Complex number to a complex power may be real

Not only that, but if we look at the quintessential complex number, the imaginary i, then you can verify that i ⁱ is real.

To prove this, recollect Euler's formula:

where e is, of course, the base of natural logarithms. Substitute t = p/2. Since

we get

From here, i ⁱ = e^i·i·p/2 = e^-p/2, which is not large but still very real.

(For a more serious discussion of the properties of complex numbers, see the collection of pages starting with Algebraic Structures of Complex Numbers.)

As can be seen from the correspondence below, the assertion that i ⁱ = e^i·i·p/2 = e^-p/2 is not 100% accurate; it was not meant to be. On page 15 of C. Zwikker's classic The Advanced Geometry of Plane Curves and Their Applications, we read (he uses j instead of i, as is customary in engineering science)

To my chagrin, I could not identify this person of taste.

In a more recent, but no less distinguished Impossible? by J. Havil, we find a quote from the famous Augustus De Morgan:

The quote serves to introduce The Power of Complex Numbers chapter, where the author also refers to a 1921 article from American Mathematical Monthly (28 (3), pp. 114-116) by H. S. Uhler who gave an approximate value for the constant

Hi,

congratulations on an excellent web-site. I just have some thoughts to add to the contents of the page entitled "Complex number to a complex power may be real". I enjoyed reading that article and was quite fascinated by the fact that i ⁱ = e^{- p/2}. I mean there is nothing magical in the derivation but these curious results of complex numbers always seem awe inspiring. After some thinking I realized that this is actually just another formulation in disguise of an expression which has dazzled me for years, namely

since this implies that

Taking square roots on both sides

in other words

Taking the i-th power on both sides yields the sought result

Cheers
Tobias Brandt

P.S. Looking over this again I realized that this is essentially the same as you had done. I'm sorry to waste your time but to me the explicit statement of e^i·p + 1 = 0 just connected this new fact with something I have seen as rather mystical for a long time.

Alexander,

I've been enjoying your web site.  Well done!

Your page titled "Complex number to a complex power may be real" caught my attention, because the example you use, while simple, is still not well known. What is even less well known is that i ⁱ actually has an infinite number of real values. This follows by choosing the more general value of

in your derivation, where n may be any integer. The result is

from which it follows that

Each value of n produces a different real value.

So which value for n is right? They all are. Complex valued expressions often have the property of being multiple valued. We typically adopt a so-called "principle value" for practical purposes (the value for n = 0 in this case). People familiar with functions like arcsine will recognize this.

It's important to note, however, that even simple real valued expressions can have multiple values. When we write

(sqrt() always takes the principle, positive value),

we mean that x^1/2 has two values. The expression i ⁱ just happens to have infinitely many values.

These two examples of powers, and the number of values that result, are actually very closely related. In fact, they both arise from the same source. To see this, consider the expression x^y in general, where x and y are any two complex numbers. How many values does it have?

When x = 0, the result is trivial. There is only one value, since 0^y = 0 for any y (but 0).

For non-zero x, a method similar to that used for i ⁱ can be used to find out how many values x^y has.

I'll use the formula

where ln |x| is the real-valued natural log of the real number |x|, the magnitude of x. arg(x) is the angle of the ray from the origin to x in the complex plane relative to the positive real axis.

This new formula is just an application of Euler's formula in reverse, as seen here:

Now back to x^y.  We have

There are two terms in this result. The first exponential gives a single complex number. It can be evaluated further using Euler's formula. The second term, in general, has infinitely many values, as before. Therefore, the product of these two terms has infinitely many values.

But notice what happens in the special case where y is real and rational. In this case, the sequence of values with respect to n eventually repeats after a finite number of steps (predicting how many leads to other interesting questions). There are only a finite number of values for x^y in this case.

One example of the special case occurs with y = 1/2 (the square root, again). The second term above alternates between +1 and -1, yielding the two values of opposite sign (infinitely repeated) that we all know, of course. Isn't that interesting?

Bob Rasmussen

Hello,

Congratulations on your excellent site!

While browsing, I found a mistake which I thought you might wish to correct. In the article "Complex number to a complex power may be real", Bob Rasmussen said "To see this, consider the expression x^y in general, where x and y are any two complex numbers. How many values does it have?

When x = 0, the result is trivial. There is only one value, since 0^y = 0 for any y (but 0)."

Although it may be trivial, I'm certain that we do not want to claim that 0^y = 0 if, say, y = -1. However, I believe that a reasonable argument can be given for taking 0^y = 0 when the real part of y is positive.

Regards,
 David Cantrell

Complex Numbers

1. Algebraic Structure of Complex Numbers
2. Division of Complex Numbers
3. Useful Identities Among Complex Numbers
4. Useful Inequalities Among Complex Numbers
5. Trigonometric Form of Complex Numbers
6. Real and Complex Products of Complex Numbers
7. Complex Numbers and Geometry
8. Remarks on the History of Complex Numbers
   • First Geometric Interpretation of Negative and Complex Numbers
9. Complex Numbers: A Dynamic Tool
10. Cartesian Coordinate System
11. Fundamental Theorem of Algebra
12. Complex Number To a Complex Power May Be Real
13. One Can't Compare Two Complex Numbers
14. Problems
    • Product of Diagonals in Regular N-gon