The fastest animal in the world, according to the Age-old Greek legend, was the charlatan Atalanta. Although she was a acclaimed huntress who alike abutting Jason and the Argonauts in the chase for the aureate fleece, she was acclaimed for her speed, as no one could defeat her in a fair footrace. But she was additionally the afflatus for the aboriginal of abounding agnate paradoxes put alternating by the age-old philosopher Zeno of Elea: about how motion, logically, should be impossible.

To go from her starting point to her destination, Atalanta charge aboriginal biking bisected of the absolute distance. To biking the actual distance, she charge aboriginal biking bisected of what’s larboard over. No bulk how baby a ambit is still left, she charge biking bisected of it, and again bisected of what’s still remaining, and so on, ad infinitum. With an absolute cardinal of accomplish appropriate to get there, acutely she can never complete the journey. And hence, Zeno states, motion is impossible: Zeno’s paradox. Here’s the unintuitive resolution.

The oldest “solution” to the absurdity was done from a absolutely algebraic perspective. The affirmation admits that, sure, there ability be an absolute cardinal of all-overs that you’d charge to take, but that anniversary new jump got abate and abate than the above-mentioned one. Therefore, as continued as you could authenticate that the absolute sum of every jump you charge to booty adds up to a bound value, it doesn’t bulk how abounding chunks you bisect it into.

For example, if the absolute adventure is authentic to be 1 assemblage (whatever that assemblage is), again you could get there by abacus bisected afterwards bisected afterwards half, etc. The series ½ ¼ ⅛ … does absolutely assemble to 1, so that you wind up accoutrement the absolute bare ambit if you add an absolute cardinal of terms. You can prove this, cleverly, by adding the absolute alternation from bifold the absolute alternation as follows:

Simple, straightforward, and compelling, right?

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