Understanding Transcendal Numbers: Importance and Examples

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In summary, transcendental numbers are numbers that are not roots of any polynomial with integer coefficients and are therefore not algebraic numbers. They are important because almost all real numbers are transcendental, including e and pi which are both examples of transcendental numbers.
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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].
 
  • #3
They are important in that almost all real numbers are transcendental.
 
  • #4
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.
 
  • #5
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.
 
  • #6
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.
 

FAQ: Understanding Transcendal Numbers: Importance and Examples

What are transcendental numbers?

Transcendental numbers are real numbers that cannot be expressed as the root of any polynomial equation with integer coefficients. In other words, they are numbers that cannot be written as fractions or roots of integers.

How are transcendental numbers different from algebraic numbers?

Algebraic numbers are real numbers that can be expressed as the root of a polynomial equation with integer coefficients. Transcendental numbers, on the other hand, cannot be expressed in this way.

Who discovered transcendental numbers?

The concept of transcendental numbers was first introduced by mathematician Joseph Liouville in 1844. However, it was not until 1873 that mathematician Georg Cantor proved the existence of transcendental numbers.

Can transcendental numbers be calculated?

No, transcendental numbers cannot be calculated exactly. They can only be approximated to a certain degree of accuracy using numerical methods.

What is the importance of transcendental numbers?

Transcendental numbers are important in mathematics because they help to bridge the gap between algebraic and geometric reasoning. They also play a key role in the development of calculus and other branches of mathematics.

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