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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?

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    $\begingroup$ I think it would simplify your question a bit to just ask "Is $\textit{0}$ purely imaginary?" $\endgroup$
    – curious
    Nov 9, 2014 at 1:16
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    $\begingroup$ But my question is why would I consider only one classification 0+0i and ignore the others $\endgroup$
    – chris
    Nov 9, 2014 at 1:19
  • $\begingroup$ "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". $\endgroup$
    – user247327
    Feb 25, 2020 at 16:20

3 Answers 3

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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.

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    $\begingroup$ from my understanding zero can also be considered real? $\endgroup$
    – chris
    Nov 9, 2014 at 1:30
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    $\begingroup$ so then my question is why can I consider only the definition that suits my needs? $\endgroup$
    – chris
    Nov 9, 2014 at 1:38
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    $\begingroup$ 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. $\endgroup$
    – Steve Kass
    Nov 9, 2014 at 1:53
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    $\begingroup$ 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. $\endgroup$
    – chris
    Nov 9, 2014 at 2:00
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    $\begingroup$ @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. $\endgroup$ Nov 9, 2014 at 3:20
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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.

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    $\begingroup$ $\ldots$ and $0$ is unique in being both purely real and purely imaginary. $\endgroup$
    – Thumbnail
    Nov 9, 2014 at 10:06
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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.

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