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Is Zero Considered a Pure Imaginary Number (as 0i)?

Date: 12/02/2003 at 15:36:39
From: Andreas
Subject: is zero purely imaginary as well as real?

Representing complex numbers in the coordinate plane, the horizontal 
axis is the real axis, the vertical is the pure imaginary axis.  They 
intersect at "0" and "0i".  Does that intersection belong to both sets 
of numbers?  Or is it only real, but not pure imaginary?

I thought the pure imaginary numbers were numbers that were not real.
I find one definition which says they consist of: "all numbers whose
squares are negative" which excludes zero.  Another says: "all numbers
bi, where b is any real number."



Date: 12/02/2003 at 16:22:36
From: Doctor Peterson
Subject: Re: is zero purely imaginary as well as real?

Hi, Andreas.

Logically, one could make an argument that 0 is neither real nor 
imaginary, since is has neither an imaginary nor a real part.  But 
excluding it from either the real or the imaginary axis would be 
extremely awkward; so we define "purely imaginary" in a negative way, 
not as a number that HAS only an imaginary part, but as one that DOES 
NOT have any (non-zero) real part:

  Purely Imaginary Number
    http://mathworld.wolfram.com/PurelyImaginaryNumber.html 

  A complex number z is said to be purely imaginary if it has no
  real part, i.e., R[z] = 0.  The term is often used in preference to
  the simpler "imaginary" in situations where z can in general
  assume complex values with nonzero real parts, but in a
  particular case of interest, the real part is identically zero.

Therefore, 0 is considered to be a (pure) imaginary number.  Your 
second definition is the better one.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 07/25/2015 at 12:26:42
From: Matthew
Subject: What is the definition of a purely real number

Above, Doctor Peterson said, "excluding [zero] from either the real or 
the imaginary axis would be extremely awkward." 

Why would it be awkward? Much like zero is neither positive nor negative 
because it sits in the center of the Cartesian plane, why can't it be 
neither imaginary nor real when it sits in the center of the imaginary 
plane?

According to Wolfram's MathWorld, a purely imaginary number is a complex 
number z that has no real part, i.e., R[z] = 0. Shouldn't it follow that 
a purely real number is a complex number z that has no imaginary part, 
i.e., C[z] = 0?

If I apply MathWorld's definition of a purely imaginary number together 
with my definition of a purely real number, the number zero would be both 
purely imaginary and purely real. Is it acceptable for a number to be 
both purely imaginary and purely real at the same time? Why or why not?

What is the definition of a purely real number? I tried to search the 
Internet for a definition of what a purely real number is. I managed to 
find some with the same definition as I have given, but they did not 
address the "conflict" with the number zero.



Date: 07/25/2015 at 19:52:55
From: Doctor Peterson
Subject: Re: What is the definition of a purely real number

Hi, Matthew.

As Matthew wrote to Dr. Math
On 07/25/2015 at 12:26:42 (Eastern Time),
> According to Wolfram's MathWorld, a purely imaginary number is a complex 
> number z that has no real part, i.e., R[z] = 0. Shouldn't it follow that 
> a purely real number is a complex number z that has no imaginary part, 
> i.e., C[z] = 0?

Yes, in fact this would seem to be obvious, more so than the definition of 
a pure imaginary number: we all know that 0 is a real number, and "pure 
real" should not mean anything different.

It's interesting that MathWorld does not bother to define "pure real." I 
presume, again, that this is because, when used, it means nothing 
different from "real," which of course is very commonly used. (In my 
experience, "pure" is usually used, rather than "purely," so I'll stick 
with that.)

> I tried to search the Internet for a definition of what a purely real 
> number is. I managed to find some with the same definition as I have 
> given, but they did not address the "conflict" with the number zero.

I don't see any real "conflict" in saying that zero is both pure 
imaginary and pure real. Zero is clearly on the real axis, so it is a real 
number; and it is clearly on the imaginary axis, so it should be a pure 
imaginary number. It is the intersection of the two. This is not much 
different from saying that a square is both a rhombus and a rectangle.

The awkwardness I referred to is in part due to the fact that in math, we 
tend to name things inclusively; a word like "irrational," emphasizing 
what something is NOT, is unusual. The word "imaginary" was initially used 
for "non-real" numbers (and it is still used that way at times); Gauss 
introduced the word "complex" for the ENTIRE complex plane, reserving 
"imaginary" to refer to the ENTIRE imaginary axis. These usages are more 
consistent with usual mathematical practice (so that we can talk about 
properties that apply to those entire sets). To require an "imaginary" 
number to be anything on the axis EXCEPT 0 would mess that up.

