So, I was completing a unit circle and I noticed that every angle that had pi as a factor in the numerator, i.e. 3pi/6 (or pi/3), for ALL denominators of 1, 2, 3, 4, and 6, ROUNDED UP. I plugged the numbers into excel and used the 'mid' and 'find' functions to pull up the 5th decimal place to the right. Also, Excel uses Pi to 15 decimal places. For ALL the standard angles that number turned out to be an 8 or a 9.
I decided to try pi factors with denominators of 5, and things got really interesting.
I grant that some of these numbers are repeated; 3pi/3=4pi/4=6pi/6=pi, etc.
| Leading | Pi | Over | Result | 5th Decimal Place |
|---|---|---|---|---|
| 1 | 3.141592654 | 6 | 0.523598776 | 9 |
| 2 | 3.141592654 | 6 | 1.047197551 | 9 |
| 3 | 3.141592654 | 6 | 1.570796327 | 9 |
| 4 | 3.141592654 | 6 | 2.094395102 | 9 |
| 5 | 3.141592654 | 6 | 2.617993878 | 9 |
| 6 | 3.141592654 | 6 | 3.141592654 | 9 |
| 7 | 3.141592654 | 6 | 3.665191429 | 9 |
| 8 | 3.141592654 | 6 | 4.188790205 | 9 |
| 9 | 3.141592654 | 6 | 4.71238898 | 8 |
| 10 | 3.141592654 | 6 | 5.235987756 | 8 |
| 11 | 3.141592654 | 6 | 5.759586532 | 8 |
| 12 | 3.141592654 | 6 | 6.283185307 | 8 |
| 1 | 3.141592654 | 4 | 0.785398163 | 9 |
| 2 | 3.141592654 | 4 | 1.570796327 | 9 |
| 3 | 3.141592654 | 4 | 2.35619449 | 9 |
| 4 | 3.141592654 | 4 | 3.141592654 | 9 |
| 5 | 3.141592654 | 4 | 3.926990817 | 9 |
| 6 | 3.141592654 | 4 | 4.71238898 | 8 |
| 7 | 3.141592654 | 4 | 5.497787144 | 8 |
| 8 | 3.141592654 | 4 | 6.283185307 | 8 |
| 1 | 3.141592654 | 3 | 1.047197551 | 9 |
| 2 | 3.141592654 | 3 | 2.094395102 | 9 |
| 3 | 3.141592654 | 3 | 3.141592654 | 9 |
| 4 | 3.141592654 | 3 | 4.188790205 | 9 |
| 5 | 3.141592654 | 3 | 5.235987756 | 8 |
| 6 | 3.141592654 | 3 | 6.283185307 | 8 |
| 1 | 3.141592654 | 2 | 1.570796327 | 9 |
| 2 | 3.141592654 | 2 | 3.141592654 | 9 |
| 3 | 3.141592654 | 2 | 4.71238898 | 8 |
| 4 | 3.141592654 | 2 | 6.283185307 | 8 |
| 1 | 3.141592654 | 1 | 3.141592654 | 9 |
| 1 | 3.141592654 | 5 | 0.628318531 | 1 |
| 2 | 3.141592654 | 5 | 1.256637061 | 3 |
| 3 | 3.141592654 | 5 | 1.884955592 | 5 |
| 4 | 3.141592654 | 5 | 2.513274123 | 7 |
| 5 | 3.141592654 | 5 | 3.141592654 | 9 |
| 6 | 3.141592654 | 5 | 3.769911184 | 1 |
| 7 | 3.141592654 | 5 | 4.398229715 | 2 |
| 8 | 3.141592654 | 5 | 5.026548246 | 4 |
| 9 | 3.141592654 | 5 | 5.654866776 | 6 |
| 10 | 3.141592654 | 5 | 6.283185307 | 8 |