What is the Midway Point?

 

Here’s a neat mathematical riddle to use on your friends to prove your genius bona fides. It sounds unsolvable until, well, you hear the solution.

 

Question:

 

What is the exact middle point between zero and infinity?

 

In other words, on this number line from negative infinity to positive infinity, what is the halfway point between zero and positive infinity on the right?

 

Any guesses?

 

Hmm?

 

Seems kinda impossible to figure out, right? At first I thought so, because infinity, that sideways-number-eight, is not really a number, like 3, 17/50, or π^cubed is a number. Yeah, 3 and 17/50 have exact locations on the number line, and even though π^cubed, like pi itself, is not an exactly defined number (it is an irrational number whose decimal expression goes on, it has been proven, forever), it pretty much has an exact location on the number line. But infinity is not a specific number but an idea. A mathematical concept. So it really doesn’t have a location on the number line, except a vague neighborhood that lives ever, ever, ever rightward as you heading that way down the number line.

 

Hint #1 (minor):

 

So the trick is not to think of the question spatially. Not as in the case of 18 inches being the midway point of a yard, or 500 meters the halfway point of a kilometer.

 

Think of numbers themselves, as in types of numbers.

 

Any guesses?

 

Hmm?

 

Hint #2 (major):

 

Every number on the number line can be expressed as a reciprocal. A reciprocal of a number is one-over-that-number. The reciprocal of x is 1/x. The reciprocal of 3 is 1/3. The reciprocal of 17/50 is 50/17. The reciprocal of π^cubed is 1/π^cubed.

 

So what’s the halfway point between zero and infinity?

 

Answer: 1

 

The reciprocal of 1 is 1/1, or 1. 1 is its own reciprocal. But for every single number greater than 1, from 1.0000000000000001 to a googolplex (10 raised to the power of 1 with 100 zeros following it), there is a corresponding reciprocal. Every single one. And that reciprocal is LESS than 1. Every number greater than 1 has a reciprocal less than 1. Therefore, 1 is the midway point between zero and infinity. Not physically, as in a spatial distance sense, but in the number of actual numbers that occupy the intellectual space between 0 and 1 and 1 and infinity.

 

Q.E.D., as they say.

 

Now go and riddle your most intelligent friend.