If you think about a number line, multiplying 2 by -1 takes you to -2. Multiply it again by -1 and its back at 2.
If you think of the arrow from 0 to 2, all you did was rotate that arrow by 180 degrees to point along the negative axis and back again.
Multiplication by -1 is already a rotation of 180 degrees!
All were doing now is extending that concept to 90 degrees by imagining a second line perpendicular to the original number line.
Two 90 degree rotations need to get to -1 to complete the 180 degree rotation we already expect in normal multiplication.
Giving it the symbol i, this means definitionally i * i = -1. It has to because -1 flips us around the other way on the number line.
That means i is the square root of negative 1.
Any values that use i to store information, even time, could be called “imaginary time”. Really it’s just constantly oscillating between the real and imaginary spaces like a constantly spinning arrow.
ComicalMayhem@lemmy.world 1 month ago
waves are related to circles: if you have a line and anchor it at one end, when you rotate it the other end of the line, it draws a circle, but if the paper you’re drawing it on moves to one side at a constant speed, you’ll get a wave. Alternatively, if you plot where the other end of the line is as time passes (for example, every second or every minute), you’ll get a wave. you can do this in reverse too.
it’s helpful to convert to circles. from a regular wave, at 0 you don’t know if the wave will go up or down without further information. 0 on a circle will correspond to one of two spots, either the very top or the very bottom, and if you know which direction the circle is rotating, you can tell what the related wave will do next.