e^(i*pi) + 1 = 0 Real or Imaginary
Name: Roger F.
There is still some debate. Is e^(i*pi) + 1 = 0 real?
Just would like some opinions.
Complex numbers are really "weird". The algebra of complex numbers is a
very powerful tool indispensable in many areas of physics, chemistry, and
mathematics. However, they are not intuitive. For example, the complex
quantity "i" raised to the "i-th" power: i^i = 0.207879576...,
which is real (it contains no "i"'s on the right hand side of the "=" mark.
It is probably irrational.
Regarding the almost mystical relation: e^(i*pi) + 1 = 0 (there really
is no debate). It is REAL.
Why? Because the right hand side of the equation is "0" which is clearly
real (it contains no "i's"). To further illustrate the breakdown of
"intuition", consider a rearrangement of:
e^(i*pi) + 1 = 0, specifically: e^(i*pi) = -1. It also turns out that:
e^-(i*pi) = -1
Note the sign of the exponent has changed signs. This is "obvious" HA!!
e^(i*pi) * e^-(i*pi) = (-1)*(-1) = +1 as it has to be to obey the
multiplication rules of exponents.
As an aside, I strongly recommend that any student considering a career
in science and/or engineering take a course (or courses) in complex number
theory after calculus. Without a good grasp of the algebra and calculus of
complex numbers it is difficult to make much progress in understanding
Whether you call it real depends on how you define the word "real". I
personally see it as using an imaginary expression to produce a real value.
A less confusing example is "i^4 - 1 = 0".
Dr. Ken Mellendorf
Illinois Central College
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Update: June 2012