You know the law of diminishing returns thingy? I want to know how it is possible to ever touch anything...?

Question

The idea being - say you move your fingers closer together. They start off a certain distance apart. They then get half the distance, then three quarters, then four fifths, etc. With the law of diminishing returns the fraction will get smaller and smaller and smaller - but even when they are 999999999/1000000000ths of the way there, they still wouldn't be touching. Surely the fraction just gets smaller, but the fingers never actually get there? How does anyone ever touch anything? Come to mention it, how does anything ever happen?

Answer 1

First, it is not really a "law" but rather a concept that applies to some situations, but not all.

An example is: you spread one pound of seed on your plot of land and harvest 100 pounds of grain. But if you spread two pounds of, you won't get 200 pounds of grain. Your "margin of return" is diminishing.

Second, I think you are refering not to "dininishing returns" but to Zeno's (5th century BC) Paradox.

His example was: Achilles and the tortoise had a race. Achilles could run ten times as fast as the tortoise, but the tortoise had a hundred yard start.

Achilles runs the hundred yards, but the tortoise is now 10 yards
ahead.

Achilles runs the 10 yards, but the tortoise is now 1 yard ahead.

Achilles runs the yard, but the tortoise is now 1/10 yard ahead.

Achilles runs the 1/10th yard, but the tortoise is now 1/100 yard
ahead, and so on.

Zeno's question to his colleagues (which they were unable to settle) was how can Achilles overtake the tortoise?

In our times we are able to resolve this problem because
we have the concept of the "limit of an infinite series."

It is easy to show that

100 + 10 + 1 + 1/10 + 1/100 + 1/1000 + .....
= 111.111111....
= 111 + 1/9 yards (exactly)

So however many fractions (or decimal places) you continue the
calculation, its value cannot exceed 111 and 1/9 yard.

This is where Achilles overtakes the tortoise, and there is no paradox or contradiction.

The ancient Greeks did not have our ideas about limits, so in their logic the problem could not be solved.

Answer 2

The law of diminishing returns doesn't apply to everything, and one of those is touching fingers. If you set out to cut the distance by a half every time you move, you can succeed in never touching the object you're moving toward, but even then you would run up against the electrical charges that keep molecules apart and would seem to be touching even if you weren't technically touching.

Answer 3

What jayleites wrote is accurate. To put it another way, the sum of an infinite number of infinitely small numbers is a finite number. If the sum is finite, you can reach the end. In other words, the fingers will touch, that last step will get you across the finish line, etc.

Answer 4

The law of diminishing returns is just a lame math game.
In reality, do you ever move your fingers that way? No. You either move them together or you don't.
Nature doesn't use math to do all the wonderful things it does. Math is how we analyze and predict things that are already happening fine without it.

Answer 5

*chuckles* Welcome to Zeno's Paradox- and I strongly encourage you to look up the fine details in Wikipedia because the math and explaination is far too long for Yahoo's Answers.

Bring your calcuator, paper and writing utinsels- you'll need to make notes- good luck