A geometric interpretation of arctan.

The other day I was looking at a string of calculations which were supposed to come out to zero. Instead, the conclusion of these calculations was written as:

Realizing that not only must this be wrong, but *severely* wrong, I plug it into a calculator just to make sure. The results are:

1. Via google.com: 0.

2. Via wolframalpha.com: -5.551115123125783×10^{-17} is the answer given while typing it in, though after hitting “enter”, we get 0, but we also get a whole bunch of series expansions and continued fractions, which is odd.

3. Via MATLAB: -5.551115123125783e-17

4. Via Python: -5.5511151231257827e-17, which is just one extra digit on wolfram and MATLAB’s output.

First, the right answer is in fact 0. Remember that *arctan(x) *takes the ratio of the opposite and adjacent sides of a right triangle (remember, side-angle-side completely determines a triangle), and returns the angle of the triangle, in radians. Then it follows that since a triangle has pi radians total, and the right angle uses up pi/2 of those radians, we have the following identity for all nonzero *x*:

It follows, then, that

though this is one of the most… interesting ways I have ever seen the number 0 written.

Tomorrow: more on why the calculators were wrong, and a rough discussion of exactly how computers calculate with floating point numbers.

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