Math 2.0

Number theory Theorem

Irrationality of e

Builds on eSum of the reciprocals of the factorials

Statement

The number ee is irrational.

Remarks

Assume e=a/be = a/b with positive integers a,ba, b. Then x=b!(e−∑n=0b1n!)x = b! \big( e - \sum_{n=0}^{b} \frac{1}{n!} \big) is an integer. But the series for ee gives 0<x<∑k=1∞(b+1)−k=1b≤10 < x < \sum_{k=1}^{\infty} (b + 1)^{-k} = \frac{1}{b} \le 1. No integer lies strictly between 00 and 11. This short proof is due to Joseph Fourier.

History

Leonhard Euler wrote the first proof in 1737. It was published seven years later.

Reading path

To understand this page from the ground up, read these 28 pages in order. The foundations come first.

Show the path as a graph
Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionSum of a finite geometric series, statement, no evidence levelSum of a finite geome…statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionSqueeze theorem, statement, no evidence levelSqueeze theoremstatementPartial sum, definition, no evidence levelPartial sumdefinitionPower series, definition, no evidence levelPower seriesdefinitionThe sequence r^n tends to zero for |r| < 1, example, no evidence levelThe sequence r^n tend…exampleRadius of convergence, definition, no evidence levelRadius of convergencedefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionSum of a series, definition, no evidence levelSum of a seriesdefinitionComparison test, statement, no evidence levelComparison teststatementGeometric series, example, no evidence levelGeometric seriesexampleRatio test, statement, no evidence levelRatio teststatementRatio formula for the radius of convergence, statement, no evidence levelRatio formula for the…statementThe exponential series has infinite radius of convergence, statement, no evidence levelThe exponential serie…statementExponential function, definition, no evidence levelExponential functiondefinitione, object, no evidence leveleobjectSum of the reciprocals of the factorials, example, no evidence levelSum of the reciprocal…exampleIrrationality of e, statement, no evidence levelIrrationality of estatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Geometric sequence definition
  5. Limit of a sequence definition
  6. Monotone sequence definition
  7. Series definition
  8. Sum of a finite geometric series statement
  9. Monotone convergence theorem for sequences statement
  10. Convergent sequence definition
  11. Squeeze theorem statement
  12. Partial sum definition
  13. Power series definition
  14. The sequence r^n tends to zero for |r| < 1 example
  15. Radius of convergence definition
  16. Convergent series definition
  17. A series of nonnegative terms converges iff its partial sums are bounded statement
  18. Absolute convergence definition
  19. Sum of a series definition
  20. Comparison test statement
  21. Geometric series example
  22. Ratio test statement
  23. Ratio formula for the radius of convergence statement
  24. The exponential series has infinite radius of convergence statement
  25. Exponential function definition
  26. e object
  27. Sum of the reciprocals of the factorials example
  28. Irrationality of e statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Proof_that_e_is_irrational, Fourier's proof
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Usese, Sum of the reciprocals of the factorials

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