Galileo's Paradox of Infinity Calculator

Your details

The smallest natural number in your comparison range.
The largest natural number in your comparison range.
A specific natural number to demonstrate the bijection: this tool shows n and its partner square n squared.
Natural numbers in range
100

Count of integers from range start to range end

Perfect squares in range10
Squares as % of naturals0.1%
Square of sample n49
Is sample n a perfect square?No
Squares per 100 naturals (at your range end)10
Natural numbers in range100
Perfect squares in range10

Finite range shows 10 squares vs 100 naturals, but both sets are equally infinite.

  • In your range, only 10 out of 100 numbers are perfect squares (10.0%), yet the bijection f(n) = n² pairs every natural number with a unique square.
  • Your sample n = 7 maps to 49 under the bijection. This pairing is one-to-one and covers every perfect square exactly once.
  • As numbers grow larger, squares become increasingly sparse. Near 100, roughly 10.00 out of every 100 naturals are squares. Yet the correspondence never breaks: every natural still has a square partner.
  • Galileo noticed this and concluded that 'less than', 'equal to', and 'greater than' cannot apply to infinite collections. Cantor later proved both sets share the same infinite cardinality, called aleph-null.

Next stepTry increasing the range end to 10,000 to see the density of perfect squares drop below 1%, while the bijection remains perfect.

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