Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne e^(i*pi) + 1 = 0 Real or Imaginary
Name: Roger F.
Status: student	
Age:  N/A
Location: N/A
Country: N/A
Date: 8/4/2004

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!! since: 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 beyond calculus.

Vince Calder

Roger F.,

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
Physics Instructor
Illinois Central College

Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (, or at Argonne's Educational Programs

Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory