Algebraic vs. Transcendental

Attribute	Algebraic	Transcendental
Definition	Can be expressed as the root of a polynomial equation with integer coefficients	Cannot be expressed as the root of a polynomial equation with integer coefficients
Examples	Integers, rational numbers, algebraic numbers	π, e, √2
Complexity	Generally easier to work with and manipulate	Often more complex and harder to analyze
Properties	Follow algebraic rules and operations	May exhibit more unpredictable behavior

Definition

Algebraic and transcendental are two classifications of numbers in mathematics. Algebraic numbers are solutions to polynomial equations with integer coefficients, while transcendental numbers are not. In other words, algebraic numbers can be expressed as roots of polynomials, whereas transcendental numbers cannot be expressed in this way.

Examples

Examples of algebraic numbers include integers, fractions, square roots, and cube roots. These numbers can be written as solutions to polynomial equations. On the other hand, examples of transcendental numbers include pi (π) and e (Euler's number). These numbers cannot be expressed as roots of any polynomial equation with integer coefficients.

Properties

Algebraic numbers have some interesting properties. For example, the sum, difference, product, and quotient of two algebraic numbers are also algebraic. Additionally, the square root of an algebraic number is also algebraic. Transcendental numbers, on the other hand, do not have these closure properties. The sum, difference, product, and quotient of two transcendental numbers may be algebraic or transcendental.

Cardinality

The cardinality of algebraic numbers is countable, meaning that there exists a one-to-one correspondence between the set of algebraic numbers and the set of natural numbers. This is because each algebraic number can be uniquely identified by its coefficients in a polynomial equation. In contrast, the cardinality of transcendental numbers is uncountable, meaning that there is no such correspondence with the set of natural numbers. This is due to the infinite and non-repeating nature of transcendental numbers.

Complexity

Algebraic numbers are generally easier to work with computationally compared to transcendental numbers. This is because algebraic numbers can be represented exactly using finite expressions, such as radicals or fractions. On the other hand, transcendental numbers often require infinite or non-repeating decimal expansions for their representation, making them more complex to manipulate in calculations.

Applications

Algebraic numbers are commonly used in various branches of mathematics, such as number theory, algebra, and geometry. They play a fundamental role in solving polynomial equations and understanding the properties of algebraic structures. Transcendental numbers, on the other hand, have applications in fields like analysis, physics, and engineering. They often arise as solutions to transcendental equations and have special significance in the study of functions and their properties.

Special Numbers

Some numbers are both algebraic and transcendental. For example, the square root of 2 is algebraic because it is a solution to the polynomial equation x^2 - 2 = 0. However, the square root of 2 is also transcendental, as proven by the Lindemann–Weierstrass theorem. This theorem states that if α is a non-zero algebraic number, then e^α is transcendental. Therefore, the square root of 2 is both algebraic and transcendental.