This exploration leads calculus students, discussing the work in trios or quartets, to formulate Euler's Identity: e^(ipi)+1=0, a beautiful equation holding 5 important numbers (0,1,e,pi,i the imaginary root of -1) and the basic operations of adding, multiplying, and exponentiation.  When can you give this to students?  After they know the basic trig derivatives & quotient rule.  There are other lessons where you prove the same formula with infinite series for sin x & cos x, but this introduces the equation earlier in the course of calculus, to inspire students to study some of beautiful topics in Calculus C and Complex Analysis in college.  Hat tip to Dr. Roberto Martinez for showing me this proof, which I put in worksheet form.


  • Mathematics > General
  • Mathematics > Calculus
  • Mathematics > Trigonometry

Education Levels:

  • Grade 11
  • Grade 12
  • Higher Education
  • Graduate
  • Undergraduate-Upper Division
  • Undergraduate-Lower Division


e^(ipi)+1 trig identities quotient rule pi imaginary root of -1 complex numbers complex analysis



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