The meaning of i^i

Imaginary numbers, like most things at the peripherals of our grasp of reality, interest me, since they appear to have their own rules at times. There was a video blog some time ago that tried to reckon with \large i^i and considered what it equalled.

One calculation I observed came from De Moivre’s identity, \large e^{i\theta} = \cos\theta + i \sin\theta. If you let \theta = \frac{\pi}{2}, you get e^{i\frac{\pi}{2}} = 0 + i = i.

Now raising both sides of the equation e^{i\frac{\pi}{2}} = i to the power i, you obtain the result: e^{i\cdot i\frac{\pi}{2}} = i^i, and, since i\cdot i = i^2 = -1, you get: e^{\frac{-\pi}{2}} = i^i.

This is a notable equation because the left hand side are all real numbers, and the right hand side are all complex. The result of the right-hand side is about 0.20788.

But did this crack that mystery? Well, if you consider the origin of the derivation was from De Moivre’s identity, then it would just compel one to think that not only is \frac{\pi}{2} a solution, but all angles co-terminal to it. i^i therefore has infinitely many answers if we are to rely on De Moivre’s Theorem.

e to the Pie Eye

Not knowing enough about the math of complex numbers, I find the equation eπi = -1 intriguing. This is apparently the famous Euler’s Identity. I mean, how exactly do they get a real number out of an imaginary one without squaring? I had gotten some insight by playing with my calculator. I have a good one which is able to compute e^{\pi i} without choking on it. It helps that it understands complex math, like many new calculators do these days.

So, I tried something using an idea such as  \lim_{x \to e} x^{\pi i}, where x was allowed to approach Euler’s constant. The way I “approached” e was a tad kludgy, but useful: I began with a “2”, computed the result, then “2.7”, adding the next successive decimal, computed that, and kept the process going for several more decimal places. My results could be tabulated as:

x Result
2 -.5702332490 + 0.8214828311 i
2.7 -.9997752846 + 0.02119859190 i
2.71 -.9999540533 + 0.009585998981 i
2.718 -.9999999469 + 0.0003257333283 i
2.7182 -.9999999955 + 0.00009457240125 i
2.71828 -1.000000000 + 0.000002112790593 i

So, as you can probably suss out, as we add successive digits to the base the base approaches e, the real part of the number becomes -1, while the imaginary part of the number slowly melts away to nothing. By the time we get to 2.718281828 (the value of e seen on most calculator displays), we get the number: -1.000000000 + 0.0000000001202932385 i. Like many things we associate with the understanding of Euler’s constant, it seems the limit (in my case, in the form of playing with my calculator) is one approach to understanding Euler’s identity. There is another, better approach.

Of couse, the next lesson I feel the need to proceed to, is, what the heck is all this about e^{\theta i} = \cos \theta + i \sin \theta?