/sci/ - Science & Math

>>15567861
>>15568838
Continued:

Indeed, it turns out that the complex numbers perfectly fill a plane in the same way the "reals" perfectly fill a line. But unlike a Cartesian plane where you make two real number lines perpendicular, the complex numbers have the added benefit that rotations (necessarily, transformations *between* the x and y terms) are encoded in the pure algebra. Multiplying by 1 is a 0° rotation (multiplying by the identity changes nothing). Doing it with -1 is 180°, as discussed before. Necessarily, because i*i = -1, multiplying i must correspond to exactly half a 180° rotation, ie: 90°. It's standard to go counterclockwise.
Thus, the expression i*i = -1 is logically equivalent to saying "two left turns face you backwards". In the same way -1 can mean "backwards," i just means "left."

If the idea of "left" apples strikes you as odd, it should. But no more than "negative" apples. But I want to stress the point that none of this is "forced." It's very natural and actually quite simple once you get the hang of it. It's one of those constructions that's so elegant it looks more like a discovery than an invention.

>>15516946
Imagine being filtered by the concept of "left."