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My son was busily memorizing digits of $\pi$ when he asked if any power of $\pi$ was an integer. I told him: $\pi$ is transcendental, so no non-zero integer power can be an integer.

After tiring of memorizing $\pi$, he resolved to discover a new irrational whose expansion is easier to memorize. He invented (probably re-invented) the number $J$:

$$J = 6.12345678910111213141516171819202122\ldots$$

which clearly lets you name as many digits as you like pretty easily. He asked me if $J$ is transcendental just like $\pi$, and I said it must be but I didn't know for sure. Is there an easy way to determine this?

I can show that $\pi$ is transcendental (using Lindemann-Weierstrass) but it doesn't work for arbitrary numbers like $J$, I don't think.

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@Fixee: Off-topic, but by "π is transcendental, so no non-zero power can be an integer" do you mean integer powers, real powers, or powers in general? –  InterestedGuest Mar 5 '11 at 22:09
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@Adrian: 6 was his age, and $J$ because it's the first letter of his name. –  Fixee Mar 5 '11 at 22:12
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I see; you've got a bright six-year-old on your hands, there :) –  Adrian Petrescu Mar 5 '11 at 22:18
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When he was 3, he asked if 5 was even. I told him, "No, even means you can cut it in half." Long pause.... "But 5 is 2 1/2 plus 2 1/2." He's been like this ever since he could talk. His mother worries about his dating prospects, however. –  Fixee Mar 5 '11 at 23:10
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2 Answers 2

up vote 26 down vote accepted

This is a transcendental number, in fact one of the best known ones, it is $6+$ Champernowne's number.

Kurt Mahler was first to show that the number is transcendental, a proof can be found on his "Lectures on Diophantine approximations", available through Project Euclid. The argument (as typical in this area) consists in analyzing the rate at which rational numbers can approximate the constant (see the section on "Approximation by rational numbers: Liouville to Roth" in the Wikipedia entry for Transcendence theory).

An excellent book to learn about proofs of transcendence is "Making transcendence transparent: an intuitive approach to classical transcendental number theory", by Edward Burger and Robert Tubbs.

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Let me throw in here Ivan Niven's wonderful little book Irrational Numbers. It is quite readable by anyone with a mathematical interest. I think it's a Carus Monograph. –  MPW Dec 31 '13 at 19:40
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This is just the Champernowne constant plus $6$. Since the Champernowne constant is transcendental, so is this number (as $6$ is, of course, algebraic).

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Thanks! I had no idea it had a name (and I did try Googling, without success). Is there a straightforward way to confirm its transcendence? –  Fixee Mar 5 '11 at 22:13
    
@Fixee: Don't know, but it was proven transcendental back in 1937. –  Arturo Magidin Mar 5 '11 at 22:18
    
Curious question: Does the fact that this is normal base 10 imply it is transcendental? Or can we have normal algebraic numbers? –  Eric Naslund Mar 5 '11 at 22:27
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@Eric Naslund: No, normality is not known to imply transcendence. According to Wikipedia, it is a conjecture that irrational algebraic numbers are normal, though no irrational algebraic number has ever been proven normal. –  Arturo Magidin Mar 5 '11 at 22:31
    
Hmm... You just uncovered a typo in Wikipedia! (Somewhere they correctly date the result to 1937; elsewhere, they claim the transcendence was proved in 1961, in Mahler's book I link to in my answer.) –  Andres Caicedo Mar 5 '11 at 22:32
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