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
Because each step is smaller. your first step is 0.1, then 0.01, then 0.001 and so on, eventually the step is infinitely small. This is a good example of the key difference between pure math and something like engineering or physics. Theoretically, you never approach 2. In reality, the steps will get so small that you can’t actually measure them. You won’t be able to accurately take the step, or determine if you’ve actually stepped. You’ll get to the point where you don’t know if you’re at 1.111, 1.113, or 1.105.
Just like there are infinite numbers there are also infinite places after the decimal. Tens, hundreds, thousands,…., Trillions, etc. for ever.
If you reframe your question to begin: “If I start at 1, and my goal is to get to 10/9”, then you’ll perfectly describe [Zeno’s paradox](https://en.wikipedia.org/wiki/Zeno%27s_paradoxes). (10/9 is 1.11111111…)
The more usual form of that paradox discusses starting at 0, goal is 1, and moving forward by 50% of the remaining distance every time, eg terms ending on 1/2, 3/4, 7/8, 15/16, 31/32, and so on. This describes an infinite series that is said to “converge to 1”, and yours converges instead to 10/9.
There are a number of infinite series that converge to a number: despite always increasing, they’ll never increase enough. There’s also series that converge to a number despite alternating increasing and decreasing, such as the [alternating harmonic series](https://en.wikipedia.org/wiki/Harmonic_sum#Alternating_harmonic_series).
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.
Another way to think about it is, in order to reach 2 from some lower number x, you eventually have to take a step that’s at least as large as the distance from x to 2, which is 2-x. With the procedure you described it’s like you’re actively avoiding taking a large enough step: if you’re at x = 1.1, you’re 0.9 away, but taking a step that’s too small to even get to 1.2, which is only 0.1 away. And so on; the next step is lower than 1.12, then the next one is like you’re trying to stay lower than 1.112, etc.
You’ll never reach a destination if you’re always taking steps that are chosen specifically to be smaller than the remaining distance.
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