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The PROOF: e and pi are transcendental

Mathologer・24 minutes read

The video explains complex transcendence proofs for e and pi, breaking them down into seven levels of enlightenment that require calculus knowledge to understand in detail. The proofs involve showing that e and pi cannot be expressed as fractions or solutions of polynomial equations with integer coefficients, ultimately highlighting the transcendental nature of these numbers.

Insights

  • The video aims to simplify and animate the complex transcendence proofs for e and pi, requiring some calculus knowledge and dividing the mathematical quest into seven levels of enlightenment.
  • By using intricate polynomial manipulations and the gamma function identity, the proof for e involves ensuring that specific polynomials lead to contradictions, while for pi, Lindermann's equation introduces complex numbers to create contradictions, all based on David Hilbert's work refined by Steinberg and Redheffer.

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Recent questions

  • What are transcendental numbers?

    Numbers beyond irrational, like e and pi.

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Summary

00:00

"Proving e and pi are transcendental"

  • e and pi are proven to be transcendental numbers, a step beyond being irrational.
  • Transcendence proofs for individual numbers like e and pi are complex and unfamiliar.
  • The aim is to simplify and animate transcendence proofs for e and pi.
  • The video is ambitious and not suitable for all viewers, requiring some calculus knowledge.
  • The mathematical quest is divided into seven levels of enlightenment.
  • Level one involves proving that e is irrational using factorials and infinite sums.
  • The proof shows that e cannot be expressed as a fraction, leading to a contradiction.
  • The contradiction arises from the fact that e can be approximated by fractions better than fractions can approximate e.
  • Level two involves proving that e is not the solution of a non-trivial quadratic equation with integer coefficients.
  • The proof leads to the conclusion that e is a quadratic irrational number.
  • Level three aims to prove that e is not the solution of any non-trivial polynomial equation with integer coefficients.
  • The proof involves showing that e and its powers can be very close to fractions simultaneously.
  • The gamma function identity is used to relate e to integers and aid in the transcendence proof.

15:51

"Transcendental Proof: Integer Coefficients in Polynomial"

  • The numerator e cubed times A can be written as an almost integer plus a tiny bit.
  • The integral can be split into two parts: from x=0 to 3 and beyond 3 to infinity.
  • Choosing p(x) carefully ensures the first part is small and the second part is an integer.
  • The first part is small due to the huge n! in the denominator and the finite chunk of the integral.
  • The polynomial p(x) must contain x^n bits to ensure the factorial beats them.
  • Shifting the integral to the left by 3 units ensures the second part is an integer.
  • Choosing p(x) with a factor (x-3)^n guarantees the shifting by 3 results in a special p(x+3).
  • Making p(x)=(x-1)^n times (x-2)^n times (x-3)^n ensures an integer plus a tiny thing equals 0.
  • Raising the powers in p(x) from n to n+1 and choosing n+1 as a large prime ensures the blue number is not 0.
  • The polynomial p has integer coefficients despite being full of complex demons, leading to a contradiction argument to prove pi is transcendental.

33:12

Transcendence of e and pi explained

  • Lindermann's equation involves complex numbers B, C, and D, obtained by integrating from a complex number to infinity, leading to a green sum that is an integer, divisible by n+1, providing a contradiction similar to the proof for e.
  • The transcendence proofs for e and pi are based on David Hilbert's work, refined by Steinberg and Redheffer, with the video being an ambitious Mathologer project that took hundreds of hours to create, aiming to help people appreciate the transcendental nature of e and pi.
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