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These are no fractions, are they? (Posted on 2006-09-05) Difficulty: 3 of 5
You are probably quite certain that the following numbers are irrational. But can you prove it?

a = 0.149162536496481...

b = 0.2357111317192329...

c = 0.1248163264128256...

See The Solution Submitted by JLo    
Rating: 2.5000 (2 votes)

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re: Now that we know these are no fractions... Comment 15 of 15 |
(In reply to Now that we know these are no fractions... by JLo)

Hmm...you are definitely going into an uncharted territory here...Here are some interesting facts I've found about transcendental numbers.

 

In late 18th century the question was: are there any non-algebraic numbers? The existence of non-algebraic (transcendental) numbers was first proved in 1844 by Liouville. He "engineered" the first transcendental number now called Liouville's constant:

 

L=10-1!+10-2!+10-3!..=0.110001000000000000000001000...

 

The first number to be proven transcendental without having been specifically constructed for that purpose was e, by Hermite in 1873. In the same year, Cantor proved that the transcendental numbers are non denumerable, whereas the algebraic numbers are denumerable. In 1898 Borel established that "almost all" real numbers are transcendental!

Proving that a given number is transcendental can be extremely difficult. pi was proved to be transcendental by Lindemann in 1882. Hilbert's seventh problem, resolved by Gelfond in 1934, concerned the transcendence of certain numbers, including e^pi and  2^sqrt(2). ln(2) and sin(1) were proved to be transcendental by Hardy and Wright as recently as 1979. Until now it is not known whether pi*e and pi+e are transcendental: however it is known that at least one of them is.

 

Since "almost all" real numbers are transcendental, then a randomly selected number on the numerical axis has zero probability of being algebraic. By rephrasing this, I came up with a "theorem" which I'm sure is "almost" true-

 

Theorem:

 

Any irrational number that is constructed without involving radicals and polynomial equations is transcendental.

 

This, of course, includes all JLo numbers as a particular case.

Edited on September 21, 2006, 5:48 pm
  Posted by Art M on 2006-09-21 17:23:15

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