“I think perhaps, that the mathematical function of something infinitely approaching zero can be expressed in everyday terms as a question gradually approaching stupid. Then, again, I suck at math…”
Party Store, just want you to know, I now quote you in my signature at another discussion board I participate in. (I change it every few weeks.) If there were a prize for this, you would have won it.
-FrL-
This has been adumbrated in previous posts, but just to make it clear, the numerator and denominator don’t necessarily change and approach anything.
If one or both are functions, then you can look at what the value of the function is as it approaches some point at which it is undefined. And take the limit (which is most definitely not the “value of the function” at that point, since the function’s undefined). As has been made clear in examples above, simply knowing that the numerator or denominator function approaches zero is not enough to tell you what the limit is .
So “one divided by zero” is simply that, and doesn’t approach infinity on its own (and even as a function, it may go positive or negative).
I am not to sure about this. Every calculator and software that I have capable of returning infinity as a result will return undefined if you try to divide by zero.
The real projective line. Thank you, come again.
We say a temperature is “hot” or “cold” to the extent that it deviates from our comfortable temperature (approximately 25 Celsius or 77 Farenheit). So a temperature which is twice as cold is twice as far away from our comfortable temperature. Thus, -77 Farenheit is twice as cold as 0 Farenheit, and 103 Farenheit is twice as hot as 90 Farenheit.
Many years ago when I wondered about 0/0, I plotted y = 1/x and observed the same as you.
My question, why don’t we say 0/0 = “plus/minus infinity”
(similar to how we say if x = y^2, then y = plus/minus square-root of x) (hmm… i’m sure some doper is gonna tell me this isn’t strictly correct…)
OK, that’s a geometric object. How are you defining multiplication and division on it? Presumably for all points that map to real numbers, it’s the equivalent of normal multiplication and division on the reals. Do you define infinity times any non-zero number as infinity? (I can’t think of any other sensible definition). And what is infinity times zero?
For cardinal numbers, as I said earlier, it makes sense for infinity times zero to be zero, but I can’t think of a sensible definition here. If you make it anything other than zero or infinity, then I think multiplication can’t be associative:
If zero x infinity = n,
then:
2 x (zero x infinity) = 2n
but:
(2 x zero) x infinity = n
First of all, the two situations aren’t analogous. The first has limits and the second has two actual solutions.
On top of that, why would a graph of 1/x tell you anything about 0/0? It’ll tell you about 1/0 maybe, but not 0/0. Your suggestion could only remotely apply to a nonzero number over 0.
No, it’s an algebraic object, as in algebraic geometry. As for the multiplication, I didn’t say it was a ring. You asserted that mathematicians never use “x/0 = infinity”.
In this particular case, functions are defined on RP[sup]1[/sup] and people study the polology. How do you study the behavior of f at infinity on the projective line? Look at f(1/x) as x varies in a neighborhood of zero.
To Infinity… and Beyond!
(I have nothing of value to add, but I had to state the above for your amusement.)
I haven’t come across the term “polology” before, and I can find a definition in my usual mathematics reference works.
And I still don’t think that there is a sensible meaning for x/0 on the projective real line, because x/0 = infinity would mean that infinity times 0 = x. However, that would be true for all x except zero and infinity, so infinity times 0 would not be unique, and uniqueness is a uiseful property for algebraic operations.
It’s the study of the nature and location of the poles of functions.
Didn’t I just say that it wasn’t a ring?