What Are Imaginary Volumes in Complex Numbers and Algebras?

[samsara15]

Imaginary numbers enable one to envision a lot of ideas. But what kind of numbers/algebras would enable us to work with imaginary volumes? Volumes, by definition, always seem to be positive, since any cubes are. What kind of numbers would give/allow a more complex picture?

 

[Svein]

The volumes you are familiar with exist in ℝ³. If you introduce imaginary numbers, the corresponding volumes would be in ℂ³. So a complex cube could be: Side "length": 1+ i. "Area": (1+i)*(1+i) = (1 + 2i + i²) = 2i. "Volume": (1+i)*(1+i)*(1+i) = 2i*(1+i) = -2+2i.