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 April 22nd, 2014, 06:14 AM #1 Newbie   Joined: Apr 2014 From: USA Posts: 24 Thanks: 1 Fibonacci numbers The number 0,112358...( the digits are the Fibonacci numbers) is rational/irational.How motivate? Thanks. April 22nd, 2014, 06:37 AM   #2
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Quote:
 Originally Posted by Roli The number 0,112358...( the digits are the Fibonacci numbers) is rational/irational.How motivate? Thanks.
I can't think of an easy proof (though I'm sure it's irrational). April 22nd, 2014, 03:20 PM   #3
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Hello, Roli!

Quote:
 The number 0,112358...( the digits are the Fibonacci numbers) is rational/irrational?

If the decimal is constructed like this:

$\displaystyle \begin{array}{cccccccccccc} 0. & 0&1\\ & 0&0&1 \\ & 0&0&0&2 \\ & 0&0&0&0&3 \\ & 0&0&0&0&0&5 \\ & 0&0&0&0&0&0&8 \\ & 0&0&0&0&0&0&1&3 \\ & 0&0&0&0&0&0&0&2&1 \\ & 0&0&0&0&0&0&0&0&3&4 \\ \hline 0. & 0&1&1&2&3&5&9&5&4 & ... \end{array}$

its value is a rational number!

Let $\displaystyle F_0 = 0,\;F_1=1,\;F_2 = 1,\;F_3 = 3,\;F_4 = 5,\;\text{ . . .}$

$\displaystyle \begin{array}{cccccc}\text{Then we have:} & X &=& \frac{F_0}{10^1} + \frac{F_1}{10^2} + \frac{F_2}{10^3} + \frac{F_3}{10^4} + \frac{F_4}{10^5} + \cdots \\ \text{Multiply by }\tfrac{1}{10}\!: & \frac{1}{10}X &=& \qquad\;\;\;\frac{F_0}{10^2} + \frac{F_1}{10^3} + \frac{F_2}{10^4} + \frac{F_3}{10^5} + \cdots \end{array}$

$\displaystyle \text{Subtract: }\;\;\frac{9}{10}X \;=\; \frac{F_0}{10} + \frac{F_1-F_0}{10^2} + \frac{F_2-F_1}{10^3} + \frac{F_3-F_2}{10^4} + \frac{F_4-F_3}{10^5} + \cdots$

$\displaystyle \text{We have: }\;\; \frac{9}{10}X \;=\; 0 + \frac{1}{10^2} + \frac{F_0}{10^3} + \frac{F_1}{10^4} + \frac{F_2}{10^5} + \cdots$

$\displaystyle \qquad\qquad\;\;\; \frac{9}{10}X \;=\; \frac{1}{100} + \frac{1}{10^2}\underbrace{\left(\frac{F_0}{10^1} + \frac{F_1}{10^2} + \frac{F_2}{10^3} + \cdots \right)}_{\text{This is }X}$

$\displaystyle \qquad\qquad\;\;\;\frac{9}{10}X \;=\;\frac{1}{100} + \frac{1}{100}X$

$\displaystyle \;\;\; \frac{9}{10}X - \frac{1}{100}X \;=\;\frac{1}{100}$

$\displaystyle \qquad\qquad \frac{89}{100}X \;=\;\frac{1}{100}$

$\displaystyle \text{Therefore: }\:X \;=\;\frac{1}{89}$ April 22nd, 2014, 08:35 PM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,617 Thanks: 2608 Math Focus: Mainly analysis and algebra Very nice! April 23rd, 2014, 01:42 AM #5 Math Team   Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Yes , very nice but note that the number derived by my friend Soroban is not the same number as the OP. I'm with CRG , the OP number is irrational. The addition method of Soroban would work if we increased the exponent of the denominator appropriately , whenever appropriate , for example , When you get to F(7) you increase the exponent in the denominator by 1 (in the Soroban derivation) When you get to F(8 ) you increase the exponent in the denominator by 2 (in the Soroban derivation) Last edited by agentredlum; April 23rd, 2014 at 01:44 AM. Reason: F(8) registered as smiley April 23rd, 2014, 01:54 PM #6 Member   Joined: Apr 2014 From: norwich Posts: 84 Thanks: 9 - The proof the second poster gave seems correct to me. Why is it wrong ? April 23rd, 2014, 01:56 PM   #7
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Quote:
 Originally Posted by William Labbett The proof the second poster gave seems correct to me. Why is it wrong ?
I don't think anything is wrong with it, but I don't think that number is the one the OP intended. Tags fibonacci, numbers Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post johnr Number Theory 4 October 8th, 2013 05:45 PM icemanfan Number Theory 18 March 13th, 2012 10:00 AM zolden Number Theory 12 January 26th, 2009 02:27 PM toejam Number Theory 2 January 4th, 2009 09:04 AM Fra Real Analysis 1 March 21st, 2008 10:50 AM

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