11.3 Transcendence of π and e

Transcendental numbers like π and e can't be roots of polynomial equations with rational coefficients. This makes them uncountable and dense in the real numbers. They're key players in the realm of numbers we can't construct with just a compass and straightedge.

Proving π and e are transcendental involves clever uses of Galois theory and contradiction. These proofs show how tricky these numbers are and why they're so special in math. They also connect to big questions about geometry and number theory.

Transcendental Numbers and Their Properties

Definition and Characteristics

Top images from around the web for Definition and Characteristics

• 

  Quarks: Is That All There Is? · Physics View original

  Is this image relevant?

• 

  Graphs of the Sine and Cosine Functions · Precalculus View original

  Is this image relevant?

• 

  Quarks: Is That All There Is? · Physics View original

  Is this image relevant?

• 

  Graphs of the Sine and Cosine Functions · Precalculus View original

  Is this image relevant?

1 of 2

Top images from around the web for Definition and Characteristics

• 

  Quarks: Is That All There Is? · Physics View original

  Is this image relevant?

• 

  Graphs of the Sine and Cosine Functions · Precalculus View original

  Is this image relevant?

• 

  Quarks: Is That All There Is? · Physics View original

  Is this image relevant?

• 

  Graphs of the Sine and Cosine Functions · Precalculus View original

  Is this image relevant?

1 of 2

• Define transcendental numbers as real or complex numbers that are not algebraic, meaning they are not roots of any non-zero polynomial equation with rational coefficients
• Highlight the uncountability of transcendental numbers, implying there are infinitely more transcendental numbers than algebraic numbers
• Explain the density of the set of transcendental numbers in the real numbers, meaning between any two real numbers, there exists a transcendental number $(\pi, e)$
• Mention that transcendental numbers are not constructible with a compass and straightedge, meaning they cannot be constructed using a finite number of steps involving only a compass and straightedge

Algebraic Operations with Transcendental Numbers

• State that the sum, difference, product, and quotient of a transcendental number and an algebraic number is always transcendental
• Provide examples demonstrating the closure of transcendental numbers under these operations $(\pi + 1, e - 2, 3\pi, e/4)$
• Explain that these properties help identify new transcendental numbers based on known ones
• Discuss the implications of these properties in the study of transcendental number theory and its applications

Proving the Transcendence of π

Proof by Contradiction

• Assume π is algebraic and let K be the splitting field of the minimal polynomial of π over Q
• Explain that the field extension K/Q is a Galois extension and its Galois group G = Gal(K/Q) is a finite group
• Consider the field extension K(e^(2πi/n))/K for some positive integer n and note that this extension is cyclotomic and has degree φ(n), where φ is Euler's totient function
• Show that the compositum of K and Q(e^(2πi/n)) is a Galois extension of Q with Galois group isomorphic to a subgroup of G × (Z/nZ)^×, which is finite

Contradiction and Conclusion

• Demonstrate that as n increases, the degree of the cyclotomic extension grows without bound, contradicting the finiteness of the Galois group
• Conclude that the assumption of π being algebraic leads to a contradiction, proving that π is transcendental
• Discuss the significance of this proof in establishing the transcendental nature of π
• Highlight the role of Galois theory in the proof and its importance in solving problems related to transcendental numbers

Transcendence of e

Proof by Contradiction

• Assume e is algebraic and let L be the splitting field of the minimal polynomial of e over Q
• Explain that the field extension L/Q is a Galois extension and its Galois group H = Gal(L/Q) is a finite group
• Consider the field extension L(2^(1/n))/L for some positive integer n and note that this extension is radical and has degree n
• Show that the compositum of L and Q(2^(1/n)) is a Galois extension of Q with Galois group isomorphic to a subgroup of H × (Z/nZ), which is finite

Contradiction and Conclusion

• Demonstrate that as n increases, the degree of the radical extension grows without bound, contradicting the finiteness of the Galois group
• Conclude that the assumption of e being algebraic leads to a contradiction, proving that e is transcendental
• Discuss the significance of this proof in establishing the transcendental nature of e
• Compare and contrast the proofs of transcendence for π and e, highlighting their similarities and differences

Implications of Transcendental Numbers in Mathematics

Geometric Consequences

• Discuss the impact of the transcendence of π and e on the solvability of certain geometric problems, such as squaring the circle or constructing regular polygons with a compass and straightedge
• Explain how the transcendental nature of these numbers limits the possibilities of constructing certain geometric objects using classical tools
• Provide historical context for these geometric problems and their significance in the development of mathematics

Connections to Other Mathematical Concepts

• Explore the relationship between the transcendence of π and e and the concept of periods, which are complex numbers that can be expressed as definite integrals of algebraic functions over algebraic domains
• Discuss the connections between the transcendence of π and e and the study of special functions, such as the gamma function and the zeta function, which have important applications in number theory and mathematical physics
• Highlight the interdisciplinary nature of transcendental number theory and its relevance to various branches of mathematics

Further Research and Open Questions

• Emphasize the power and versatility of Galois theory in solving problems related to transcendental numbers, as demonstrated by the proofs of the transcendence of π and e
• Discuss how the existence of transcendental numbers, exemplified by π and e, highlights the richness and complexity of the real number system and its extensions
• Identify open questions and areas for further research in transcendental number theory, such as the transcendence of other mathematical constants $(\ln 2, \zeta(3))$ and the development of new techniques for proving transcendence
• Encourage the exploration of the implications of transcendental numbers in other areas of mathematics and their potential applications in science and engineering