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Complex Number Paradox


An improper use of the symbol sqrt(-1) for the imaginary unit leads to the apparent proof of a false statement.

sqrt(-1)=sqrt(-1)
(1)
sqrt((-1)/1)=sqrt(1/(-1))
(2)
(sqrt(-1))/(sqrt(1))=(sqrt(1))/(sqrt(-1))
(3)
sqrt(-1)·sqrt(-1)=sqrt(1)·sqrt(1)
(4)
-1=1.
(5)

The reason for the fallacy is that sqrt(-1) 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

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References

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.

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Complex Number Paradox

Cite this as:

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

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