transcendental number
Noun: A transcendental number is a specific type of irrational number. It is a real number that is not a root of any non-zero polynomial equation with integer (or, equivalently, rational) coefficients. This means it cannot be expressed as a solution to any finite algebraic equation involving only integers and basic operations (addition, subtraction, multiplication, division, and exponentiation by whole numbers). All transcendental numbers are irrational, but not all irrational numbers are transcendental.
The term is used in the field of mathematics, specifically number theory and analysis, to classify numbers based on their algebraic properties. - Pi (π) and Euler's number (e) are the most famous examples of transcendental numbers. - Proving that a specific number is transcendental is often very difficult. - While there are infinitely many transcendental numbers, it is often challenging to identify them.
- Transcendental Number Theory: This is a major branch of number theory dedicated to studying the properties of transcendental numbers, including questions about their independence and approximation by rational numbers.
- Liouville's Theorem: An important early result that provided the first proof of the existence of transcendental numbers by constructing examples (Liouville numbers).
- Transcendental (adjective): Of or relating to a transcendental number. ()
- Algebraic Number (noun): The conceptual opposite; a number that is a root of a non-zero polynomial with integer coefficients. (e.g., √2, which solves x² - 2 = 0).
- Non-algebraic number (This is a descriptive synonym but is less commonly used as a formal term than "transcendental number").
- Irrational Number: A broader category encompassing all real numbers that cannot be expressed as a simple fraction. Transcendental numbers are a proper subset of irrational numbers. (Example: √2 is irrational but algebraic, not transcendental).
- Lindemann–Weierstrass Theorem: A key theorem that established the transcendence of π and e.
- Hilbert's Seventh Problem: A famous problem posed by David Hilbert in 1900, asking about the transcendence of numbers like 2^(√2), which was later solved.
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an irrational number that is not algebraic