Math 2.0

Number theory Theorem

Transcendence of pi

Builds on Transcendental element

Statement

The number π\pi is transcendental over ℚ\mathbb{Q}.

Remarks

The statement follows from the Lindemann–Weierstrass theorem: eαe^\alpha is transcendental for every nonzero algebraic number α\alpha. Indeed, if π\pi were algebraic, then iπi \pi would be algebraic and nonzero, but eiπ=−1e^{i \pi} = -1 is algebraic. So π\sqrt{\pi} is also transcendental, and it is not constructible.

History

Charles Hermite proved the transcendence of ee in 1873. Ferdinand von Lindemann proved in 1882 that eαe^\alpha is transcendental for every nonzero algebraic α\alpha, and so that π\pi is transcendental. Karl Weierstrass proved a more general statement in 1885.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionField extension, definition, no evidence levelField extensiondefinitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionTranscendental element, definition, no evidence levelTranscendental elementdefinitionTranscendence of pi, statement, no evidence levelTranscendence of pistatement
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. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Subring definition
  6. Unit of a ring definition
  7. Polynomial evaluation definition
  8. Field definition
  9. Root of a polynomial definition
  10. Subfield definition
  11. Field extension definition
  12. Algebraic element definition
  13. Transcendental element definition
  14. Transcendence of pi statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem, Transcendence of e and π
  2. Web https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem, lead section
About this page
Tag0G0F7FGCGP
URIm2://0G0F7FGCGP
Aliaseslindemann-s-theorem
Kindstatement, theorem
WritingW0 stub
MSC11J81
StatusThis copy of the site has no build data. Run m2 status 0G0F7FGCGP.
UsesTranscendental element

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