Undetermined vs. Undefined

Question

This often comes up in precalculus and calculus, that is sometimes an expression will be said to undefined while at other times undetermined. What is the fundamental difference between the two?

For example, division by zero $x/0$ is said to be undefined, while $0/0$ is said to be undetermined.

I had a thought: Is it that you cannot extract more information from an expression that is in indeterminate form but can with an expression in undetermined form? — Bunny, Aug 02 '15 at 21:34
I feel like I wrote an answer for this question already today... — Asaf Karagila, Aug 02 '15 at 21:53
I think there will come a time when we must decide to create a tag of $0/0$ questions or create a bot to delete them, because it is getting quite repetitive. — Aloizio Macedo, Aug 03 '15 at 00:08

score 4 · Accepted Answer · answered Aug 02 '15 at 21:36

4

Division by zero is undefined in every case.

In calculus, the phrase “$0/0$ is an indeterminate form” means that you have a limit of the form $$ \lim_{x\to a}\frac{f(x)}{g(x)} $$ where $$ \lim_{x\to a}f(x)=0 \qquad\text{and}\qquad \lim_{x\to a}g(x)=0 $$ but $f(x)/g(x)$ is defined in a set having $a$ as a limit point (usually, but not necessarily, a punctured neighborhood of $a$) and nowhere you do $0/0$, which makes no sense. In this case you can apply no standard theorem on limits and the limit, if existing, must be computed with some different technique than simply substituting the value $a$.

Some say that the value of the fraction $0/0$ (no reference to limits) is undetermined, but this has no real usefulness.

answered Aug 02 '15 at 21:36

egreg

238,574

$0^n$ for $n<0$ is undefined and $0^0$ is undetermined, right? – Display Name Jul 29 '19 at 16:25
@GoldDiggingProgrammer $0^0=1$, by definition of power with exponent $0$. In the context of limits, one has to remember that "a limit of the form $f(x)^{g(x)}$ with both function having limit $0$ has no general theorem available”, so it's common to say in the context of limits that $0^0$ is an indeterminate form. – egreg Jul 29 '19 at 17:35
Thank you very much! – Display Name Jul 30 '19 at 05:50

score 3 · Answer 2 · answered Aug 02 '15 at 21:35

3

$x/y=z$ means that $zy=x$, i.e. $x/y$ is the number you multiply by $y$ to get $x$. For $1/0$, there is NO number you can multiply by zero to get 1. For $0/0$, there is no UNIQUE number you can multiply by zero to get 0.

answered Aug 02 '15 at 21:35

vadim123

82,796

score 1 · Answer 3 · answered Aug 02 '15 at 22:03

Without looking too deeply into it, I would first propose that undefined is more along the lines of not solvable. For ex: x/0 is not solvable until we know the definition of x. Undetermined is not yet solved. 0/0 is not yet solved, but we can use L'Hopital's rule to (often) eventually solve it.

(I'd like to caveat that this is an initial impression, so make of the answer what you will)

Undetermined vs. Undefined

3 Answers3