If I start at 1 and my goal is to get to 2, I could move to 1.1 and be a little bit closer to 2. Then move to 1.11, and be even closer. Then 1.111, so on and so forth. I understand that with each move, the step forward is smaller than the previous one. But it’s moving forward nonetheless. How is it that I can forever move forward, but never get there?
In: Mathematics
This is a common idea in mathematics, and the basis of quite a lot of mathematical paradoxes. The only answer I can think to give you’re question is that yes, you are right, there is an infinite amount of numbers between 1 and 2, or between 0.1 and 0.2, or between 0.000004 and 0.000005 etc etc.
There are a lot of mathematical concepts related to this idea such as the asymptotic curve or convergent series.
I’m sure I’m going to get many of the details about this story wrong, but I was once told about an ancient Greek philosopher who posed the following paradox:
An old man and a strong Spartan warrior have a race. The Spartan can run twice as fast as the old man so, being the honourable warrior he is, he gives the man a 100m head start. The race begins, and since the Spartan moves twice as fast as the man, when the man reaches the 150m mark, the Spartan is at the 100m mark. When the man reaches 175m, the Spartan is at 150m. Every time the man moves forward, the distance is halved, but logically, this means the Spartan will not ever overtake him despite the fact that he is running faster.
It took hundreds of years for someone to come up with a mathematical proof that explains the paradox.
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