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/