1How to measure "endless"
You can't count it — so you pair it up instead
You can never finish counting something endless. So how do you tell if two endless piles are the same size? You only need two ideas. Watch each one:
Counting goes on forever
1, 2, 3, 4… never stops. Whatever number you reach, you can always add one more. That endless ladder of counting numbers is our measuring stick.
Pair them one-to-one
Same size = a perfect match. If every number on the left can grab exactly one partner on the right with nobody left lonely, the two piles are the same size — even if you can't finish either one.
2Two endless piles to test
The line-them-up pile vs the squeeze-them-in pile
We'll try to match the counting numbers to two different endless piles. They look very different up close:
The even numbers
2, 4, 6, 8… You can write them in a neat numbered list, one after another.
Decimals between 0 and 1
0.418…, 0.073…, 0.999… — endless decimals packed into a tiny gap. Can they even be listed?
3Your turn — run the pairing machine
Match the counting numbers to the even numbers
Tap to add pairs: 1 grabs 2, 2 grabs 4, 3 grabs 6… Keep going as long as you like. Watch for anyone left without a partner.
Tap "Add a pair" to start matching.
4Now try the squeeze-them-in pile
Make a list of EVERY decimal between 0 and 1 🔍
Switch the machine to the decimals. Here is a numbered list that claims to hold every single decimal between 0 and 1. Before we test it, make your guess.
Guess before you run it
You try to make a numbered list that contains every decimal between 0 and 1. Can it be done so nothing is left off?