Imaginary Power

This blog has included many articles on imperial economic and political power, a reality that must be understood in order to make sense of the world. As something to consider as a contrast over the holiday period, following is a note on imaginary power taken from the work of Swiss genius, Leonhard Euler.

Firstly, start with Euler's famous formula:

e^ix = cosx + i sinx

For x = π, this produces the even more striking result

e^iπ + 1 = 0

(Nobody has constructed a simpler or more concise - some would say beautiful - equation, one that includes so many of the fundamental operations and core elements of mathematics: unity, zero, e, π, i, equality, addition, exponentiation and multiplication)

But, for x = π/2, Euler's formula can be used to answer the question: what is i, the so-called imaginary unit (the square root of -1), to the power of i?

In other words, what is iⁱ? This is a particularly weird question if translated as 'an imaginary number to the power of an imaginary number', stranger perhaps than asking: what colour is 5.6 and is it different from the flavour of 6.5?

Using x = π/2, Euler's formula becomes

e^iπ/2    = cos π/2 + i sin π/2

          = 0 + i

so

e ^iπ/2 = i

in which case, raising both sides of the equation to the power i gives

iⁱ        = (e^iπ/2)ⁱ

          = e^{i.i. π/2}

          = e^-π/2

So, it turns out that

          iⁱ = 0.2079 (to 4 decimal places)

and it is a real (and a positive) number.

But that is not the end of the process, because i = e^iπ/2 is just one of an infinite number of values for the equation

e^ix = cosx + i sinx

since sin π/2 , sin 5π/2 , sin 9π/2 , etc, are also equal to 1, while the respective cosx values are all equal to 0 (note these are measured in terms of radians, and that π/2 radians is 90 degrees).

So the general equation for i is:

i = e^i(4n+1)π/2

For example, when

          n= 0,             iⁱ = 0.2079…

          n = -1           iⁱ = 111.3178…

          n = 1            iⁱ = 3.3882… x 10^-4

etc, for positive and negative integer values of n

This odd result follows from the nature of i. I think that it is best to think of i as a mathematical operator that, when applied to itself squared, as  (i x i) or i², results in minus 1, rather than in the unnecessarily mysterious form of an 'imaginary' unit or number.

Perhaps surprisingly, Euler has not been considered to be one of the founders of Rastafarianism, despite his understanding of i to the power of i.

Tony Norfield, 14 December 2013


Note added on 4 July 2014: there are several ways of deriving the above results, some of which can be found on Wikipedia and in other Internet sources. However, the method used above was taken from a book by Y E O Adrian, The Pleasures of Pi, e and Other Interesting Numbers, World Scientific Publishing, 2006, pp. 205-207.