i to the i Power

Note: This page uses the HTML <sup> tag extensively. This tag may be supported only in Internet Explorer 3+, Netscape Navigator 4+, and Opera 3.5+. If you're using any other browser, the superscripts may not appear correctly, which will make this page difficult to read correctly. (Also, superscripted images are apparently not supported in IE5 under Windows, so the superscripted pi characters won't appear correctly in that configuration.) If you can't read this page, I apologize; I generally prefer my pages to be viewable on any platform, but in this case using superscripts makes the page vastly more readable for those who can view them.

Question: What is ii, expressed as a power of e?


Since we want to express it as a power of e, we can start by saying that

ex = ii

for some x. So

x = ln(ii)

x = i ln(i)

Now, i is the square root of -1, so

x = i ln(-11/2)

x = (i/2) ln(-1)

But as demonstrated in the sidebar,

ln(-1) = i(pi)


x = (i/2) . i(pi)

x = (i . i . (pi)) /2

but i . i is -1, so

x = -(pi)/2


ii = e-(pi)/2

(Note that this is a real number! It's about 0.208.)


Sidebar: What is ln(-1)?

Euler's Equation states that

ei(pi) + 1 = 0


ei(pi) = -1


i(pi) = ln(-1)


Thanks to Peter Hartman for showing me this elegant derivation.