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Irrational numbers vs. Transcendental numbers 
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#1
Mar804, 08:05 PM

P: 47

It would seem that an irrational number would have to be a transcendental number. If a transcendental number is a number which goes on infinitely and never repeats, then all irrational numbers would have to be transcendental, because if they repeated then you could find a fraction doing the whole x = .abcdef... 1000000x=abcdef.abcdef... and so forth.
Are there any counterexamples? 


#2
Mar804, 08:15 PM

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First off:
Irrational means 'cannot be expressed as a fraction'. For example [tex]\sqrt{2}[/tex] is an irrational number. Transcendental means is not a solution to any equation that contains only rational numbers. For example, [tex]\pi[/tex] is a transcendental number. [tex]\sqrt{2}[/tex] is not a transcendental number. 


#3
Mar804, 08:18 PM

P: 47

Thanks for your time! 


#4
Mar804, 08:29 PM

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Irrational numbers vs. Transcendental numbers
Sorry, let me me more clear.
Transcendental numbers can never be expressed as the roots of polynomial equations. (Equations involving addition, multiplication, subtraction or division. No infinite series, logarithms, or trig functions.) 


#5
Mar804, 08:30 PM

P: 47

So is the lack of any repeating trademark to irrational numbers as a whole? 


#6
Mar804, 09:56 PM

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Well, "lack of any repeating" isn't a good description, but that's essentially it, yes.



#7
Mar904, 10:11 PM

P: 140

Isn't that also (indirectly) stating that trancendental numbers appear in their own definition?



#8
Mar1004, 12:13 AM

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#9
Mar1004, 08:49 AM

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#10
Mar1311, 07:44 PM

P: 5

What is the relationship between continued fractions for irrational numbers and how exactly does this differ from the continued fractions of transcendental numbers?
Also wondering: transcendental functions (trig and log functions, infinite series): completely impossible to construct out of polynomial functions with 100% accuracy? 


#11
Jun1911, 10:53 PM

P: 460

1/sqrt(2) rationalize the denominator...very easy. 1/pi rationalize the denominator...you become FAMOUS! 


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