e^(i*pi) = -1: pi = 0 ?

Date: 10/17/97 at 11:29:15
From: John K. Koehler
Subject: Weird tricks

Dr. Math,

I know that from a certain trig identity we get the equation 
e^(i*pi) = -1. I have been playing around with this equation and have 
found some disturbing things. I hope that you can help me.

  First:

     e^(2*i*pi) = 1
      e^(-2*pi) = 1  (raised both sides to i and 1^n is 1)
          -2*pi = 0 (took the ln of both sides and ln(1) = 0)
             pi = 0 !

  Second:

           i*pi = ln(-1)   {1}
         ln(-n) = ln(-1) + ln(n) 
     therefore by {1} 
         ln(-n) = i*pi + ln(n)

  Finally:

            i^2 = -1 
       e^(i*pi) = i^2
        e^(-pi) = i^(2i)
      e^(-pi/2) = i^i

      e^(-pi/2) is a real number and therefore so is i^i

Can you please tell me why I get such ridiculus results?  
I asked my calc2 teacher last year - but all he could tell me 
was that imaginary numbers don't work like reals, and I still 
don't see how that would change anything.

Thanks in advance,
John

Date: 10/17/97 at 13:08:36
From: Doctor Bombelli
Subject: Re: Weird tricks

Well, John, not all the results you have are ridiculous!  In fact, you 
and your teacher are both right; complex numbers don't always work 
like real numbers, but sometimes they do.

Complex numbers can be written in many equivalent ways:

   z = x+iy
   z = re^*(it)  where t is the angle the point (x,y) makes with the 
                 positive x axis.
   t is called the argument of z--arg(z)
   z = r[cos(t)+i sin(t)]

In fact, the definition of e^(it) is cos(t)+i sin(t).

This is why we have e^(-ipi) = cos(pi)+i sin(pi)= -1.

We would like w = log(z) and z = e^w to match up, just as for real 
numbers.  

Let z = re^(it) and w = u+iv.

    z = e^w = e^(u+iv) = e^u*e^(iv) and z is also re(^it)

Match the real part and the "i part" (the imaginary part).

    r = e^u and e^(it) = e^(iv)

So  u = ln(r) and v can be any value with cos(v) = cos(t) 
[since e^(it) is cos(t)+i sin(t)].

So we say, for complex numbers, ln(z)=ln(r)+i arg(z)+i2kpi  
(k an integer). In fact, ln(z) has many values.

We also define, in this way, that

  z^a = e^[a ln(z)] when a is complex.  

(Check that this works for real numbers a, also.)

Note that 1^a = e^[ln(1)] = e^[0+i2kpi], so only one of the answers 
is e^0 = 1.  

Here is your problem:

     e^(2*i*pi) = 1
      e^(-2*pi) = 1  (raised both sides to i and 1^n is 1)
          -2*pi = 0 (took the ln of both sides and ln(1) = 0)
             pi = 0

In steps 2 and 3 you don't have one-to-one functions any more.  
(The reason e^x = 1 implies x = 0 is because the real logarithm 
function is one-to-one.  It is kind of like why x^3 = 1 gives x = 1 
and x^2 = 1 gives x = 1 or -1.  The cube root is one-to-one, but 
the square root isn't.)

Now in your second experiment:

           i*pi = ln(-1)   {1}
         ln(-n) = ln(-1) + ln(n) 
     therefore by {1} 
         ln(-n) = i*pi + ln(n)

This is okay.
ln(-n) = ln(n) + i pi + i 2kpi from the definition above.

In your third experiment:

            i^2 = -1 
       e^(i*pi) = i^2
        e^(-pi) = i^(2i)
      e^(-pi/2) = i^i

      e^(-pi/2) is a real number and therefore so is i^i

i^i is e^[i ln(i)], and ln(i) is ln(1) + i pi/2 + i 2kpi

So...  i^i = e^[0 + i^2 pi/2+ i^2 2kpi] = e^[- pi/2 - 2kpi]  
which is a real number.

I commend you on playing around with this stuff.  That is how new 
mathematics is learned, on all levels.

-Doctor Bombelli,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/