Transcendental Number

Definition

A transcendental number is a type of real or complex number that is not a root of any non-zero polynomial equation with rational coefficients. This distinguishes them from algebraic numbers, which can be expressed as solutions to such polynomial equations. Transcendental numbers are significant because they highlight the limitations of algebraic methods and introduce a more complex layer to the understanding of number theory and mathematics.

5 Must Know Facts For Your Next Test

1. Transcendental numbers are uncountably infinite, meaning there are vastly more transcendental numbers than algebraic numbers.
2. Famous examples of transcendental numbers include $\pi$ and $e$, both of which have essential roles in mathematics and various applications.
3. The proof that $\pi$ is transcendental was provided by Ferdinand von Lindemann in 1882, marking a significant milestone in mathematics.
4. Transcendental numbers demonstrate that not all real numbers can be solved using basic algebraic methods, challenging traditional views on number properties.
5. The existence of transcendental numbers was first established by Joseph Liouville in the 19th century through his work on approximating real numbers.

Review Questions

• How do transcendental numbers differ from algebraic numbers, and what implications does this have for number theory?
  • Transcendental numbers differ from algebraic numbers in that they cannot be expressed as roots of any non-zero polynomial equation with rational coefficients. This distinction has significant implications for number theory as it shows that there are types of numbers that exist beyond the scope of polynomial equations. This creates a broader understanding of mathematical concepts and challenges mathematicians to develop new methods and tools to explore these unique numbers.
• Discuss how transcendental numbers like $\pi$ and $e$ have influenced mathematical theory and applications.
  • $\pi$ and $e$ are crucial examples of transcendental numbers that have heavily influenced both mathematical theory and practical applications. $\pi$, for instance, is fundamental in geometry, particularly in calculations involving circles, while $e$ is central to calculus and complex analysis, appearing in exponential growth and decay models. The recognition of these constants as transcendental has led to deeper investigations into their properties, contributing to advances in analysis, number theory, and applied mathematics.
• Evaluate the historical development of the concept of transcendental numbers and its impact on modern mathematics.
  • The historical development of transcendental numbers began with Joseph Liouville's work in the 19th century, where he demonstrated their existence. This was further advanced by Lindemann's proof of the transcendence of $\pi$. The recognition and study of these numbers have profoundly impacted modern mathematics by expanding the understanding of number classification and highlighting the limitations of algebraic methods. As mathematicians explore deeper into transcendence theory, new questions arise about the nature and properties of different classes of numbers, fostering ongoing research and discovery.

Related terms

Algebraic Number: An algebraic number is a number that is a solution to some polynomial equation with integer coefficients, making it distinct from transcendental numbers.

Irrational Number: An irrational number is a number that cannot be expressed as a simple fraction, meaning it cannot be represented as the quotient of two integers.

Pi ($\pi$): Pi is an example of a transcendental number, representing the ratio of the circumference of a circle to its diameter and cannot be the root of any polynomial with rational coefficients.