The square root of -1 is a special number we call i
and it is a member of a set of numbers called the
imaginary numbers.  The imaginary numbers are
the set of all numbers that are the square root of
negative numbers.
                                              - The Math Forum



Q: Why bother with such a seemingly frivolous
and useless abstraction as i?

A: Albeit abstract, i is neither frivolous nor useless,
as is demonstrated below,

For most human tasks, real numbers (or even
rational numbers) offer an adequate description
of data. Fractions such as 2/3 and 1/8  are
meaningless to a person counting stones, but
essential to a person comparing the sizes of
different collections of stones.

Negative numbers such as -3 and -5 are
meaningless when weighing the mass of an
object, but essential when keeping track of
monetary debits and credits.

Similarly, imaginary numbers have essential
concrete applications in a variety of sciences
and related areas such as signal processing,
control theory, electromagnetism, quantum
mechanics, cartography, and many others.
                                                -wikipedia



Despite their utility, it is the highly abstract
nature of imaginary numbers that we tend to
struggle with, to resist.

For most of us this level of abstraction represents
a barrier that separates us from the world of higher
mathematics; while for others, as shown in the video
below, the abstraction of imaginary numbers is "cool".

It represents the first real and welcome challenge
on the road to a career in mathematics, and more
importantly, to a glimpse at truths that few can
comprehend.






The soundtrack of the above is an excerpt from a
BBC Radio 4 5 Numbers podcast.

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