# Is 0 an imaginary number?

#### rbj

this is a question of mine because of an edit to the Wikipedia article Imaginary number .

the funny thing is that i couldn't find, in three of my old textbooks a clear definition of "imaginary number". (they were pretty good at defining "imaginary part", etc.)

i understand that that zero lies on both the real and imaginary axis. is 0 both a real number and an imaginary number? we know, certainly, that there are complex numbers that are neither purely real nor purely imaginary. but i've always previously considered that an imaginary number had to square to be a real and negative number (not just non-positive). clearly we can (re)define a real number as a complex number with imaginary part that is zero (meaning that 0 is a real number) but, if one were to define an imaginary number as a complex number with real part zero, that would also include 0 among the pure imaginaries.

what is the complete and formal definition of "imaginary number" (outside of the Wikipedia reference or anything derived from it)?

#### rohanprabhu

nice question..
if we define 'imaginary numbers' as 'complex numbers having a real part as 'zero' and a non-zero imaginary part'.. 0 doesn't fit in this description. But by, convention and for theoretical symmetry , we'll have to define 'real numbers' in pretty much the same way, and hence 0 would neither be a purely imaginary number or a purely real number.

In the argand plane, if we define imaginary numbers as the numeral vectors having a component only along 'y-axis' and real numbers as the numeral vectors having a component only along 'x-axis'.. then 0 still falls in neither group, as '0' doesn't have a component along any axis.

If we modify the same definition to something like: imaginary numbers are the numbers who lie solely on the y-axis and real number are the numbers who lie solely on the x-axis, '0' lies on both, the y-axis and the x-axis.

This is something like the question: "Is 0 positive or negative?".

Also, this number is controversial on the argand plane [imho] because, it has an indeterminate amplitude [0/0].

I realize that I was less confused before i read this thread :D

#### rbj

nice question..
if we define 'imaginary numbers' as 'complex numbers having a real part as 'zero' and a non-zero imaginary part'.. 0 doesn't fit in this description. But by, convention and for theoretical symmetry , we'll have to define 'real numbers' in pretty much the same way, and hence 0 would neither be a purely imaginary number or a purely real number.

In the argand plane, if we define imaginary numbers as the numeral vectors having a component only along 'y-axis' and real numbers as the numeral vectors having a component only along 'x-axis'.. then 0 still falls in neither group, as '0' doesn't have a component along any axis.

If we modify the same definition to something like: imaginary numbers are the numbers who lie solely on the y-axis and real number are the numbers who lie solely on the x-axis, '0' lies on both, the y-axis and the x-axis.

This is something like the question: "Is 0 positive or negative?".
something like it, but not quite. i think it is well established that 0 is real and that 0 is neither positive nor negative (it is both non-negative and non-positve). so the question is, is 0 both purely real and purely imaginary or just purely real? i used to think the latter, but now i don't know.

Also, this number is controversial on the argand plane [imho] because, it has an indeterminate amplitude [0/0].
you mean angle or arg( )?

I realize that I was less confused before i read this thread :D
same for me.

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#### Hurkyl

Staff Emeritus
Gold Member
I can't imagine any good reason to exclude 0 from being imaginary, and I would expect if I asked any of my colleagues, they would say that 0 is imaginary.

#### Gib Z

Homework Helper
The definition I learned of a purely imaginary number is a complex number whose real part is zero. Zero fits this description, so yes I would say it was an imaginary number.

#### daveb

Well, since 0 is the additive identity element in both the real field and the complex field, then I would think it is complex, and also real. Of course, this criteria then means that 1 is also both real and complex as the multiplicative identity element, meaning you have to extend this to all numbers, which also makes sense, since the reals are a subfield of complex numbers. Thus, all reals are also complex. Now, if you're asking if 0 is strictly complex, then by this logic, the answer would be no.

#### CRGreathouse

Homework Helper
I'd say yes: 0 is the unique imaginary real number.

#### yasiru89

Depends on which field is in question really. A simple matter of definition. As the complex system is an extension of the real system; within it zero will mean the quantity 0 + 0i
The addition sign is of course purely arbitrary, particularly in this case. If you think of it literally zero will fall into many categories just like one- i.e. 1 is NOT and yet at times IS prime. An imaginary number is basically of the form ki, where i is the imaginary unit V-1 and k is a real quantity, zero inclusive. The alternating degree property of i makes it this simple of course. The zero itself can be 0 = 0i but that won't matter since under no circumstances is division by zero allowed(strictly, even in the extended theory of limits)!

#### mathwonk

Homework Helper
i think all numbers are imaginary.

