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Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
Complex functions paradoxes
Infinity paradoxes
Set theory paradoxes
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Complex functions paradoxes

There are many cases with complex functions which appear to be paradoxes. The familiar rules of algebra and trigonometry of real numbers may break down when applied to complex numbers. We may not, for example, use the rule , when a and b are negative, for otherwise we would have
.
There is another example of An improper use of SQRT(-1)

Complex sine (cosine) functions could be equal to 2 or 5. Is it a paradox? Look at  Sin(z) = 2

arctg(1) =pi/4. If to consider arctg(x)=arctg(x) = integral dx/1+x*xand forget that complex logarithm is a multi-valued function, you will get "arctg(1) = 0". Look at Paradox of Bernoulli and Leibniz

Swiss mathematician John Bernoulli constructed a beautiful sophism with the chain of arguments, pretending to prove that Ln(-z) = Ln(z) for any . Look at Bernoulli's sophism

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An improper use of SQRT(-1)

Another example as an improper use of = i leads to an error.

i = i                                    (1)

=

-1 = 1                               (2)

The reason for the fallacy (2) is that rule for computing the quotient of radicals does not apply to .

by Tetyana Butler

Complex functions Tutorial
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.
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