# Can a set containing 0 be purely imaginary?

A purely imaginary number is one which contains no non-zero real component.

If I had a sequence of numbers, say $\{0+20i, 0-i, 0+0i\}$, could I call this purely imaginary?

My issue here is that because $0+0i$ belongs to multiple sets, not just purely imaginary, is there not a valid case to say that the sequence isn't purely imaginary?

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I think it would simplify your question a bit to just ask "Is $\textit{0}$ purely imaginary?" – curious Nov 9 '14 at 1:16
But my question is why would I consider only one classification 0+0i and ignore the others – chris Nov 9 '14 at 1:19

0 is both purely real and purely imaginary. The given set is purely imaginary. That's not a contradiction since "purely real" and "purely imaginary" are not fully incompatible. Somewhat similarly baffling is that "all members of X are even integers" and "all members of X are odd integers" is not a contradiction. It just means that X is an empty set.

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$\ldots$ and $0$ is unique in being both purely real and purely imaginary. – Thumbnail Nov 9 '14 at 10:06

A complex number is said to be purely imaginary if it's real part is zero. Zero is purely imaginary, as it's real part is zero.

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This is really important in fields like ordinary differential equations, where you look at having no purely imaginary eigenvalues for it to be hyperbolic – Alan Nov 9 '14 at 1:23
from my understanding zero can also be considered real? – chris Nov 9 '14 at 1:30
Yes, a complex number is real if it's imaginary part is zero. So zero is also real. – Seth Nov 9 '14 at 1:33
so then my question is why can I consider only the definition that suits my needs? – chris Nov 9 '14 at 1:38
By definition, $0$ is purely imaginary. The fact that $0$ has other properties (it is real; it is nonnegative; it is rational; it is an integer; it is algebraic; it is divisible by every prime number) does not mean you can’t use the property you need. Similarly, is $\{-2,4\}$ a set of even numbers? Yes. The number $-2$ is not only even, but it’s also negative. The fact that it’s negative doesn’t mean you can’t use the fact that it’s even. Some sets defined by properties overlap. – Steve Kass Nov 9 '14 at 1:53