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A math lesson from 4chan
(sh.itjust.works)
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this post was submitted on 11 Jan 2024
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The 180 deg rotation indicates multiplying by negatives. It’s a good analogy to represent change to the opposite side. Which multiplying with negatives does, the number goes from one side of 0 on the number line to the other side.
It was used in my. complex numbers class - multiply by i means rotate 90 degrees on complex plane
and multiplying by i twice means you rotate 90 degrees twice. So 180 degrees. Multiplying by i twice means multiplying by i^2 = -1.
So multiplying by -1 is a rotation by 180 degrees
I think the problem with this example is, while it provides a simple/visual thing to help people wrap their head around the idea of it, it still doesn't really explain it.
When you turn around, you are spinning around a central point, in a circular motion, and when you have a circle, you will always end up back at the start. You start at 0°, and if you keep adding degrees, eventually you will hit 365°, and one degree more puts you back to 0° off from where you started. You do not have to reverse direction to get back to 0°
But with negative numbers you're trying to explain a line, and that line goes in both directions infinitely. There is no point on the positive side of the line where you get teleported back to 0. You start at 0, keep adding numbers, and it's going to keep going down that line forever until you start subtracting.
The problem is usually more about the person not understanding what a negative number actually represents. They just think of them as regular numbers that happen to exist on the other side of 0 on the number line. So multiplying by a negative number gets treated the same as multiplying by a positive, just with a little dash next to the result.
Real numbers are 1-dimensional in that they all fit in just one continuous line, so there are only 2 directions you can face in that line - forwards and backwards - hence only 2 possible values for rotation (quite literally it's a binary option).
So when mapping the rotation around on a plane (i.e. a 2D rotation) to a 1D rotation, because the 2D rotation is a range of possibilities you need to pick 2 and only 2 positions out of the infinite possibilities and they both must obbey they rule that you rotate in 2D from one to the other one you end up facing the opposite direction.
As it so happens any 2 numbers for a 2D angle of rotation that differ by 180 degrees or PI radians obbey both rules so any such pair of numbers can be used. Because in 2D rotations, there is the property that any rotation angle is equivalent to any angle which differs from it by a multiple of +/- 360° that gives you more rotation value pairs which have different numbers but represent the same rotation.
For simplicity, the tendency is to use 0° and 180°, but anything that obbeys the rules above would work, say -270 and +90, or 96 and 276 as long as you can form a straight line in the 2D plane for that rotation passing both angles and the center of rotation you can use them to express 1D rotations.