Incredible Numbers Incredibly Close to Modest Rationals: Liouville’s Theorem and the transcendence of \(\sum\nolimits_{n = 1}^\infty {{{10}^{ - n!}}} \)
  • Edward B. Burger
  • Robert Tubbs


As we begin our journey into the theory of transcendental numbers, we are immediately faced with a nearly insurmountable obstacle: A transcendental number is defined not by what it is but rather by what it is not. What will become apparent as we develop the classical theory of transcendental numbers is that every demonstration of the transcendence of a particular number is indirect—a number is shown to be transcendental by showing that it is not algebraic.


Rational Number Rational Approximation Algebraic Number Minimal Polynomial Decimal Digit 
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Copyright information

© E.B. Burger and R. Tubbs 2004

Authors and Affiliations

  • Edward B. Burger
    • 1
  • Robert Tubbs
    • 2
  1. 1.Department of MathematicsWilliams CollegeWilliamstownUSA
  2. 2.Department of MathematicsUniversity of Colorado at BoulderBoulderUSA

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