Mathematics Database Programming Web Design Price List     Possibly the greatest paradox is that mathematics has paradoxes... Bernoulli's sophism Paradox of Bernoulli and Leibniz Paradox of even (odd) and natural numbers Paradox of Hilbert’s hotel Ross-Littlewood paradox Paradox of wizard and mermaid Paradox of enchantress and witch Paradox of Tristram Shandy Barber paradox Achilles and tortoise   With finite sets, a part is always smaller than the whole. But with infinite sets one part of the set can be just as large as the whole. Often it looks as a paradox, but from the mathematical point of view there is no paradox.

Some of natural numbers are perfect squares, such as 1, 4, 9, 16, 25 ... . Each natural number is an exact square root of one perfect square. There are as many perfect squares as there are natural numbers. This can be seen by pairing the natural numbers with the perfect squares to show that there is a one-to-one correspondence between the two sets:

 1, 2, 3, 4, 5, ... n, ...     ... ... 1, 4, 9, 16, 25, ... ...

This is sometimes known as Galileo's Paradox, as it was shown by the famous Italian scientist.

by Tetyana Butler Top  Complex analysis is studying the most unexpected, surprising, even paradoxical ideas in mathematics. The familiar rules of algebra and trigonometry of real numbers may break down when applied to complex numbers. Free Lessons Contact us