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Transcendental numbers, such as π and e, are not solutions to any polynomial equation with rational coefficients, setting them apart from algebraic numbers. This text delves into their distinctive properties, the history of transcendental number theory, and the methods used to prove their transcendence. It also explores the diversity of transcendental numbers, including Liouville numbers and non-computable numbers, and their occurrence in various mathematical contexts.

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## Definition of Transcendental Numbers

### Distinction from Algebraic Numbers

Transcendental numbers cannot be expressed as a simple fraction of two integers, distinguishing them from algebraic numbers

### Irrationality

Transcendental numbers are also irrational, meaning they cannot be expressed as a simple fraction of two integers

### Countability

The set of transcendental numbers is uncountably infinite, indicating that they are more numerous than algebraic numbers

## Evolution of Transcendental Number Theory

### Historical Contributions

The study of transcendental numbers has a rich history, marked by the contributions of many mathematicians

### Key Discoveries

Important discoveries in the study of transcendental numbers include the proof of their existence and the identification of specific examples

### Significance of Cantor's Work

Georg Cantor's work on the countability of algebraic numbers versus the uncountability of real numbers highlighted the prevalence of transcendental numbers

## Properties of Transcendental Numbers

### Non-Algebraic Roots

Transcendental numbers cannot be the root of any non-zero polynomial equation with integer coefficients

### Irrationality of Real Transcendental Numbers

All real transcendental numbers are irrational

### Output of Algebraic Functions

When used as an input to a non-constant single-variable algebraic function, transcendental numbers will also yield a transcendental output

## Proving the Transcendence of Numbers

### Complex Mathematical Proofs

The transcendence of various numbers has been established through complex mathematical proofs

### Key Theorems

The Lindemann-Weierstrass theorem and the Gelfond-Schneider theorem have been instrumental in proving the transcendence of specific numbers

### Examples of Proven Transcendental Numbers

Numbers such as e, π, and various trigonometric and logarithmic functions have been proven to be transcendental

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