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I have heard of a statement like this:

A car can technically never run out of gas (when still moving) if the driver uses half of the gas left each time.

Is this possible (mathematics wise)?

Kamil Jarosz
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SaltMeter
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    Look up Zeno's paradoxes. https://en.wikipedia.org/wiki/Zeno%27s_paradoxes – Deepak Mar 11 '16 at 01:09
  • This reminds me of Keats' Paradox (that's what I call it). He talks about pictures of lovers painted on a pot, and says that for eternity they will be close to each other, yet they will never be able to approach closer. The moral of the story is, you can have an infinity of something, yet end up having only a finite amount of content in it. Keats' paradox refers to an infinite sum of zeros, and says it's zero, which is a significant observation since in extended real number systems, the equality $0.\infty=0$ is often used compared to $0.\infty=1$. – Sarvesh Ravichandran Iyer Mar 11 '16 at 01:21
  • Zeno has nothing whatever to do with this. At some point you have one molecule of gasoline. When you divide it in half, you no longer have gasoline. Why on earth are people upvoting the comment of @Deepak? Surely even mathematicians understand that gasoline is a complex molecule and not an infinitely divisible substance. – user4894 Mar 11 '16 at 17:54
  • @user4894 I'm afraid it is you that are missing the point here. This has a lot to do with Zeno's paradoxes (specifically the Dichotomy paradox). Even if gasoline were infinitely subdivisible, that paradox (of halves) would apply. And on the macroscopic scale, gasoline is, to all intents and purposes, infinitely subdivisible. It is only at the molecular scale that the discrete nature of matter begins to, well, matter. And even then, it is not that dissimilar from the paradox. The question of space quantisation is a tricky one, and the Planck length may be a natural lower limit on distance. – Deepak Mar 12 '16 at 03:07

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You are overthinking this. Yes, in draining a 100 litre tank full of gasoline, you can imagine that infinitely many events occur: At some point in time for example, the tank will be (1) 1/2 full, and (2) 1/4 full, and (3) 1/8 full, and so on. But we can measure the rate at which the tank is being emptied, in units of say litres per hour, and calculate precisely when the tank will be emptied even if infinitely many of the above "events" had occurred in the interval. Nothing "paradoxical" here.

Dan Christensen
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Assume that the car engine is perfect and proportionally converts gas into distance. Then the answer is yes and no. Using the series $$ \sum_{j=1}^\infty \frac{1}{2^j} = 1 $$ The car never runs out of gas since infinitely many terms are non-zero, but only travels a finite distance since the sum is finite.

This only works if you view gasoline as continuous matter instead of discrete particles.

Henricus V.
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"A car can technically never run out of gas (when still moving) if the driver uses half of the gas left each time."

A more practical restatement: If you reduce the speed of the car as a function of time like $v(t) = v(0)\exp(-a t)$, then the car will always keep on moving, yet the fuel consumption will be bounded.

Count Iblis
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