Navigation Panel: Previous | Up | Down to First Subsection | Forward (skipping subsections) | Graphical Version | PostScript version | U of T Math Network Home
University of Toronto Mathematics Network
Answers and Explanations
Do "Imaginary Numbers" Really Exist?
An "imaginary number" is a multiple of a quantity called "i"
which is defined by the property that i squared equals -1.
This is puzzling to most people, because it is hard to imagine
any number having a negative square. The result: it is tempting to
believe that i doesn't really exist, but is just a convenient
This isn't the case. Imaginary numbers do exist. Despite their name,
they are not really imaginary at all. (The name dates back to when
they were first introduced, before their existence was really understood.
At that point in time, people were imagining what it would be like to have
a number system that contained square roots of negative numbers, hence the
name "imaginary". Eventually it was realized that such a number system
does in fact exist, but by then the name had stuck.)
Before discussing why imaginary numbers exist, it's helpful to think about why
we're even asking the question. Why is it so hard to accept that there
could be numbers with negative squares?
One has to come to terms with the things that seem so puzzling and
confusing about this concept and see that they are not really so
unreasonable after all, before one can move on to accept
the existence of imaginary numbers. Having done that, we can
move on to seeing why they exist, and what relevance they have.
Therefore, we will address the following questions (you may select
any of the items below to see the explanation):
This page last updated: September 1, 1997
Original Web Site Creator / Mathematical Content Developer:
Current Network Coordinator and Contact Person:
Any Wilk - firstname.lastname@example.org
Go backward to Does "Infinity" Exist?
Go up to Answers and Explanations -- Index
Go down to first subsection Imaginary Numbers: More Reasonable than they First Appear
Go forward to Does the Number e Have Special Meaning? (skipping over subsections)
Switch to graphical version (better pictures & formulas)
Access printed version in PostScript format (requires PostScript printer)
Go to University of Toronto Mathematics Network