Treasure Hunts
e^ipi = -1 So What?

Introduction

Now we take out our calculators and push sin 3 and the calculator displays ____________. How does the calculator do this? Sine, cosine, natural log, tangent and log are all buttons. What does the calculator have to be programmed to do? It is a little known fact except amongst struggling BC calculus students that infinite series such as x-(x^3/3!) + (x^5/5!)+... are used to approximate functions such as the sine and obtain their values.

Guiding Question

1. What do calculators use to obtain these values of transcendental and periodic functions? Can you do it without a calculator?
2. e^ipi = -1 Why is this of any importance or even of interest?

Questions

  1. What do calculators use to calculate values of periodic functions such as the sine?
  2. What do calculators use to calculate values of transcendental functions such as log x?
  3. How can we get more accurate values for these functions?
  4. What is e? What is i? What is pi?
  5. Why is e^ipi + 1 = 0? Can it be proven?
  6. Does all of this tie together in some way?
  7. Why is the equation e^ipi + 1 = 0 interesting at all?


Internet Resources

e^(i Pi) = -1 Euler's formula
Ask Dr. Math
Taylor Series
Taylor Series #2
Significance of the number e
Information on imaginary numbers
e^(i Pi) + 1 = 0
Question Corner about e^(i pi)

Activities

  1. Research and describe series to approximate sine, cosine, tangent, ln, and e^x.
  2. Write a paragraph explaining the relation between series and functions
  3. Research e^ipi + 1 = 0 and write your own conclusion.
  4. Answer all the questions above

Tom Swartz, Teacher, and Billy Cheung, AP student, collaborated to develop this web hunt.
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