Galileo's Paradox Calculator
Galileo's Paradox Calculator illustrates that up to any number there are far fewer perfect squares than integers, even though the two infinite sets can be matched one-to-one. Enter an upper limit.
Formula
Perfect squares ≤ N = ⌊√N⌋
- Although squares thin out among the integers, every integer n pairs with its square n².
- This one-to-one pairing shows both infinite sets have the same size (cardinality).
N = 100
Inputs
- Upper Limit (N): 100
100 integers but only 10 perfect squares (1,4,9,…,100), yet they can be matched one-to-one.
Frequently asked questions
What is Galileo's paradox?
That there seem to be fewer perfect squares than integers, yet each integer pairs with exactly one square, so the sets are the same size.