Transcendental Numbers

"...Pi wasn't the only transcendental number. In fact there was an infinity of transcendental numbers. More than that, there were more transcendental numbers than ordinary numbers, even though pi was the only one of them she had heard of. In more ways than one, pi was tied to infinity."

These beautiful lines are from the book, Contact by Carl Sagan.

Even though there are infinite transcendental numbers most of us have heard of only e and pi.
e = 2.718281828459045...
Pi = 3.1415926535...

Transcendental numbers are irrational. That is, they cannot be expressed as the ratio of two integers.

The abc book of e : mathwithbaddrawings.com

But all irrational numbers are not transcendental.

Its because, transcendental numbers cannot be expressed as a solution of a polynomial equation. In other words they are not the solution of any polynomial equation with integer coefficients.

So square root of 2 is an irrational number but not transcendental.

In this Numberphile video a simple proof is shown that pi is transcendental. The proof was given by Ferdinand von Lindemann in 1882.

1. He first proved that e^a is transcendental where a is nonzero. In 1873 Charles Hermite had already proved that e is transcendental.

2. Next, he used Roger Cotes' identity (famously but inaccurately known as Euler's identity). And using proof by contradiction he proved that i(pi) and pi are transcendental.

mathwithbaddrawings.com

There is a conjecture in transcendental theory which indirectly implies that above equation is the only nontrivial relation between e, pi and i.

Hilbert's seventh problem is also about transcendental numbers:

If a is an algebraic number such that a>1 and b is an irrational algebraic number, is a^b necessarily transcendental?

It was eventually proved and now known as Gelfond-Schrieder theorem in 1934.

Here are some of the numbers and functions that are proved to be transcendental.

wikipedia.org