>>13756511The imaginary unit is the archetypal example of how to use group theory to extend the number line.
The number line itself is bidirectional, so we're always talking about groups where the cyclic group of order 2 is a normal subgroup. After itself, the next smallest example is , which gives rise to the complex numbers. You can also pick [spoiler:lit]C_2 \times C_2[/spoiler:lit] and get the split-complex or hyperbolic numbers, but nobody cares about those. Skipping over 6, when you have a group of order 8, you have three best choices for how to extend your field: (boring), quaternions (adding in weird new elements j and k which also square to -1), or the group which represents rotations of a square (i squares to -1, j and k square to 1).
