What is a transcendental number?

Question

I came across some numbers which were called transcendental numbers. What are they exactly I want with explanation and eg

Did you try googling the term for a start? – Tobias Kildetoft Mar 24 '17 at 10:54 — Tobias Kildetoft, Mar 24 '17 at 10:54

score 2 · Answer 1 · answered Mar 24 '17 at 10:56

Trancedental numbers are numbers which are not the zero of a polynomial with coefficients in $\Bbb Q$. Actually there is a notion of transcendental numbers over any field, but it is usually referring to $\Bbb Q$ when the field is not specified.

For example, $\sqrt{2}$ is not transcendental, since it is a zero of the polynomial $x^2-2$. On the other hand, both $e$ and $\pi$ are transcendental, a difficult thing to prove.

score 2 · Accepted Answer · edited Dec 22 '23 at 19:23

2

Transcendental numbers are those numbers which are not the root of a non-zero polynomial with integer coefficients.

The most famous example of a transcendental number is $\pi$

$\sqrt2$ is an irrational number but is not transcendental as it is the solution of the equation $x^2-2 = 0 $

You might be wondering how to prove $\pi$ is transcendental.

It can be proved by contradiction, suppose $\pi$ is algebraic, then $\pi i$ will also be algebraic as $i$ is algebraic and by Lindemann–Weierstrass theorem, we know that $e^{\pi i} = -1$ will be transcendental which is a contradiction as this is the solution of the equation $x^2 -1 = 0$

edited Dec 22 '23 at 19:23

Ajay

109

answered Mar 24 '17 at 11:02

JamesBond101

36

Rational can be reduced to integer coefficients – JamesBond101 Mar 24 '17 at 11:47
Ahh, of course. Not sure why I mentally had the polynomials be with leading coefficient 1. – Tobias Kildetoft Mar 24 '17 at 11:49
Wait so $\sqrt {2}$ is not transcendental? I swear I have read somewhere that it is transcendental, but the way you explained it makes perfect sense$\ldots$ – George N. Missailidis Aug 18 '17 at 14:32

What is a transcendental number?

2 Answers2