminate.

\displaystyle\lim_{f(x),g(x)\rightarrow0}f(x)^{g(x)} is indeterminate.


Forms like 0^0 or 0/0 are only indeterminate if you are talking about limits, but the actual number 0^0 is undefined - there is no such number.

 

[chroot]

master_coda,

I must repectfully disagree. 0^0 is not undefined; it is over-defined. It could have a number of different acceptable values, and thus is indeterminate.

Infinity - infinity is undefined, because we cannot assign even one acceptable value to it.

- Warren

 

[quantumdude]

Does it even make sense to speak of indeterminate forms such as 0⁰ apart from limits? I've never seen done.

Infinity - infinity is undefined, because we cannot assign even one acceptable value to it.

In the context of limits, infinity-infinity is indeterminate. It could represent any real number.

 

[master_coda]

master_coda,

I must repectfully disagree. 0^0 is not undefined; it is over-defined. It could have a number of different acceptable values, and thus is indeterminate.

Infinity - infinity is undefined, because we cannot assign even one acceptable value to it.

- Warren

Indeterminate does not mean "this can have multiple values". It means that we do not have enough information to determine the value.

Given a limit that produces something like 0^0 or 0/0 or infinity-infinity, we say that the limit is "indeterminate" because just knowing the limiting behavior of individual parts of the limit does not give us enough information to determine the actual value of the limit. That doesn't mean that the limit has multiple values.

If a function or symbol actually has multiple values, then you say that it has multiple values, not that it is indeterminate. If f(x)=x^2 you do not say that the inverse of f is indeterminate.

 

[chroot]

Hmm well, okay, I need you guys to help me make my definitions more precise...

At any rate, I've *always* heard 0^0 described as indeterminate, master_coda. You are the first to disagree.

- Warren

 

[quantumdude]

At any rate, I've *always* heard 0^0 described as indeterminate, master_coda.

Same here. Every calculus book I have ever seen refers to the following as indeterminate:

0/0
(+/-)(infinity/infinity)
0*(infinity)
0⁰
1^infinity
infinity-infinity

But as I said I have never seen any of those things discussed outside the context of limits. I wonder if it makes any sense to do that.

 

[master_coda]

Hmm well, okay, I need you guys to help me make my definitions more precise...

At any rate, I've *always* heard 0^0 described as indeterminate, master_coda. You are the first to disagree.

- Warren

Well, the first time people encounter 0/0 (or other "indeterminate" things) is in a calculus class, when learning about limits. So calling them indeterminate is usually appropriate. The fact that 0/0 itself (not a limit of the form 0/0) is undefined is usually never brought up, so a lot of people get this idea that things like 1/0 are undefined while things like 0/0 are indeterminate. But everybody knows better than to divide by zero anyway, so it's not a big issue.

Besides, I don't usually bring it up when people use the words "undefined" and "indeterminate" incorrectly; it's usually pretty clear what they mean, no matter what word they use. As long as nobody uses 0^0 as if its a real number, justifying it by saying that "it's not undefined, just indeterminate", then I don't really care which word they use. But you corrected someone else, so I felt compelled to mention it.