How this calculator works
Within a finite inclusive interval L through N, count squares by counting integer roots from ceil(sqrt(L)) through floor(sqrt(N)). Their finite proportion becomes small as N grows. Yet the unbounded map n to n squared pairs each positive integer with exactly one positive square. The finite count and infinite pairing answer different questions; a finite sample does not calculate an infinite probability.
Formula
Finite square count = floor(sqrt(N)) - ceil(sqrt(L)) + 1. Integer count = N - L + 1. Pairing: n ↔ n², with inverse sqrt(s) on perfect squares.
Worked example
Between 1 and 100 there are 10 squares, a fraction of 10%. The number 7 pairs with 49. For the full infinite sets, every positive square still has exactly one positive root.
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Frequently asked questions
Are there fewer squares than integers?
Inside a fixed finite bound, usually yes. The two unbounded sets have equal cardinality because the squaring map is a bijection.
Does the table prove a statement about infinity?
No finite table proves it. The algebraic inverse on every square establishes the pairing for all positive integers.
Why reject 12 in square mode?
It is not a perfect square of an integer. The pairing is between positive integers and positive integer squares.