Two versions of modulo arithmetic: On the clock and on the complex unit circle.

 

We are used to the clock face, though I wonder for how much longer with so many people using their phones as time pieces. We have the numbers 1 through 12, with 12 at the "high noon" position, and also 60 hash marks for counting both minutes and seconds. The idea of clockwise movement comes from Northern Hemisphere sundials, because a Southern Hemisphere sundial travels counter-clockwise, the 12 being pointed due south instead of due north.

 

60 minutes is a remnant of the base 60 number systems used first in ancient Sumeria, then Babylon and passed down to us in how we cut up circles into 360 degrees.

 

  

Counting on the complex unit circle starts at 1, rightmost point on the real axis. Multiplying complex numbers has the physical representation of multiply the lengths and adding the angles.

 

On the complex unit circle, all lengths are 1, so it is just about adding the angles, where we count the positive x-axis as 0. The unit circle means the radius is 1, the diameter is 2 and the circumference is 2𝝅. 

 

I have labeled four of the points as 1, -1, i and -i, as they lie exactly on the real axis or the imaginary axis. The other eight points are given in the form e^(ni𝝅/6), where n is a number between 1 and 11, and if n = 2k, the form is e^(ki𝝅/3). When n = 12, we get e^(2i𝝅), which means we have gone full circle, and can identify the point with e^0 = 1.

 

Most people learn degrees long before they learn radians, so degrees feel more "natural" to us, but 360 degrees was an arbitrary decision based on how many divisors 360 has.

 

Divisors of 360: 1 & 360, 2 & 180, 3 & 120, 4 & 90, 5 & 72, 6 & 60, 8 & 45, 

9 & 40, 10 & 36, 12 & 30, 15 & 24, 18 & 20 

 

Let's use the definition of natural that means "based on nature". While it is a human concept, 𝝅 = circumference/diameter, and those two measurements are based on the nature of circles, so I would argue that radians are more natural than degrees. While perfect circles are hard to find in nature, if you pour a little oil into a still pan of water, the oil droplets will do their best to form circles, because a circle is the two-dimensional shape maximizes surface area and minimizes the perimeter. The surface area will be 𝝅r² and the perimeter 2𝝅r. No other shape will give us a better ratio.

 

Whether we start at the top and move clockwise around the circle starting at 12 or start at the right and move counterclockwise starting at 0, these are both representations of ℤ12, the most commonly used cyclic group around the world.