Top ten transcendental numbers

Everyone loves a top ten list, and what’s better than a top ten list about numbers? (I’m reminded of David Letterman’s top ten numbers between one and ten from September 22, 1989.) So, on the heels of my previous posts about algebraic and transcendental numbers (here and here), here’s my list of the…

Top Ten Transcendental Numbers

1. $0.1100010000000000000000010\ldots= \sum_{k=1}^{\infty}\frac{1}{10^{k!}}$ (Liouville, 1851): the first known transcendental number not expressed as a continued fraction.

2. ${e}$ (Hermite, 1873): the first non-contrived example of a transcendental number.

3. ${\pi}$ (Lindeman, 1882): use the Lindemann-Weierstrass theorem (below) and Euler’s identity, ${e^{\pi i}=-1}$ . This showed that it is impossible to square the circle.

Lindemann-Weierstrass Theorem (1882/1885). If ${a_{1},a_{2},\ldots,a_{m}}$ are distinct algebraic numbers and ${b_{1},\ldots,b_{m}}$ are nonzero algebraic numbers, then ${b_{1}e^{a_{1}}+b_{2}e^{a_{2}}+\cdots+b_{m}e^{a_{m}}\ne 0}$ .

4. ${\sin(1)}$ : use the Lindemann-Weierstrass theorem and the fact that $\sin z=\frac{e^{iz}-e^{-iz}}{2i}$ .

5. ${\ln(2)}$ : use the Lindemann-Weierstrass theorem and the fact that ${\ln x}$ is the inverse function for ${e^{x}}$ .

Hilbert’s 7th problem (1900). If ${a}$ and ${b}$ are algebraic numbers with ${a\ne0,1}$ and ${b}$ not rational, then ${a^{b}}$ is transcendental.

6. ${e^{\pi}}$ (Gelfond, 1929): Gelfond solved Hilbert’s 7th problem in the special case that ${b}$ has the form ${\pm i\sqrt{r}}$ with ${r}$ a positive rational number. Notice that ${e^{\pi}=(-1)^{-i}=i^{-2i}}$ . (Gelfond’s result also shows that ${i^{i}}$ is a real transcendental number.)

7. ${2^{\sqrt{2}}}$ (Siegel, 1930): Siegel saw how to extend Gelfond’s result to the case where ${b}$ is a real quadratic irrational.

8. ${2^{\sqrt[3]{2}}}$ (Gelfond and Schneider, 1934): Gelfond and Schneider independently solved the full version Hilbert’s 7th problem.

9. Champernowne constant (can you see the pattern in the digits?),
${0.1234567891011121314151617181920\ldots}$ (Mahler, 1937).

10. Chaitin constant, ${\Omega}$ (Chaitin, 1975): an example of a noncomputable, hence transcendental, number.

Interesting fact: In a lecture in 1920 David Hilbert said that he would probably live to see a solution to the Riemann hypothesis, that the younger audience members would probably see a proof of Fermat’s last theorem, but he doubted that anyone in the room would see a proof that $2^{\sqrt{2}}$ is transcendental. As you probably know, the Riemann hypothesis is still unsolved and Fermat’s last theorem was proved in 1994.

Before we start feeling too confident in our ability to identify transcendental numbers, let’s take a look at a few numbers that are not yet known to be transcendental.

${e-\pi}$ and ${e+\pi}$ : since the algebraic numbers form a field and ${\frac{1}{2}((e-\pi)+(e+\pi))=e}$ is transcendental, we know that at least one of these two must be transcendental. But we do not know which one. (Seriously, does anyone believe that either one is algebraic?) R. J. Lipton calls this a mathematical embarrassment.

Similarly, we don’t know whether ${\pi^{e}}$ , ${e^{e}}$ , and ${\pi^{\pi}}$ are transcendental.

${\gamma}$ : Euler-Mascheroni constant (there is still no proof that ${\gamma}$ is irrational!)

13 Comments

Chris says:

November 4, 2010 at 1:14 pm

I think you mean e-π and e+π in the 3rd to last paragraph.

Dave Richeson says:

November 4, 2010 at 1:23 pm

Thanks for catching that. I fixed it.

sherifffruitfly says:

November 4, 2010 at 1:27 pm

First, Chaitin’s omega isn’t *one* number, but rather a *class* of numbers.

Second, any such list that doesn’t have e as first is sort of silly.

Dave Richeson says:

November 4, 2010 at 1:30 pm

I chose to leave out the details of Chaitin’s constant(s) and give a link to a description instead.

The list is chronological. That’s why e is second.

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Baylink says:

December 31, 2010 at 8:14 pm

In case you’re wondering about the traffic, NPR’s Monkey See picked you as one of their Top 20 Top 10 Lists in 2010. I’m glad they did: while I can tell I’ll understand about as much here as at Language Log, it’ll be fun to try. :-)

Dave Richeson says:

December 31, 2010 at 10:01 pm

Thanks! (Hi, everyone!)

I actually saw the story in NPR’s Facebook stream. I did a double-take when I got to #13.