An improper use of the symbol for the imaginary unit leads to the apparent proof of a false statement.

			(1)
			(2)
			(3)
			(4)
			(5)

The reason for the fallacy is that is not an ordinary (real) square root, hence the rule for computing the quotient of radicals does not apply to it.

This entry contributed by Margherita Barile

Eves, H. An Introduction to the History of Mathematics, 3rd ed. New York: Holt, Rinehart, and Winston, p. 385, 1969.

Gardner, M. Mathematical Puzzles and Diversions. New York: Simon and Schuster, p. 144, 1959.

CITE THIS AS:

Barile, Margherita. "Complex Number Paradox." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/ComplexNumberParadox.html