There are infinitely many real numbers between 0 and 1. Are there twice as many between 0 and 2, or are the two amounts equal?

Question

I know the actual technical answer. I’m looking for a witty parallel that has a low chance of triggering an infinite “why?” procedure in a child.

In: 1335

Answer 1

Given that this is targeted towards a child, I wonder if this might work as a visualization:

Partially inflate a balloon and use a sharpie to draw a number line representing 0-1 on it, with a number of regularly spaced reals marked on it as tick marks.

Now, blow the balloon up further so that the interval becomes twice as long, now representing 0-2, and of course the number of marked reals remains the same (corresponding to the r -> 2r pairing).

The child may reasonably object that these numbers now are more spaced out (“you can fit more numbers into 0-2”), but then you can draw an additional ticks between each number and deflate the balloon back to original size to show how they still all fit into the 0-1 interval (corresponding to the r->0.5r pairing).