Transcendental numbers

Definition

Transcendental numbers are numbers that are not the root of any non-zero polynomial equation with rational coefficients. This makes them distinct from algebraic numbers, which can be represented as solutions to such polynomial equations. Transcendental numbers play a critical role in understanding the limits of constructible numbers and the impossibility of certain geometric constructions.

5 Must Know Facts For Your Next Test

1. Famous examples of transcendental numbers include $$ ext{e}$$ (the base of natural logarithms) and $$ ext{π}$$ (pi), both of which cannot be expressed as the root of any polynomial with rational coefficients.
2. Transcendental numbers are uncountably infinite, which means there are far more transcendental numbers than algebraic numbers.
3. The first proof that a specific number is transcendental was provided by Ferdinand von Lindemann in 1882 when he proved that $$ ext{π}$$ is transcendental.
4. While all constructible numbers are algebraic, not all algebraic numbers are constructible; some transcendental numbers arise from geometric constructions that cannot be achieved with compass and straightedge.
5. The study of transcendental numbers highlights limitations in geometric constructions, particularly demonstrating that certain lengths and areas cannot be constructed using traditional methods.

Review Questions

• How do transcendental numbers differ from algebraic numbers in terms of their relationship to polynomial equations?
  • Transcendental numbers differ from algebraic numbers in that they cannot be the solution to any non-zero polynomial equation with rational coefficients. Algebraic numbers can be expressed as roots of such polynomials, meaning they can be precisely calculated or approximated through algebraic methods. In contrast, transcendental numbers cannot be represented this way, showcasing a deeper layer of complexity within number theory and its implications for geometric constructions.
• Discuss the implications of transcendental numbers on geometric constructions and why they matter in this context.
  • Transcendental numbers have significant implications for geometric constructions because they demonstrate limits on what can be achieved using classical methods like compass and straightedge. For instance, certain lengths associated with transcendental numbers cannot be constructed geometrically. This leads to important conclusions about which geometric problems can or cannot be solved using traditional techniques, thereby influencing the foundations of geometry and mathematical reasoning.
• Evaluate how the understanding of transcendental numbers has evolved and its impact on both mathematics and other fields.
  • The understanding of transcendental numbers has evolved significantly since their initial identification, particularly following important proofs such as Lindemann's work on $$ ext{π}$$. This evolution not only advanced pure mathematics but also influenced fields such as computer science and physics where these concepts underpin various theories and applications. The acknowledgment of transcendental numbers reshapes our understanding of numerical systems and opens up inquiries into functions, calculus, and even complex systems in nature.

Related terms

Algebraic numbers: Numbers that are roots of non-zero polynomial equations with rational coefficients, including integers, fractions, and some irrational numbers.

Constructible numbers: Numbers that can be obtained from the integers using a finite number of operations involving addition, subtraction, multiplication, division, and the extraction of square roots.

Geometric constructions: Procedures for drawing shapes, angles, and figures using only a compass and straightedge, which are constrained by the types of numbers that can be constructed.