Background: I am in 8th grade and I like to study around advanced mathematical subjects. However, I do not know enough to be sure in my conjectures. Therefore, I would like your help.
I have a question regarding 0. Namely, 0's reciprocal. Of course, it sounds ridiculous- For 0 to have a reciprocal, another number must be multiplied by 0 to equal 1. However, here is how a reciprocal looks: X/1 *1/x = x/x = 1
Let's say that x = 0
0/1 * 1/0 = 0/0 = 1?
Now here are a few problems. 1/0, or, any number divided by 0, is undefined. However, through previous research and conjectures, I have hypothesized that x/0 is infinity if x is not 0. Therefore, assuming I'm correct, 1/0 is infinity. However, let us continue. 0/1 = 0, and 0 * infinity = 0/0 which I said equals to 1. What I am suggesting is that 0 to the 0 power is 1, since to get to 0/0, you need 0^x * 0^-x, which is 0^x-x, or 0^x/0^x. Now, since 0 to any number which is not 0 is simply 0, we have 0/0. Now, if I would follow the limit concept, let me make this equation:
f(x) = 0/x
As you may guess, this is simply a horizontal line, the x axis. Therefore, if we use the limit concept, 0/0 is simply 0. But, if we say that 0/0 is 1, then 0 would have a reciprocal -> infinity.
0/1 * 1/0 = 0/0 = 1 or
0 * infinity = 0/0 = 1
Please review my theory and tell me if I am, at the very least, somewhat correct.
I know very little of Calculus, so please do not expect any grand mathematical knowledge from me.
EDIT : I suppose that I don't want an actual answer as to my original assumption that 0's reciprocal is infinity, but rather an answer as to whether my basis for this assumption is correct: Is 0/0 = 1 and is 1/0 = infinity?
I would like a review of my theory/process in the comments if you can provide it.
Thank you!
Gil Keidar