Understanding Transcendal Numbers: Importance and Examples

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Discussion Overview

The discussion revolves around the concept of transcendental numbers, their definition, significance, and examples, particularly focusing on well-known numbers such as e and π. The scope includes theoretical explanations and clarifications regarding their properties and importance in mathematics.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant requests an explanation of transcendental numbers and their importance.
  • Another participant defines transcendental numbers as those not being roots of any polynomial with integer coefficients, distinguishing them from algebraic numbers.
  • It is noted that while all transcendental numbers are irrational, not all irrational numbers are transcendental, with an example of √2 provided.
  • One participant claims that almost all real numbers are transcendental.
  • There is a question regarding whether e and π are included among transcendental numbers, with a subsequent affirmation that they are indeed transcendental.
  • A participant mentions that e and π are roots of certain equations, but clarifies that they are not roots of any polynomial with rational coefficients.

Areas of Agreement / Disagreement

Participants generally agree on the definition of transcendental numbers and the status of e and π as transcendental. However, there is some uncertainty regarding the nature of equations that e and π may satisfy.

Contextual Notes

There are unresolved aspects regarding the specific types of equations that transcendental numbers may satisfy, as well as the implications of their properties in broader mathematical contexts.

ziad1985
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Can someone please explain what are Transcendal Numbers are?
and why they are so important?
 
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A transcendental number is a number which isn't a root of any polynomial with integer coefficients. It's therefore not an algebraic number (of any degree).

All transcendental numbers are irrational but the converse is not true. For example, [itex]\sqrt 2[/itex] is irrational but is a solution of [itex]x^2 = 2[/itex].
 
They are important in that almost all real numbers are transcendental.
 
That means it should include e and Pi ?
I read one time that e and pi are belived to be a root of some kind of an equation.
 
ziad1985 said:
That means it should include e and Pi ?
I read one time that e and pi are belived to be a root of some kind of an equation.
e and pi are both transcendental.
 
e and pi certainly are the roots of 'some kinds of equation', just not any polynomial over Q. They are roots if [itex]x^2-(e+\pi)x+e\pi[/itex], for instance.
 

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