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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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Transcendental numbers cannot be expressed as a simple fraction of two integers, distinguishing them from algebraic numbers
Transcendental numbers are also irrational, meaning they cannot be expressed as a simple fraction of two integers
The set of transcendental numbers is uncountably infinite, indicating that they are more numerous than algebraic numbers
The study of transcendental numbers has a rich history, marked by the contributions of many mathematicians
Important discoveries in the study of transcendental numbers include the proof of their existence and the identification of specific examples
Georg Cantor's work on the countability of algebraic numbers versus the uncountability of real numbers highlighted the prevalence of transcendental numbers
Transcendental numbers cannot be the root of any non-zero polynomial equation with integer coefficients
All real transcendental numbers are irrational
When used as an input to a non-constant single-variable algebraic function, transcendental numbers will also yield a transcendental output
The transcendence of various numbers has been established through complex mathematical proofs
The Lindemann-Weierstrass theorem and the Gelfond-Schneider theorem have been instrumental in proving the transcendence of specific numbers
Numbers such as e, π, and various trigonometric and logarithmic functions have been proven to be transcendental
00
Transcendental numbers are not solutions to any ______ equation with ______ coefficients.
polynomial
rational
01
Numbers like π and e are transcendental and cannot be written as a fraction of two ______.
integers
02
Although all transcendental numbers are ______, not all ______ numbers are transcendental.
irrational
irrational
