Expert: Paul Klarreich - 3/3/2011

Question
I have been told that the square root of -1, i, has no sign.
ie it is not positive or negative. Also that -i=-1*i, and +i=+1*i.
Therefore I conclude that +i=i.
If this is true how can i not have a sign?
Surely any number x is the same thing as +x.
Is this not true?

Answer
Questioner: Padraic
Country: United Kingdom
Category: Advanced Math
Private: No
Subject: Sign of i
Question: I have been told that the square root of -1, i, has no sign.
ie it is not positive or negative. Also that -i=-1*i, and +i=+1*i.
Therefore I conclude that +i=i.
If this is true how can i not have a sign?
Surely any number x is the same thing as +x.
Is this not true?
..........................................
Your example demonstrates:

1. The importance of careful use of vocabulary in studying mathematics.
2. Why you need the services of a teacher. (I'm not volunteering -- I am retired.)

Positive and negative do not apply to imaginary numbers.  Use these rules for the

PLUS SIGN : the thing above the = on the keyboard.

   If it is followed by a real constant, as in + 865.5, it means 'positive.'
   If it is followed by anything else, as in + (.....), it means 'addition.'

MINUS SIGN : to the left of the = on the keyboard.

   If it is followed by a real constant, as in - 865.5, it means 'negative.'
   If it is followed by anything else, as in - (.....), it means either 'subtraction' or 'the opposite of," whichever seems to fit the context.

So when you see  ' - i ', you read 'the opposite of i', because 'i' is not a real constant.

And  + i, means something is added to i.  If there is nothing visible to add it to, assume it means:

0 + i.

NOW THEN, you wrote:

"I have been told that the square root of -1, i, has no sign."

whoever said that should be ignored.  He is no mathematician.  The proper phrasing is:

"the square root of -1, denoted i, is neither positive nor negative."

In fact, inequalities have no meaning in imaginary numbers, either.

Best wishes for your study of complex analysis -- it is a fascinating branch of mathematics.