This is the answer to a common question about why we need to define the square root of -1 at all.
Well the answer rests on the fact that you want all algebraic equations to be solvable. If you consider the set of all real numbers, equations such as
x 2 + 3 = 5
can be solved. The solution above is of course sqrt(2).
However if you want all algebraic equations to be solvable the real number system is insufficient. For example the equation:
x 2 + 1 = 0
does not have a real
solution. It turns out that the only addition you need to make to the
real number system is to introduce this funny quantity called
i and say that numbers of the form
a + i*b are to be called complex numbers. With this addition
every algebraic equation is solvable.