Transcendental Numbers

Definition

Transcendental numbers are a special class of real or complex numbers that cannot be the root of any non-zero polynomial equation with rational coefficients. Unlike algebraic numbers, which can be expressed as solutions to such equations, transcendental numbers extend the concept of number beyond algebraic constructs, highlighting the limitations of classical constructions and proofs of impossibility.

5 Must Know Facts For Your Next Test

1. Famous examples of transcendental numbers include $$e$$ (the base of natural logarithms) and $$\pi$$ (the ratio of a circle's circumference to its diameter).
2. Transcendental numbers are uncountably infinite, meaning there are far more transcendental numbers than algebraic ones.
3. The proof that $$\pi$$ is transcendental was established by Ferdinand von Lindemann in 1882, marking a significant milestone in mathematics.
4. While all transcendental numbers are irrational, not all irrational numbers are transcendental; for example, $$\sqrt{2}$$ is an algebraic irrational number.
5. Transcendental numbers play a crucial role in understanding the limitations of classical geometric constructions, as they cannot be constructed using just a compass and straightedge.

Review Questions

• How do transcendental numbers differ from algebraic and constructible numbers, and what implications does this have for classical geometric constructions?
  • Transcendental numbers differ from algebraic numbers in that they cannot be expressed as roots of any polynomial with rational coefficients, while algebraic numbers can. Constructible numbers are even more restricted; they must be obtainable through finite steps using only a compass and straightedge, which excludes many transcendental numbers from being constructed. This distinction highlights the limitations faced by mathematicians in solving classical construction problems, such as squaring the circle or doubling the cube.
• Discuss the significance of proving the transcendence of $$\pi$$ and how it relates to classical construction problems.
  • The proof of $$\pi$$ being transcendental has far-reaching implications in mathematics, particularly in relation to classical construction problems. It definitively shows that constructing a square with an area equal to that of a circle (squaring the circle) is impossible with just a compass and straightedge. This was a significant outcome because it settled a long-standing question in geometry and emphasized the constraints imposed by the nature of transcendental numbers on what can be achieved through classical methods.
• Evaluate how Cantor's diagonal argument contributes to our understanding of transcendental numbers and their relationship to algebraic numbers.
  • Cantor's diagonal argument reveals that there are infinitely more real numbers than natural numbers, which implies that while algebraic numbers are countable, transcendental numbers are uncountably infinite. This difference in cardinality highlights not only the existence of transcendental numbers but also their density within the set of real numbers. Understanding this relationship enhances our grasp of mathematical concepts related to infinity and limits, illustrating why transcendental numbers hold a unique place in the broader landscape of mathematics.

Related terms

Algebraic Numbers: Numbers that are roots of non-zero polynomial equations with rational coefficients; this category includes all integers, rational numbers, and some irrational numbers.

Constructible Numbers: Numbers that can be constructed using a finite number of steps with a compass and straightedge, which limits their properties to algebraic numbers.

Cantor's Diagonal Argument: A mathematical proof demonstrating that the set of real numbers is uncountable and larger than the set of natural numbers, leading to the existence of transcendental numbers.