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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?

Seth
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chris
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    I think it would simplify your question a bit to just ask "Is $\textit{0}$ purely imaginary?" – curious Nov 09 '14 at 01:16
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    But my question is why would I consider only one classification 0+0i and ignore the others – chris Nov 09 '14 at 01:19
  • "Imaginary", in mathematics, applies to [I]number[/I]. I would object to calling a set "imaginary". It is, of course, a "set of imaginary numbers". – user247327 Feb 25 '20 at 16:20

3 Answers3

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

Seth
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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 09 '14 at 01:23
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    from my understanding zero can also be considered real? – chris Nov 09 '14 at 01:30
  • Yes, a complex number is real if it's imaginary part is zero. So zero is also real. – Seth Nov 09 '14 at 01:33
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    so then my question is why can I consider only the definition that suits my needs? – chris Nov 09 '14 at 01:38
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    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 09 '14 at 01:53
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    I get where you're coming from, but in my head being real and also purely imaginary is a contradiction. thats a better answer to my question than this answer. enough thinking for me for one day. – chris Nov 09 '14 at 02:00
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    @chris: You may be taking the word 'imaginary' too seriously. Imaginary numbers aren't any more unreal or nonexistent than say negative or irrational numbers. The name 'imaginary' is just a label. – Michael Shaw Nov 09 '14 at 03:20
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    I wasn't taking it literally, but more so confused at the idea of a purely imaginary number also being real. – chris Nov 09 '14 at 03:34
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    @chris Maybe as an analogy, consider the empty set. All of the following are true: It contains only even numbers. It contains only odd numbers. It contains only purple monkeys. – user253751 Nov 09 '14 at 11:03
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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.

user191016
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As other answers say, zero is purely real as well as purely imaginary. Let me give an intuitive explanation based on the graphical representation of a complex number.

In the Argand plane, the points on the real axis represent complex numbers which are "purely real" as their imaginary part is zero. On the other hand, points on the imaginary axis denote complex numbers whose real part is zero and hence they are "purely imaginary".

We know that, the real and imaginary axis meet at the origin which represents the complex number $0+0i$. As this point simultaneously lies on the real as well as the imaginary axis, we say that zero is both purely real and purely imaginary.

Vishnu
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