Pi is transcendental

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Pi is transcendental

• Roland van der Veen

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• Transcendental numbers
• e is irrational
• e is transcendental
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• Symmetric polynomials
• Pi is transcendental
• Open problems

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Transcendental numbers

• A complex number c is algebraic (over Q) if it is
  a root of a polynomial with rational
  coefficients.
• c is transcendental if it is NOT algebraic.
• Almost all numbers are transcendental.
• e is transcendental (Heremite 1873).
• p is transcendental (Lindemann 1882).

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General theorems

• Let a and b be algebraic numbers.
• Theorem (Gelfond, Schneider 1934)
• If a ? 0,1 and b is irrational, then ab is
  transcendental.
• Example ep i-2i is transcendental.
• Theorem (Baker 1966)
• a log(b) is transcendental.

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e is irrational (Lambert 1761)
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e is transcendental (Heremite 1873)
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We will show that
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BREAK
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Elementary symmetric functions
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Theorem on symmetric polynomials

• Let p be a symmetric polynomial (invariant under
  permutation of its variables). Then p can be
  written as a polynomial in the elementary
  symmetric functions.

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p is transcendental (Lindemann 1882)
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We will show that
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The first term is not divisible by p
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The second term is divisible by p
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Open problems