#### rbj

I can't imagine any good reason to exclude 0 from being imaginary, and I would expect if I asked any of my colleagues, they would say that 0 is imaginary.
could you do that, Hurk? and if someone can find a nice reference that says so explicitly, that would be helpful. i know that this is an issue of definition, and i can't understand why neither my 1974 calc book (Seeley) (in the Complex Numbers chapter) nor my applied engineering math book (Kreyszig) nor my my complex variables book (Levinson & Redheffer) makes that clear. i know it's just a definition, but you would think someone would have weighed in on this.

we had a similar thing regarding the definition of "Nyquist frequency". i have a solid opinion about the proper definition, but there exists a textbook i respect greatly that has that definition wrong.

thanks guys for any assistance in nailing this down.

#### rbj

BTW, Hurk, this ambiguity supports you a bit in our little dispute we had a while back regarding the appropriateness of the terms "real numbers" and "imaginary numbers". whereas i still believe that actual physical quantities are still real (and when we measure them, the best we can do is measure rational approximations to those quantities), this real and imaginary 0 thing muddies the waters a little.

so, being an enjunear, i see it a little differently than wonk. there are some numbers that i see as not imaginary.

#### Howers

The imaginary number is defined as a + bi = 0, where a=0 and b=real. zero is real, so zero is defined as imaginary if thats how we define 0i. Just as 5 is natural, rational, and real. Not all numbers belong exclusively to one set.

#### disregardthat

This is a matter of definition.

I don't know if there is an official definition of an imaginary number, but wiki define it this way:

"In mathematics, an imaginary number (or purely imaginary number) is a complex number whose squared value is a real number not greater than zero"
As zero squared is zero, it fits the definition here too.

I saw another definition:

"An imaginary number is a quantity of the form ix, where x is a real number and i is the positive square root of -1"
Zero fits like a glove.

A number can't physically exist, so I guess they are all imaginary :P

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#### belliott4488

I'd think it would depend on the use to which you're putting the term "imaginary numbers". In other words, for some uses zero is clearly an appropriate example of an imaginary number, for others it might not be. For example, I could state that "the set of imaginary numbers forms a group under the operation of addition", but that would be true iff I included zero in the set.

#### jcsd

Gold Member
It's just a matter of definition.

I would say that most would find it desribale to define zero as an imaginary number in order to make imaginary numbers a subspace of the complex numbers over the reals.

#### Gokul43201

Staff Emeritus
Gold Member
Including zero permits the imaginary numbers to possess an additive inverse. I think that might be useful to keep, as pointed out above in posts #14,15.

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#### rbj

This is a matter of definition.

I don't know if there is an official definition of an imaginary number, but wiki define it this way:

"In mathematics, an imaginary number (or purely imaginary number) is a complex number whose squared value is a real number not greater than zero"
but that's the whole point of me asking about this! that wikipedia definition was recently changed in such a way that the net meaning was changed (i linked to that change in my first post). i want to try to find other support for that change otherwise it was just one editor's POV. a month ago the wikipedia article said this: "an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number." it's meaningfully different. one definition includes 0 and the other definition does not (and the latter is what existed for years at that wikipedia article).

so without reference to the Wikipedia article, what past reference might we have (like some textbook) do we have that actually says one way or another?

#### Hurkyl

Staff Emeritus
Gold Member
so without reference to the Wikipedia article, what past reference might we have (like some textbook) do we have that actually says one way or another?
A historical note:

Because of what it erroneously connotes, it is a shame that the term imaginary is used in definition (ii). Gauss, who was successful in getting mathematicians to adopt the phrase complex rather than imaginary number, also suggested that we use lateral part of $z$ in place of imaginary part of $z$. Unfortunately, this suggestion never caught on, and it appears we are stuck with the words history has handed down to us.​
-- John H. Mathews, Russell W. Howell, Complex Analysis for Mathematics and Engineering

(Definition (ii) was the definition of the imaginary part of a number)

So, once upon a time, the word "imaginary number" actually meant what we now call a "complex number". In fact, this text doesn't seem to define the phrase "imaginary number".

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#### jcsd

Gold Member
but that's the whole point of me asking about this! that wikipedia definition was recently changed in such a way that the net meaning was changed (i linked to that change in my first post). i want to try to find other support for that change otherwise it was just one editor's POV. a month ago the wikipedia article said this: "an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number." it's meaningfully different. one definition includes 0 and the other definition does not (and the latter is what existed for years at that wikipedia article).

so without reference to the Wikipedia article, what past reference might we have (like some textbook) do we have that actually says one way or another?
Mathworld and planetmath, both have defintions which include zero FWIW.

