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   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) = . If to consider arctg(x)= and 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 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 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