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:

eit = cos(t) + i·sin(t)

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

cos(p/2) = 0 and sin(p/2) = 1,

we get

i = eip/2

From here, i i = ei·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 = ei·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)

Raise both sides of the equation j = exp(jp/2) to the j-th power. The result is jj = exp(-p/2) and, quite unexpectedly, jj is real. A Frenchman once called this formula "la plus belle formule de la mathématique".

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:

Imagine a person with a gift of ridicule. [He might say] First that a negative quantity has no logarithm; secondly that a negative quantity has no square root; thirdly that the first nonexistent is to the second as the circumference of a circle to the diameter.

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

i i ≈ 0.207 879 576 350 761 908 546 955 ...

Subject: Some ponderings
Date: Thu, 17 Jun 2000 01:32:31 +0200
From: Tobias Brandt


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

ep + 1 = 0;

since this implies that

ep = -1.

Taking square roots on both sides

ep/2 = (-1)(1/2)

in other words

ep/2 = i

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

i i = [ ep/2 ]i = e-p/2

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 ep + 1 = 0 just connected this new fact with something I have seen as rather mystical for a long time.

Subject: Infinite number of values
Date: Mon, 01 Jan 2001 01:57:22 -0800
From: Robert D. Rasmussen


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

t = p·(2·n + 1/2)

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

i = ep·(2·n + 1/2) ,

from which it follows that

i i = e -p·(2·n + 1/2) .

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

x1/2 = ±sqrt(x)

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

we mean that x1/2 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 xy 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 0y = 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 xy has.

I'll use the formula

ln x = ln |x| + i·(2·n·p + arg(x) )

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:

e ln x = e ln |x| + i·(2·n·p + arg(x) )
 = |x| e i·2·n·p ei arg(x)
 = |x| e i·arg(x)
 = |x| cos(arg(x)) + i·|x| sin(arg(x))
 = Re(x) + i·Im(x)
 = x

Now back to xy.  We have

xy = e y·ln  x
  = e y·(ln |x| + i·(2·n·p + arg(x) ) )
  = e y·(ln |x| + i·arg(x) ) e i·y·(2·n·p)

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 xy 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

Subject: Erratum in "Complex number to complex power ..."
Date: Sat, 13 Apr 2002 22:35:02 EDT
From: David Cantrell


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 xy 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 0y = 0 for any y (but 0)."

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

 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. Plane Isometries As Complex Functions
  9. Remarks on the History of Complex Numbers
  10. Complex Numbers: an Interactive Gizmo
  11. Cartesian Coordinate System
  12. Fundamental Theorem of Algebra
  13. Complex Number To a Complex Power May Be Real
  14. One can't compare two complex numbers
  15. Riemann Sphere and Möbius Transformation
  16. Problems

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