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The following is taken from Wikipedia's definition.

An imaginary number is a number whose square is less than or equal to zero.

But I also heard that

An imaginary number is a number whose square is less than zero.

Which is the correct one?


My doubt came from the fact that some mathematicians consider the imaginary numbers are on the vertical axis and the real numbers are on the horizontal axis. So the intersection point at the origin. As a result, zero is included in the imaginary number set.

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The first is dead wrong. May be a record for Wikipedia as the earliest ridiculous assertion in an entry. – André Nicolas Oct 20 '12 at 18:16
How can the square of ANY number be equal to zero unless that number IS zero in the first place? – Hawk Oct 20 '12 at 18:21
The Wikipedia page on complex numbers defines of a purely imaginary number to be "a complex number whose real part is zero" which, of course, includes $0$. – Bill Dubuque Oct 20 '12 at 18:24
The intent is "an imaginary number is a complex number whose square is a real number $\le 0$". Not everyone considers the complex number $\:0 = 0 + 0\cdot i\:$ to be imaginary. – Bill Dubuque Oct 20 '12 at 18:30
Possible duplicate of Can a set containing 0 be purely imaginary? – Najib Idrissi Nov 28 at 14:07

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