# Is 0 an imaginary number?

1. Dec 26, 2007

### 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)?

2. Dec 27, 2007

### 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

3. Dec 27, 2007

### rbj

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.

you mean angle or arg( )?

same for me.

Last edited: Dec 27, 2007
4. Dec 27, 2007

### Hurkyl

Staff Emeritus
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.

5. Dec 27, 2007

### Gib Z

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.

6. Dec 27, 2007

### 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.

7. Dec 27, 2007

### CRGreathouse

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

8. Dec 27, 2007

### 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)!

9. Dec 27, 2007

### mathwonk

i think all numbers are imaginary.

10. Dec 27, 2007

### rbj

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.

11. Dec 27, 2007

### 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.

12. Dec 27, 2007

### 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.

13. Dec 27, 2007

### 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

Last edited: Dec 27, 2007
14. Dec 27, 2007

### 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.

15. Dec 27, 2007

### jcsd

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.

16. Dec 27, 2007

### Gokul43201

Staff Emeritus
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.

Last edited: Dec 27, 2007
17. Dec 27, 2007

### rbj

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?

18. Dec 27, 2007

### Hurkyl

Staff Emeritus
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".

Last edited: Dec 27, 2007
19. Dec 27, 2007

### jcsd

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.

20. Dec 27, 2007

### 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 ...