
Possibly the greatest
paradox is that mathematics has paradoxes... 



















Galileo's
paradox



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 onetoone 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






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




