How do you define an infinitely small value that is greater than zero?
$1/\infty$ is apparently invalid because you can't divide by infinity. And $0.\bar{0}1$ is invalid because you can't put a 1 after an infinite number of zeros.
So how do use infinitely small values in equations? If I say $x$ tends to $0$ and $ x > 0$ then is x an infinitely small positive value?
If $x \rightarrow 0$; $ x > 0$ is invalid, how else could I define an infinitely small value? And if I could define $x$ as being in infinitely small value, could I then say that $ x * \infty = 1 $?
And, is there a symbol for an infinitesimal (infinitely small) positive value?
Thank you.
EDIT:
I realize that for every number there is a smaller one (e.g. $x/2$) so therefore there is no "smallest" number – I'm looking for a symbol (or equation/name/concept) that stands for any (for lack of a better word) "number" infinitely close to zero. Similar to the way in which infinity isn't really a number, but stands for any "number" infinitely large. I know $\infty$ isn't a number!
Best Answer
The short answer is that you can't. But you can define any positive value which is arbitrarily small, in other words as small as you want it to be.
This means that just like how there is no "largest number" (since for any number $N$, we know that $N + 1$ is larger), there is no smallest positive number, since for any real number $N \in \mathbb{R}$,
$$\frac{1}{N} > \frac{1}{N+1} $$