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
It entirely depends on how you count the numbers:
The most common answer to this post is that because you can pair up every number in one set with a number in the other they must be the same size. They’re both continuums.
But there’s another way to count, by measuring (using the Lebesgue measure), which gives us the more intuitive answer that the set of numbers between 0 and 2 is twice the measure of the set of numbers between 0 and 1.
This will probably only lead to more “why” questions.
Lets play the paper infinity game.
I have a sheet of paper, you have two sheets of paper. You have twice as many as me. I cut one page in half, you do the same. Now you have 3 sheets and I have 2. Now I cut a sheet in half and you do the same. I have 3 sheets and you have 4. You can do this forever, even though you started with twice as many sheets as me, you can only ever have one more sheet than me. After an infinite number of cuts, we both have an infinite number of sheets of paper, you are just 1 piece ahead of me. I can always catch up by making another cut though.
Another fun part of that, my pages get smaller a LOT faster than yours. You have double the “area” of paper as me to start. My first cut makes each sheet half the size, you require 2 cuts to do the same thing. By 3 cuts my paper is all 1/4th the size of where we started, it would take you 6 cuts to be in the same place as me. For me to get to 1/8th size I need 7 cuts, you need 14. You can always, eventually, get to the same size sheet of paper as me but it will always take you twice as many cuts to get there.
So, going one way you always only ever have one more page than me, going the other way you always take twice as long to get to the same page size as me. Either way we can both eventually get to where the other is, it just takes more work for one of us depending on what your end goal is.
You can map every number between 0 and 1 to every number in between 0 and 2 by means of a simple function: f(x) = 2x. If you can, conceptually, draw a line between every element of one set, and every element of another, they must have the same number of elements.
Infinity isn’t a number, it’s the condition of being boundless. So don’t think of the quantity of real numbers, instead consider that they are boundless in the same way.
There the same amounts.
Take any number between 0 and 1. Multiply it by two, you get a number between 0 and 2.
Take any number between 0 and 2. Multiply it by 0.5, you get a number between 0 and 1.
Since you can couple every number between 0 and 1 with a number between 0 and 2 in that way, with no number staying uncoupled on either side, you got the same amount of numbers in each interval.
Source: math teacher.
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).
The reason this is counterintuitive is because it brings into contrast two measurements of mathematical size: cardinality and volume. The interval between 0 and 2 has twice the volume, but the same cardinality.
The first thing to understand is that a single number takes up no space. The reason this is true is because we can contain it in an arbitrarily small ball. Think about 0, for instance. The interval (-0.1, 0.1) contains 0, and the volume of this interval is 0.2. The interval (-0.01, 0.01) also contains 0 and has size 0.02. We can continue this process, and squeeze 0 into a smaller and smaller ball. Now the mathematical concept of a limit comes into it. Because we can fit 0 into a ball of arbitrarily small volume, 0 itself must have 0 volume.
The thing that is hard to understand is that even though an individual number has 0 volume, if we look at all the numbers between 0 and 1, that set has volume 1. This phenomenon is one example of how our intuition between “count” and “volume” breaks down when dealing with infinite sets.
Not all infinitys are the same. Some are larger than others. What you have here is infinity times 2. The uncountable infinity between 0 and 1, and the uncountable infinity between 1 and 2.
This goes to the Hotel Paradox. Whichever number of rooms you need to fill can be filled. For every variation or increase in that number it could be x or x +1 or x^2. An infinite is uncountable and any number greater than infinite is still just infinite.
You have number 1 to the number 2. Any valid answer, the closer you get to each end point is also how much farther away you will be. Give the kid a challenge, how many numbers between 1 and 2 can they count to before the end of an hour will get the point across. 1.01, 1.02. 1.03 . . . Etc
There are not more numbers between 0 and 2 than between 0 and 1. At least when you accept the way mathematicians compare relative sizes (even for infinite amounts).
Mathematicians say for each number (x) between 0 and 1 there is a number between 0 and 2 (2x) and vice Vera’s: for each number between 0 and 2 (y) there is a number (y/2) between 0 and 1.
As long as you can find every number being co-paired in some way, like you do when counting, the amount is the same.
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