Top ten transcendental numbers

Posted by: Dave Richeson | November 4, 2010

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!)

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Posted in Math | Tags: algebraic number, Chaitin constant, Champernowne constant, e, gamma, Hilbert, pi, transcendental number

Responses

I think you mean e-π and e+π in the 3rd to last paragraph.
By: Chris on November 4, 2010
at 1:14 pm

Reply
- Thanks for catching that. I fixed it.
  By: Dave Richeson on November 4, 2010
  at 1:23 pm
  
  Reply
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.
By: sherifffruitfly on November 4, 2010
at 1:27 pm

Reply
- 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.
  By: Dave Richeson on November 4, 2010
  at 1:30 pm
  
  Reply
[...] Richeson blogs about transcendental numbersand algebraic numbers of the form trigfunction(πθ) . (Seems like this would apply the hyperbolic [...]
By: Random walks around the web « Random Walks on November 4, 2010
at 4:17 pm

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[...] un récent article du blog, l’auteur nous a fait part de son top 10 des nombres transcendants. Rappelons qu’un nombre transcendant est un nombre qui n’est racine d’aucun [...]
By: Top 10 des nombres irrationnels « Blogdemaths on November 12, 2010
at 9:05 am

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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. :-)
By: Baylink on December 31, 2010
at 8:14 pm

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- Thanks! (Hi, everyone!)
  
  I actually saw the story in NPR’s Facebook stream. I did a double-take when I got to #13.
  By: Dave Richeson on December 31, 2010
  at 10:01 pm
  
  Reply
They have misprint in number 9.
By: Riv on January 4, 2011
at 3:43 pm

Reply
- A misprint on number 9 above? I don’t see it. Can you be more specific?
  By: Dave Richeson on January 4, 2011
  at 3:58 pm
  
  Reply
[...] Top ten transcendental numbers – Lista dos 10 mais em números transcendentais. Não são números que almejam o nirvana matemático por meio de meditação transcendental e sim, números irracionais que não são raízes de uma equação polinomial com coeficientes inteiros. [...]
By: Tidbits de matemática | CyberGi on January 16, 2011
at 8:12 am

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[...] Transcendental Numbers: This list is totally undiscerning and leads to a mathematical embarrassment. [...]
By: The Top 20 Top 10s Of 2010 on February 11, 2011
at 8:43 am

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[...] Top ten transcendental numbers « Division by Zero – October 8th ( tags: transcendental numbers math list ) [...]
By: Delicious Bookmarks for October 8th through October 9th « Lâmôlabs on October 9, 2012
at 1:01 am

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Division by Zero