Mathematic principle...?

Question

Sorry to be vague but here's my question: What is the name of the principle illustrated in this paraphrased ancient Greek analogy? Two people are running one runs 2 times faster than the other. If the slower party gets a head start of 10 meters, the distance is cut in half when he reaches 15.

For fear of slaughtering it further, the jist is that the faster party can never catch up because the distance between them is always cut in half. What is the name of this principle and how it is overcome?

Answer 1

it's called Zeno's paradox.
i think the way to challenge it is to realize that when the distance gets smaller, the time needed to reach it also gets smaller.

Answer 2

Achilles and the tortoise

"You can never catch up."

"In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead." (Aristotle Physics VI:9, 239b15)
In the paradox of Achilles and the Tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox.

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The dichotomy paradox

"You cannot even start."

"That which is in locomotion must arrive at the half-way stage before it arrives at the goal." (Aristotle Physics VI:9, 239b10)
Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

The resulting sequence can be represented as:
This description requires one to complete an infinite number of steps, which for Zeno is an impossibility. This sequence also presents a second problem in that it contains no first distance to run, for any possible first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion.
This argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts. It contains some of the same elements as the Achilles and the Tortoise paradox, but with a more apparent conclusion of motionlessness. It is also known as the Race Course paradox.

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The arrow paradox

"You cannot even move."
"If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless." (Aristotle Physics VI:9, 239b5)
Finally, in the arrow paradox, we imagine an arrow in flight. At every moment in time, the arrow is located at a specific position. If the moment is just a single instant, then the arrow does not have time to move and is at rest during that instant. Now, during the following instants, it then must also be at rest for the same reason. The arrow is always at rest and cannot move: motion is impossible.
Whereas the first two paradoxes presented divide space, this paradox starts by dividing time — and not into segments, but into points. It is also known as the fletcher's paradox.

Answer 3

Zeno's paradox. And it's not a law. It's an analogy. It's an example of letting theoretical thinking overcome common sense.
It does, in a sort of weird way, apply to relativity.
The dichotomy paradox is the name of the case. Never reaching a goal due to an infinite number of halves.

The idea of halves is an excellent tool, and along with the arrow paradox, Newton used some elements to create the system of calculus.

Answer 4

I believe that you may be thinking of an asymptote. If that is the term you are looking for, you gave an incorrect example. When the faster party ran 20 meters, they would be even, and the faster party would be passing the slower party.

Answer 5

you have already have been given some solutions appropriate to the 1st one, this is basically an occasion of an arithmetic sequence. the 2d relation you chanced on became into that for any x: double x = 2x upload a million = 2x + a million upload to x^2 = x^2 + 2x + a million = (x+a million)^2 basically calculate (x+a million)^2 = (x+a million)(x+a million) = x^2 + x + x + a million) = x^2 + 2x + a million notice that this relation is genuine for any fee of x (no longer basically constructive integer values).

Answer 6

Its zenos paradox..