. But the entire aspect of the problem changed when Euler appealed to imaginary magnitudes, such as
; where
stands for
and
stands for any real number, as before. Of course an imaginary number, as its name indicates, was conceived of by its inventors as an unreal magnitude, a mere mathematical fiction having nothing but a symbolic significance. And yet, as we shall see, without imaginary numbers some of the most practical industrial problems could never have been solved.