Mathematical paradox? help!?
Ok so numbers are infinite
Positive numbers will go infinite
Negative numbers will go infinite
But if you think about it fractions and decimals are also infinite
1.01, 1.02, 1.03,....
So if in between 1 and 2 we have an infinite amount of numbers. How the heII do we come to 2 from 1?
Comments
the paradox is resolved whenever your step size is always finite.
then you can think of traversing the distance between values (like 1 to 2, or 1.01 to 1.02) as hopping from lily-pad to lily-pad in a pond -- your finite hop distance defining which lily pads you can hit.
once you let that step size become a variable function of step, then you can end up with Zeno's bridge (classic math paradox -- c.f. wikipedia if not familiar with it.)
This is a problem concerning mathematical analysis and the application of limits, and upper and lower bounds
An example of this is summing a geometric series such as
S_n = 1 + r + r^2 + r^3 + ...... + r^(n-1)
where r is a common multiple with 0 < r < 1 and where there are n terms
I don't know if you are familiar with this, but the S_n can be written
S_n = (1 - r^(n+1)) / (1 - r)
Now because 0 < r < 1, as n tends to infinity,the SEQUENCE {r^n} tends to zero. Thus, the infinite SERIES
lim (n -> ∞) S _n = 1/(1 - r)
Put r = 1/2, then
lim (n -> ∞) S _n = 1 / (1 - 1/2)
= 1 / (1/2)
= 2
Even though we have an infinite number of terms, the final result is finite. This above is an example of a convergent series. If r > 1 then the series would diverge to infinity.
The topic of infinity is also a concern of Set Theory and involves the cardinality of sets and 1-to-1
correspondence between sets. I could explain further but it will be quite long-winded and possibly not very illuminating.
(Wriggled out of THAT one!)
What I suggest is that you look up Georg Cantor
Also try looking at Hilbert's Hotel.
The Hilbert Hotel analogy to infinity is quite fascinating
I've pasted a couple o websites which you might find interesting.
The question has no meaning as the numbers you refer to are simply a notation - they have no particular properties but can be manipulated by the normal mathematical rules such as 2+2=4 in a decimal system. If you want to think of them as physical entities consider what is zero or a negative number ...let alone an imaginary one.