Ah. Then I believe my problem was in my defintions, however, I was under the impression that transcendental meant it never repeated or whatnot. I guess this is more of a symptom. so, is root 2 not transcendental because it solves x^2 = 2? Wouldn't that make pi merely irrational, as it solves various series that converge at it, like x = 1 + 1/3 - 1/5 or however it goes.Originally posted by NateTG
First off:
Irrational means 'cannot be expressed as a fraction'. For example [tex]\sqrt{2}[/tex] is an irrational number.
Transcendental means is not a solution to any equation that contains only rational numbers. For example, [tex]\pi[/tex] is a transcendental number. [tex]\sqrt{2}[/tex] is not a transcendental number.
Ah! That clears that up.Originally posted by NateTG
Sorry, let me me more clear.
Transcendental numbers can never be expressed as the roots of polynomial equations. (Equations involving addition, multiplication, subtraction or division. No infinite series, logarithms, or trig functions.)
Originally posted by On Radioactive Waves
Isn't that also (indirectly) stating that trancendental numbers appear in their own definition?
Originally posted by NateTG
Sorry, let me me more clear.
Transcendental numbers can never be expressed as the roots of polynomial equations. (Equations involving addition, multiplication, subtraction or division. No infinite series, logarithms, or trig functions.)
selfAdjoint said:It's important to add that these polynomials have integer, or rational, or algebraic coefficients. In other words you can't get transcendental numbers from polynomials with coefficients that aren't transcendental.
Irrational numbers are numbers that cannot be expressed as a ratio of two integers, while transcendental numbers are numbers that cannot be expressed as the root of a polynomial equation with integer coefficients. In other words, all transcendental numbers are irrational, but not all irrational numbers are transcendental.
The most common method to prove a number is irrational is by contradiction, assuming the number can be expressed as a ratio of two integers and showing that it leads to a contradiction. To prove a number is transcendental, one must show that it is not a root of any polynomial equation with integer coefficients.
Some examples of irrational numbers include pi (π), the square root of 2 (√2), and the golden ratio (φ). Examples of transcendental numbers include e, which is the base of the natural logarithm, and the Euler-Mascheroni constant (γ).
Irrational and transcendental numbers play a crucial role in mathematics as they provide a counterexample to the common belief that all numbers can be expressed as a ratio of two integers. They also have many applications in various fields, such as physics, engineering, and cryptography.
Yes, irrational and transcendental numbers can be approximated using decimal expansions or continued fractions. However, these approximations are not exact and will always have a margin of error. This is because irrational and transcendental numbers have infinitely many digits after the decimal point and cannot be expressed as a finite decimal or fraction.