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Possibly the
greatest paradox is that mathematics has paradoxes...
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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.
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Paradox of even and natural numbers
The
Natural numbers contain even and odd
numbers. However there are as
many even numbers as there are natural
numbers. This can be seen,
by pairing natural numbers with even
numbers to show that there is a one-to-one
correspondence between the two sets:
| 1, |
2, |
3, |
4, |
5, |
... |
n, |
... |
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... |
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... |
| 2, |
4, |
6, |
8, |
10, |
... |
2n, |
... |
Top
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Paradox of odd and natural numbers
The
Natural numbers contain even and odd
numbers. There are as many odd
numbers as there are natural numbers.
This can be seen, by pairing the natural
numbers with the odd numbers to show
that there is a one-to-one correspondence
between the two sets:
| 1, |
2, |
3, |
4, |
5, |
... |
n, |
... |
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... |
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| 1, |
3, |
5, |
7, |
9, |
... |
2n - 1, |
... |
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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
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