Because the word "imaginary" historically has two quite different usages 
(any number that is NOT real, and any number that has no real PART), when 
we need to refer to the former it is better to say "non-real," and for the
latter, "PURE imaginary." Both of these terms are clearer than merely 
"imaginary."

But you will certainly find disagreements about the meaning of "pure 
imaginary," and people who share your uncertainty about whether zero 
should be considered both imaginary and real; I found debates about it on 
various forums, as is true of just about anything that is not commonly 
taught in an explicit way. I suspect part of the reason is that 
mathematicians don't need to talk about pure imaginary numbers very often, 
because they are not closed under any operation except addition (and if we 
exclude 0, they are not even closed under addition). For that reason also, 
I don't think it matters much whether we define "pure imaginary" to 
include or exclude 0.

Looking around for definitions from reputable sources, I find this one 
that agrees with me and MathWorld on the definition, but shows a Venn 
diagram that does not include 0 (perhaps inadvertently):

    http://www.mathwords.com/i/imaginary_numbers.htm

    Imaginary Numbers, Pure Imaginary Numbers
    Complex numbers with no real part, such as 5i.

Wikipedia also currently agrees, though there has been some debate among 
its contributors on this point:

    https://en.wikipedia.org/wiki/Imaginary_number

    An imaginary number is a complex number that can be written as a
    real number multiplied by the imaginary unit i, which is defined
    by its property i^2 = -1. The square of an imaginary number bi
    is -b^2. For example, 5i is an imaginary number, and its square
    is -25. Except for 0 (which is both real and imaginary),
    imaginary numbers produce negative real numbers when squared.

    ... The term "imaginary number" now means simply a complex number
    with a real part equal to 0, that is, a number of the form bi.

This site gives a definition that implicitly excludes zero, and does not 
give zero as an example, but that could be an oversight:

    http://www.regentsprep.org/regents/math/algtrig/ato6/imagineles.htm

    An imaginary number is a number whose square is negative.

But, as I said, this issue is mostly restricted to pedagogy, where pure 
imaginary numbers are introduced in preparation for the "real" concept of 
complex numbers. It is not really important how we define imaginary 
numbers once we have the whole set of complex numbers to discuss.

Can you suggest some contexts in which this matters?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 07/26/2015 at 01:11:27
From: Matthew
Subject: Thank you (What is the definition of a purely real number)

Thank you so much, Doctor Peterson. This was a very enlightening answer. 

Accepting that the definitions of both a pure imaginary and pure real 
number were true met with much resistance inside my head. I couldn't wrap 
my mind around the concept of a number being both pure imaginary and pure 
real. And when I wasn't able to find a definition in MathWorld for pure 
real numbers, I assumed that not defining it was done intentionally to 
avoid the "conflict" with zero. Then with that assumption, I concluded 
that there can't be a pure real number, because no definition for it was 
given. I was under the impression that no number could be both real and 
imaginary. 

I now understand the mistake of the assumption and conclusion I made. It 
seems I just have to accept that zero is both pure real and pure 
imaginary, and there is nothing conflicting about that.

I became curious about pure real numbers while I was going over some 
concepts of electronics for a blog post. In electronics, as I'm sure you 
know, we use imaginary numbers in impedance -- specifically, in the 
reactive part of impedance. I remember defining certain loads to be 
purely reactive and purely resistive. Of course, I immediately equated 
them with pure imaginary and pure real. I then became curious about the 
technical definition of both terms because sometimes the technical 
definition will give more information than just what we commonly know 
about the terms. Finding a definition for pure imaginary was easy. It was 
with finding a definition for pure real numbers that caused all of this 
confusion.

Thanks again for the help, Doctor Peterson. Zero is really a very 
complicated number!



Date: 07/26/2015 at 14:22:46
From: Doctor Peterson
Subject: Re: Thank you (What is the definition of a purely real number)

Hi, Matthew.

Zero is indeed complicated! Special cases like this tend to mess up our 
thinking wherever they show up.

Another example that comes to mind is "degenerate conic sections," where 
an equation almost meets the requirements to be, say, a circle, but turns 
out to be a single point (x^2 + y^2 = 0).

In your application, you are finding that a zero impedance is not really 
resistive or reactive. In some sense, it is an extreme case of each: you 
can dial down a resistance until it reaches zero, or you can dial down an 
inductance until it reaches zero. So it belongs in each "world," yet is 
not really "of" either. Mathematically, we just like to include such a 
special case within each for consistency.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
pre
Associated Topics:
High School Definitions
High School Imaginary/Complex Numbers

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