Persdonally I think you'd find both definitions of imaginary numbers are reasonably common place if you look hard enough. But I'd think you'd find that the zero-inclusive definition would be more common place among mathematicians by virtue of the fact that there's just a little bit more to be said about them.

Obviously the imaginary numbers with zero form a group under addition, whereas the imaginary numbers without zero are not closed under addition so don't even form a groupoid

On the other hand someone might argue that the imaginary numbers under addition is a pretty uninterestign group anyway. Even with extra strucure such as order the group is isomorphic to the reals, so there's nothing to say about the imagianries as an additive group that hasn't already been said about the reals.

I'd say most of the time we think of the imaginaries as being 'orthogonal' to the reals (viewing the complex numbers as a real vector space with an appropiate inner product), in this case the imaginary numbers is the subspace of complex numbers which are orthogonal to the subspace of real numbers. This defintion includes zero as an imaginary number as it is orthogonal to all complex numbers including itself. We tend write imagianry numbers in terms of a real and imagianry components in the same way we tend to write real vectors in terms of an orthonormal basis.

At the end of the day though it really is just a matter of definition.

#### belliott4488

Thanks, jcsd - I think that summarizes the issue very well. I agree that the set of imaginaries is not in and of itself very interesting, which might be why there is not a clear preference for one or the other definition (although I'm at a loss to say why one would prefer to leave zero out - perhaps there is a reason that I haven't heard yet).

I think it's of more value to say that zero is a real number, and since they are contained within the group of complex numbers, it's also a complex number, and since they are contained within the group of quaternions, it's also a quaternion, .... and also an octonion. It seems to me somewhat sterile to ask whether or not it's pure imaginary. I could be missing something, though ...

#### rbj

A historical note:

Because of what it erroneously connotes, it is a shame that the term imaginary is used in definition (ii). Gauss, who was successful in getting mathematicians to adopt the phrase complex rather than imaginary number, also suggested that we use lateral part of $z$ in place of imaginary part of $z$. Unfortunately, this suggestion never caught on, and it appears we are stuck with the words history has handed down to us.​
-- John H. Mathews, Russell W. Howell, Complex Analysis for Mathematics and Engineering

(Definition (ii) was the definition of the imaginary part of a number)

So, once upon a time, the word "imaginary number" actually meant what we now call a "complex number". In fact, this text doesn't seem to define the phrase "imaginary number".
it's a good reference, Hurk. (along with others above.) i'm gonna leave the article alone, but was a little alarmed by a substantive change to it and didn't know, at first, which way to go.

i still think about our little argument a year or so ago about the appropriateness of the term "imaginary". i still don't think we measure raw physical quantities as imaginary or complex numbers (how much something moves, how much length or time or mass or charge or energy or whatever something is.) not to dismiss how useful, as sorta abstrations, that complex quantities are in modeling or describing something in reality. and as a signal processing engineer, i daily think about signals as complex quantities. but, in the end, i know that the physical signalling mechanism is with real quantities of "stuff" (usually a voltage) and, for that reason, still think that Gauss wasn't too far off in his naming convention.

#### slider142

Actually, you more likely deal only with rational quantities of "stuff". I doubt any of your instruments measure irrational numbers. To borrow the reasoning from your post, like complex numbers, the full machinery of real numbers just makes analysis easier.

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#### symbolipoint

Homework Helper
Gold Member
Is zero a real number? Yes.
Are real numbers also complex numbers? Yes.

If zero is a complex number, because it is also a real number, is it an imaginary number? It's imaginary componant is '0'; so does this mean that with an imaginary componant of zero, must the number ZERO be real but not imaginary? What do the mathematicians say about this?

I am not secure in my own point of view if I say that 0 is either real or imaginary. I believe it is real; I believe it is complex; but to express an imaginary number which uses coefficient on 'i' of zero, I can only say that this: 0 is real and 0*i is imaginary. 0 is real but NOT imaginary. Again, what do the mathematician experts say?

#### Hurkyl

Staff Emeritus
Gold Member
Again, what do the mathematician experts say?
Although 0 and 0i are different expressions, when used to denote complex numbers, they denote the same one. If you are logically consistent, and you think that 0i denotes an imaginary number, then you must also think that 0 denotes an imaginary number.

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#### JonF

The two definition of “imaginary numbers” depend on the context.

On is a member of the complex field defined by the operations:
(a,b)+(c,d) = (a+c,b+d)

where a,b,c, and d are reals. In this case (0,0) = 0 is clearly a member

The other is a complex number (a number in the form of a+bi) where a=0. Cleary 0 again satisfies this since 0 = 0+0i

So I would say yes.

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