2011-05-18T12:54:13Z Photos, electronics, cnc, and more Jeff Eplerjepler@unpythonic.net 2011-05-18T12:54:13Z 2011-05-18T12:54:13Z https://emergent.unpythonic.net/01305723253 From time to time, I've written little programs to find <a href="http://mathworld.wolfram.com/AlmostInteger.html">almost-integers</a> in certain forms. <p>Most recently, I hit upon the form <i>tan(i)/tan(j)</i> where <i>i</i> and <i>j</i> are both integers. I was at first surprised to see that there were in fact a lot of the form <i>tan(2x)/tan(x) ≅ 2</i>, but a little thought revealed why: These were values of <i>x</i> that were just above multiples of π. <p>Example: <pre> &gt;&gt;&gt; x=355; tan(2*x)/tan(x); x/pi 2.000000001817364 113.00000959524569 </pre> <p>Reduce <i>x</i> modulo π, and the formula becomes <i>tan(2ϵ)/tan(ϵ) ≅ 2ϵ/ϵ ≅ 2</i>. Mystery solved.