The complex numbers form a ring that is isomorphic to the cross product of the ring of real numbers and itself....
C≅R×R
For those who haven't done abstract algebra:
This means your can map a complex number to (x,y), that every complex number maps to a unique (x,y), and that if you add or multiply two complex numbers and then convert the result to (x,y), you can convert the two complex numbers to (x,y) and then multiply getting the same answer. (it is also bijective, meaning it works both ways).
Importance: Complex numbers can be dealt with as coordinates and coordinates can be dealt with as complex numbers.
That's not true: the multiplication is different. (x1,y1) × (x2,y2) would have to be (x1x2 - y1y2, x1y2 + y1x2), not (x1y1, x2y2).
The addition does work as you suggest, so a more correct formal statement would be: the complex numbers form an additive group that is isomorphic to the cartesian product of the additive group of real numbers and itself; the complex numbers also happen to form a ring (indeed a field).
The complex numbers form a ring that is isomorphic to the cross product of the ring of real numbers and itself....
C≅R×R
For those who haven't done abstract algebra:
This means your can map a complex number to (x,y), that every complex number maps to a unique (x,y), and that if you add or multiply two complex numbers and then convert the result to (x,y), you can convert the two complex numbers to (x,y) and then multiply getting the same answer. (it is also bijective, meaning it works both ways).
Importance: Complex numbers can be dealt with as coordinates and coordinates can be dealt with as complex numbers.