A real number is said to be transcendental if it is not an algebraic number. In other words, it is not a solution to any polynomial equation of the form, \[ c_n x^n + c_{n-1}x^{n-1}+\cdots+c_1 x + c_0=0 \quad c_0,c_1,\dots,c_n \in \mathbb{Z} \]

The first transcendental number that was proved to be transcendental was Liouville's number, which we will call $\alpha$. This was proved in 1844 by Joseph Liouville. I find it interesting that $\alpha$ was proved to be transcendental before the other transcendental numbers that we have all come to love (e.g. $\pi$, $e$). Anyway, \begin{aligned} \alpha &= 10^{-1!}+10^{-2!}+10^{-3!}+10^{-4!}+\cdots\\ &=10^{-1}+10^{-2}+10^{-6}+10^{-24}+\cdots\\ &=0.1+0.01+0.000001+\cdots\\ &=0.110001 000000 000000 000001 000000 000000\dots \end{aligned}

Another famous transcendental number is $e$. It is the base of the Napier logarithms, or natural logarithms. In 1665, Newton showed that \[ e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots\\ \implies e=1+1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots \] Charles Hermite proved that $e$ is transcendental in 1873.

I think one of the transcendentals that should be looked at a little more is $\pi$. "$\pi$ is the only irrational and transcendental number that occurs naturally in every society where circles are measured"(Wells, 1986, p. 48). It has been under the microscope of mathematicians for thousands of years. $\pi$ is the ratio of the circumference of a circle to it's diameter. In other words, \[ \pi=\frac{circumference}{diameter}=3.1415926535\dots \] Wait, well I just said that it was a ratio, didn't I? So, why isn't it a rational number? Here might be some area of confusion but, although I did say that it was a ratio, this is not the ratio of two integers, it is just the ratio of two real numbers.

I think that it is worthy to point out here that there have been many people who have tried to express this number as a ratio of two integers, the biggest I've seen was done by Jacob Marcelis in about 1700 when he came to the conclusion that \[ \pi=3\frac{1,008,449,087,377,541,679,894,282,184,894}{6,997,183,637,540,819,440,035,239,271,702} \] But, this is simply not true, because it was proved in 1882 by Carl Louis Ferdinand von Lindemann that $\pi$ is transcendental (Conway, 1996, p. 239). As a result of this proof the ancient "squaring the circle" problem that we talked about in the Irrational Numbers section of this website was also solved to be impossible.

There are tons of transcendental numbers that we can talk about. In fact, we can talk about transcendental numbers more than we can about rational numbers. This is an interesting thing that Georg Cantor proved. Here is the proof:

Theorem: $\{transcendental\; numbers\}$ is uncountable.
Proof: We know that $\mathbb{R}$ is uncountable and $\mathbb{Z}$ is countable. If $\mathbb{Z}$ is countable, then $\{polynomials\; with\; integer\; coefficients\}$ is also countable. Since, each such polynomial has a finite number of zeros, then $\{algebraic\; numbers\}$ must be countable. But, $\mathbb{R}=\{algebraic\;numbers\}\cup\{transcendental\;numbers\}$ and the union of two countable sets is countable. Since $\mathbb{R}$ is uncountable we conclude that either $\{algebraic\;numbers\}$ is uncountable or $\{transcendental\;numbers\}$ is uncountable. But, $\{algebraic\;numbers\}$ is countable, therefore $\{transcendental\;numbers\}$ is uncountable.

Now, there is one last thing that I want to say, basically because this is part of my final project and I set out to discover how irrational number can be expressed as an infinite sum in every situation. I have determined that this is possible, but I have not found any proof of such an existence. But, if this were the case, then there would be an uncountable number of infinite series. Although, I did find that $\displaystyle e=\sum_{n=0}^{\infty}\frac{1}{n!}$ and also $\displaystyle \pi=\sum_{n=0}^{\infty}\frac{4(-1)^n}{2n+1}$ The others I did not find a proof.