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Let us assume .5 second has passed. Now similarly let .9 seconds also passed.. Then let's say that .99 second has passed...we're still not done because 1 second hasn't passed. Then follows .999 seconds, after that .9999 seconds.....goes on and on to .9999999999999999999999999999999999.....the numbers aren't gonna give up, so 1 second is never reached because no matter how many 9s you add, it's never gonna be equal to 1. How can we solve this problem?

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closed as off topic by John Rennie, Waffle's Crazy Peanut, dmckee May 24 '13 at 17:49

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This is an example of Zeno's paradox, and it's more philosophy than physics. –  John Rennie May 24 '13 at 17:13
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thank you Brother John.. I have checked wiki articles of Zeno paradox. But was dissatisfied. Could you please explain in short how to solve this paradox. –  newera May 24 '13 at 17:15
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@newera: Are you aware of that infinite series $0.999\ldots=\sum_{n=1}^{\infty}9\cdot 10^{-n}=\frac{9}{10}\sum_{n=0}^{\infty}\left(\frac{1}{10}\right)^n=\frac{9}{10} \cdot \frac{1}{1-\frac{1}{10}}=1$ represents the same real number, namely one? –  Qmechanic May 24 '13 at 17:17
    
1/3=0.333.... so 1= 3(1/3)=3(0.333....)=0.99999..... Alternatively, you could say that 1 -0.9999...=0.000...01=limit n approaches infinity 10^-n=0 –  Dimensio1n0 Jun 6 '13 at 13:40

3 Answers 3

You reach one second by adding 9's infinitely fast. This isn't a paradox because the 9's are also getting infinitely small.

Alternatively, when you reach .9 seconds, just add another .1 seconds.

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the paradox is still not solved , brother?? The question is how can you add another .1 seconds so easily? –  newera May 24 '13 at 17:12
    
The same way that you initially added the .5 seconds so easily. If that still doesn't satisfy you, see the first part of the answer. –  Izzhov May 24 '13 at 17:13
    
Umm.. But still the number doesn't gets zero, does it? We can still go into infinite.. since infinite is infinite. –  newera May 24 '13 at 17:16
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@newera: What Izzhov suggested is right. As he mentions first, infinitely adding 9 gives 1. See QMechanic's comment on your question ;-) –  Waffle's Crazy Peanut May 24 '13 at 17:18
    
Ohh ! yeah.. true –  newera May 24 '13 at 17:20

You have just stumbled upon Zeno's Paradox.

First of all, doesn't time seem to pass second by second to you? I mean, intuitively this "paradox" is pretty thin, right? Human intuition might be wrong on some accounts, but surely not something as simple and (evolutionarily useful) as time measurement.

Look at a clock. Surely, the second hand ticks along merrily.

Physical solution:

Modern physics, especially the study of fundamental constants, yields some interesting results. For instance there is in our universe an absolute highest speed, the speed of light. There also happens to be a number of other boundraries on possible measurements in our universe.

These are called the Planck Units and a few of them are absolute smallest quantities, among those are time. A Planck Interval is about $ 10^{-43} $ seconds long. Smaller time intervals simply do not yield meaningful results when plotted into the standard model of particle physics and quantum mechanics.

So time passes in discrete 'ticks' of the universe's second hand, in tiny, finitely small, increments.

Mathy solution:

  • You start with .9 which means .1 up to 1 second. This step takes .9 seconds.
  • Then by the first iteration you get .99 with .01 up to 1. This step takes .09 seconds.
  • Then by the third iteration you get .999 with .001 up to 1. This step takes .009 seconds.
  • By the N'th iteration you have: $$ t_N = \sum_{n=0}^{N} 0.9 \times 10^{-n} $$ But on the other hand, the N'th step takes: $$ \Delta t_N = 0.9 \times 10^{-N} $$
  • Notice that the step time tends to zero as the time tends towards 1.
  • Taking the limit of both: $$ \lim_{N \to \infty} \sum_{n=0}^{N} 0.9 \times 10^{-n} = 1 $$ and similarly: $$ \lim_{N \to \infty} 0.9 \times 10^{-N} = 0 $$ meaning that you reach 1 after an infinite number of steps, and the last step, paradoxically, takes zero time to complete.
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To solve this thought problem ask yourself this:

"How on Earth did I reach 0.5 seconds to begin with?"
After all, you needed to go through 0.25 seconds and then 0.4 seconds and then 0.49 and so on. Similarly, you can apply the same thinking to all time spans. The real problem here is being unable to visualize infinity.

You're thinking, "To get to any time I have to move through 0.0(infinite 0's)1 then 0.0(infinite 0's)2 and so on. But that's an infinite number of steps to get to 1 second. How am I moving that fast?"

But you are moving that fast; looking at your watch can show that one second can exist and pass. The problem is more with numbers than with time passing. Time has no concept of our arbitrary numbering system, it just progresses.

A simplistic yet conceptually challenging point of view could be that time is not an absolute measured in numbers, it is merely the recognition of change in the universe. If the universe did not change from one moment to the next, it would be considered the same moment and no time has passed. The amount of time passing is directly related to the amount the universe has changed from your perspective. Thus we reach 1 second not by running through 0.0000000000001-0.99999999999999 seconds, but by experiencing 1 second's worth of change in our location.

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