Simplest proof that some number is transcendental?

Question

I tried googling for simple proofs that some number is transcendental, sadly I couldn't find any I could understand.

Do any of you guys know a simple transcendentality (if that's a word) proof?

E: What I meant is that I wanted a rather simple proof that some particular number is transcendental ($e$ or $\pi$ would work), not a method to prove that any number is transcendental, sorry for the confusion.

Or even a proof about transcendental numbers being 'as common' as algebraic numbers?

Answer 1

It's possible to argue fairly elementarily that Liouville's number $$ L = \sum_{k=1}^\infty 10^{-k!} $$ is transcendental, by showing directly that for every integer polynomial $$p(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ there is a decimal position in $p(L)$ that must be nonzero.

The powers of $L$ have the form $$ L^k = c_1 10^{-b_1} + c_2 10^{-b_2} + c_3 10^{-b_3} + \cdots $$ where each of the (all different) $b_i$ is a number that can be written as the sum of $k$ (not necessarily different) factorials, and $c_i$ is some integer $\le k!$ that depends on whether some of the factorials are different.

Now let $n$ be the degree of $p$. Among the $b_i$s we find numbers of the form $B_{h,n}=(h+1)!+(h+2)!+\cdots (h+n)!$, and by choosing $h$ large enough, these numbers can be arbitrarily far from anything that can be written as a sum of fewer than $n$ factorials (which are the ones the appear in the expansion of lower powers of $L$).

So if we choose $h$ large enough, we find a $B_{h,n}$ such that the only terms in $p(L)$ that contribute to the digits around decimal position $B_{h,n}$ is the product of $a_n$ with $C_{h,n}$, which is nonzero.

(Everything to the right of this is the sum of products of some $10^{-b_j}$ with a factor that is at most $n!a_n+(n-1)!a_{n-1}+\cdots+a_1$. The bound of the factor depends only on $p$, so if only we choose $h$ such that that the first $b_j$ after $B_{h,n}$ is separated by at least the length of this bound their contribution cannot reach position $B_{h,n}$.

(Similarly, we can arrange for there to be enough space to the left of $B_{h,n}$ to make room for all of $a_nC_{h,n}$ before the previous $b_j$).

Answer 2

Cantor showed a simple way to list the real algebraic numbers in a sequence $a_1,a_2,a_3,\ldots$. Every term has only finitely many terms before it, and every algebraic number will be reached after only finitely many steps. Now let us seek a transcendental number in the interval $(0,1)$. Find the first number in the above sequence that is between $0$ and $1$; call it $b_1$, and narrow down the interval from $(0,1)$ to $(b_1,1)$. Find the first term in the sequence that comes after $b_1$ and falls in our now narrower interval; callit $c_1$, and narrow down the interval from $(b_1,1)$ to $(b_1,c_1)$. Then find the first number in the sequence that comes after $c_1$ and falls in our now yet narrower interval; call it $b_2$, and narrow down the interval to $(b_2,c_1)$. And so on: the next narrowing will be to $(b_2,c_2)$ then $(b_3,c_2)$, then $(b_3,c_3)$. We have $0<b_1<b_2<b_3<\cdots < c_3<c_2<c_1<1$. Some number must be between the $b$s and the $c$, and it can't be algebraic because our process has eliminated all of those.

This argument involving narrowing was in Cantor's very first paper on uncountability, published in 1874. His "diagonal argument" came later.

http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof

Answer 3

There is a BIG difference between showing that a particular number that naturally occurs (like $e$ or $\pi$) is transcendental and showing that some number is.

The existence of transcendental numbers was first shown in 1844 by Liouville. In 1851 he proved that $\sum_{k=1}^{\infty} \frac1{10^{k!}} $ is transcendental.

In 1873, Hermite proved that $e$ is transcendental. The next year, Cantor showed that the algebraic numbers were countable, so that almost all numbers are transcendental. In 1882, Lindemann showed that $\pi$ is transcendental.

Look up more if you want.

Answer 4

The easiest proof is via Liouville's criterion.

Lemma. Suppose $\alpha$ is an irrational algebraic number. There exist integers $C,n$ such that for all integers $p/q$, $$ \left| \alpha - \frac{p}{q} \right| \geq \frac{C}{q^n}. $$

You can find a proof of the lemma on Wikipedia, and in many other sources. Now consider the number $$ \alpha = \sum_{m=0}^\infty \frac{1}{10^{m!}}, $$ sometimes known as Liouville's number. Since the decimal expansion of $\alpha$ is aperiodic, it is irrational. On the other hand, for all $r$ we have $$ \left| \alpha - \frac{\sum_{m=0}^r 10^{r!-m!}}{10^{r!}} \right| \leq \sum_{m=r+1}^\infty \frac{1}{10^{m!}} \leq \sum_{m=r+1}^\infty \frac{1}{10^{(r+1)!+m-(r+1)}} \leq \frac{2}{10^{(r+1)!}}. $$ Since $2/10^{(r+1)!}$ is so much smaller than $10^{r!}$ for large $r$, it is not hard to check that $\alpha$ doesn't satisfy the conclusion of the lemma. Indeed, suppose that it did, for some $C,n$. Then for all $r$ we must have $$ \frac{2}{10^{(r+1)!}} \geq \frac{C}{10^{r!n}} \Longleftrightarrow 10^{r!n} \geq (C/2) 10^{(r+1)!} \Longleftrightarrow 1 \geq (C/2) 10^{r! (r+1-n)}, $$ which fails for large enough $r$.

Since the conclusion of the lemma doesn't hold, one of its premises must be false. Since $\alpha$ is definitely irrational, it must be not algebraic. In other words, $\alpha$ is transcendental.